Tuesday, April 7, 2020
Moonshine, the MOG, and the Hexacode
Thursday, April 25, 2013
Note on the MOG Correspondence
In light of the April 23 post "The SixSet,"
the caption at the bottom of a note of April 26, 1986
seems of interest:
"The R. T. Curtis correspondence between the 35 lines and the
2subsets and 3subsets of a 6set. This underlies M_{24}."
A related note from today:
Friday, May 14, 2010
Competing MOG Definitions
A recently created Wikipedia article says that “The Miracle Octad Generator [MOG] is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24dimensional space….” (Clearly any array with 24 parts is so capable.) The article ignores the fact that the MOG, as defined by R.T. Curtis in 1976, is not an array of coordinates, but rather a picture of a correspondence between two sets, each containing 35 structures. (As a later commentator has remarked, this correspondence is a wellknown one that preserves a certain incidence property. See Eightfold Geometry.)
From the 1976 paper defining the MOG—
“There is a correspondence between the two systems of 35 groups, which is illustrated in Fig. 4 (the MOG or Miracle Octad Generator).” —R.T. Curtis, “A New Combinatorial Approach to M_{24},” Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 2542
Curtis’s 1976 Fig. 4. (The MOG.)
The Wikipedia article, like a similar article at PlanetMath, is based on a different definition, from a book first published in 1988—
I have not seen the 1973 Curtis paper, so I do not know whether it uses the 35sets correspondence definition or the 6×4 array definition. The remarks of Conway and Sloane on page 312 of the 1998 edition of their book about “Curtis’s original way of finding octads in the MOG [Cur2]” indicate that the correspondence definition was the one Curtis used in 1973—
Here the picture of “the 35 standard sextets of the MOG”
is very like (modulo a reflection) Curtis’s 1976 picture
of the MOG as a correspondence between two 35sets.
A later paper by Curtis does use the array definition. See “Further Elementary Techniques Using the Miracle Octad Generator,” Proceedings of the Edinburgh Mathematical Society (1989) 32, 345353.
The array definition is better suited to Conway’s use of his hexacode to describe octads, but it obscures the close connection of the MOG with finite geometry. That connection, apparent in the phrases “vector space structure in the standard square” and “parallel 2spaces” (Conway and Sloane, third ed., p. 312, illustrated above), was not discussed in the 1976 Curtis paper. See my own page on the MOG at finitegeometry.org.
Saturday, October 24, 2020
Switchin’ the Positions*
“C24 is the list of codewords of the extended
binary Golay code C_{24}. Each codeword is expressed
by a subset of the set M of the positions [1, . . . , 24]
of MOG.”
— From Shimada’s notes on computational data at
http://www.math.sci.hiroshimau.ac.jp/~shimada/
preprints/Edge/PaperEdge/compdataEdge.pdf .
* Related material — A new Ariana Grande video and . . .
a recent digital artwork, “Code Girl,” with accompanying story —
Wednesday, September 2, 2020
Space Wars: Sith Pyramid vs. Jedi Cube
For the Sith Pyramid, see posts tagged Pyramid Game.
For the Jedi Cube, see posts tagged Enigma Cube
and cuberelated remarks by Aitchison at Hiroshima.
This post was suggested by two events of May 16, 2019 —
A weblog post by Frans Marcelis on the Miracle Octad
Generator of R. T. Curtis (illustrated with a pyramid),
and the death of I. M. Pei, architect of the Louvre pyramid.
That these events occurred on the same date is, of course,
completely coincidental.
Perhaps Dan Brown can write a tune to commemorate
the coincidence.
Thursday, August 27, 2020
The Complete Extended Binary Golay Code
All 4096 vectors in the code are at . . .
http://neilsloane.com/oadir/oa.4096.12.2.7.txt.
Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —
As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the ConwaySloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexicoMOG equivalence was first discovered by M. J. T. Guy.
(Of course, any permutation of the 24 columns above produces
a version of the code qua code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)
See my post of July 13, 2020 —
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
For some related results, Google the twelfth generator:
* Sloane’s list is of the codewords as the rows of an orthogonal array —
See also http://neilsloane.com/oadir/.
Thursday, August 6, 2020
After Personalities . . . Principles
In memory of New York personality Pete Hamill ,
who reportedly died yesterday —
Seven years ago yesterday —
In memory of another New York personality, a parkinggarage mogul
who reportedly died on August 9, 2005 —
Icon Parking posts and . . .
Tuesday, July 28, 2020
Wednesday, July 22, 2020
Card
“The pattern of the thing precedes the thing.
I fill in the gaps of the crossword at any spot
I happen to choose. These bits I write on
index cards until the novel is done.”
— Vladimir Nabokov, interview,
Paris Review No. 41 (SummerFall 1967).
Another story —
Related material: Mathematics as a Black Art.
Monday, July 13, 2020
The Lexicographic Octad Generator (LOG)*
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
By Steven H. Cullinane, July 13, 2020
Background —
The Miracle Octad Generator (MOG)
of R. T. Curtis (ConwaySloane version) —
A basis for the Golay code, excerpted from a version of
the code generated in lexicographic order, in
“Constructing the Extended Binary Golay Code“
Ben Adlam
Harvard University
August 9, 2011:
000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100
Below, each vector above has been reordered within
a 4×6 array, by Steven H. Cullinane, to form twelve
independent Miracle Octad Generator vectors
(as in the ConwaySloane SPLAG version above, in
which Curtis’s earlier heavy bricks are reflected in
their vertical axes) —
01 02 03 04 05 . . . 20 21 22 23 24 > 01 05 09 13 17 21 02 06 10 14 18 22 03 07 11 15 19 23 04 08 12 16 20 24 0000 0000 0000 0000 1111 1111 > 0000 11 0000 11 0000 11 0000 11 as in the MOG. 0000 0000 0000 1111 0000 1111 > 0001 01 0001 01 0001 01 0001 01 as in the MOG. 0000 0000 0011 0011 0011 0011 > 0000 00 0000 00 0011 11 0011 11 as in the MOG. 0000 0000 0101 0101 0101 0101 > 0000 00 0011 11 0000 00 0011 11 as in the MOG. 0000 0000 1001 0110 0110 1001 > 0010 01 0001 10 0001 10 0010 01 as in the MOG. 0000 0011 0000 0011 0101 0110 > 0000 00 0000 11 0101 01 0101 10 as in the MOG. 0000 0101 0000 0101 0110 0011 > 0000 00 0101 10 0000 11 0101 01 as in the MOG. 0000 1001 0000 0110 0011 1010 > 0100 01 0001 00 0001 11 0100 10 as in the MOG. 0001 0001 0001 0001 0111 1000 > 0000 01 0000 10 0000 10 1111 10 as in the MOG. 0010 0001 0001 0010 0001 1101 > 0000 01 0000 01 1001 00 0110 11 as in the MOG. 0100 0001 0001 0100 0100 1110 > 0000 01 1001 11 0000 01 0110 00 as in the MOG. 1000 0001 0001 0111 0010 0100 > 10 00 00 00 01 01 00 01 10 01 11 00 as in the MOG (heavy brick at center).
Update at 7:41 PM ET the same day —
A check of SPLAG shows that the above result is not new:
And at 7:59 PM ET the same day —
Conway seems to be saying that at some unspecified point in the past,
M.J.T. Guy, examining the lexicographic Golay code, found (as I just did)
that weight8 lexicographic Golay codewords, when arranged naturally
in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed
at the time of his discovery, he would have identified these patterns as
those of the MOG. (Lexicographic codes have apparently been
known since 1960, the MOG since the mid1970s.)
* Addendum at 4 AM ET the next day —
See also Logline (Walpurgisnacht 2013).
Monday, May 25, 2020
The Shimada Documents
(For Harlan Kane)
From Shimada’s notes on computational data at
http://www.math.sci.hiroshimau.ac.jp/~shimada/
preprints/Edge/PaperEdge/compdataEdge.pdf —
“C24 is the list of codewords of the extended
binary Golay code C_{24}. Each codeword is expressed
by a subset of the set M of the positions [1, . . . , 24]
of MOG.”
Thursday, May 14, 2020
Art Issue*
“… the beautiful object
that stood in
for something else.”
— Holland Cotter quoting an art historian
in The New York Times on May 13
From a post of April 27, 2020 —
“The yarns of seamen have a direct simplicity,
the whole meaning of which lies within the shell
of a cracked nut. But Marlow was not typical
(if his propensity to spin yarns be excepted),
and to him the meaning of an episode was not inside
like a kernel but outside….”
— Joseph Conrad in Heart of Darkness
The beautiful object —
Something else —
* The title is a reference to other posts now also tagged Art Issue.
Saturday, May 2, 2020
Monday, April 27, 2020
The Cracked Nut
“At that instant he saw, in one blaze of light, an image of unutterable
conviction, the reason why the artist works and lives and has his being –
the reward he seeks –the only reward he really cares about, without which
there is nothing. It is to snare the spirits of mankind in nets of magic,
to make his life prevail through his creation, to wreak the vision of his life,
the rude and painful substance of his own experience, into the congruence
of blazing and enchanted images that are themselves the core of life, the
essential pattern whence all other things proceed, the kernel of eternity.”
— Thomas Wolfe, Of Time and the River
“… the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it.
(It induces A_{8} on the octad, the kernel of this action being the translation
group of the affine space.)”
— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside….”
— Joseph Conrad in Heart of Darkness
Friday, April 24, 2020
Art at Cologne
This post was suggested by a New York Review of Books article
on Cologne artist Gerhard Richter in the May 14, 2020, issue —
“The Master of Unknowing,” by Susan Tallman.
Some less random art —
Thursday, April 23, 2020
Octads and Geometry
See the web pages octad.group and octad.us.
Related geometry (not the 759 octads, but closely related to them) —
The 4×6 rectangle of R. T. Curtis
illustrates the geometry of octads —
Curtis splits the 4×6 rectangle into three 4×2 “bricks” —
.
“In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ_{1}, Λ_{2}, Λ_{3}, intersects at least one of these ‘ bricks’ —
the ‘heavy brick’ – in just four points.” . . . .
— R. T. Curtis (1976). “A new combinatorial approach to M_{24},”
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 2542.
Wednesday, February 19, 2020
Aitchison’s Octads
The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.
A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :
The Curtis octads are related to symmetries of the square.
See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —
Aitchison's octads are instead related to symmetries of the cube.
Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .
Image from Christmas Day 2005.
* http://www.math.sci.hiroshimau.ac.jp/branched/files/2018/
presentations/AitchisonHiroshima22018.pdf.
See also Aitchison in this journal.
Monday, December 23, 2019
Orbit
"December 22, the birth anniversary of India’s famed mathematician
Srinivasa Ramanujan, is celebrated as National Mathematics Day."
— Indian Express yesterday
"Orbits and stabilizers are closely related." — Wikipedia
Symmetries by Plato and R. T. Curtis —
In the above, 322,560 is the order
of the octad stabilizer group .
Wednesday, December 11, 2019
Miracle Octad Generator Structure
(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)
Tuesday, October 29, 2019
Triangles, Spreads, Mathieu
There are many approaches to constructing the Mathieu
group M_{24}. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is wellknown that …
There are 56 triangles in an 8set.
There are 56 spreads in PG(3,2).
The alternating group A_{n }is generated by 3cycles.
The alternating group A_{8 }is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M_{24}.
Some background —
A Log24 post of May 19, 2013, cites …
Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.
See also a Google search for “56 triangles” “56 spreads” Mathieu.
Update of October 31, 2019 — A related illustration —
Update of November 2, 2019 —
See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.
Update of December 20, 2019 —
Saturday, September 21, 2019
Annals of Random Fandom
For Dan Brown fans …
… and, for fans of The Matrix, another tale
from the above death date: May 16, 2019 —
An illustration from the above
Miracle Octad Generator post:
Related mathematics — Tetrahedron vs. Square.
Wednesday, March 6, 2019
Tuesday, March 5, 2019
A Block Design 3(16,4,1) as a Steiner Quadruple System:
A Midrash for Wikipedia
Midrash —
Related material —
________________________________________________________________________________
Friday, March 1, 2019
Wikipedia Scholarship (Continued)
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
Friday, February 22, 2019
Back Issues of AMS Notices
From the online home page of the new March issue —
For instance . . .
Related material now at Wikipedia —
Thursday, February 7, 2019
Geometry of the 4×4 Square: The Kummer Configuration
From the series of posts tagged Kummerhenge —
A Wikipedia article relating the above 4×4 square to the work of Kummer —
A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finitegeometry properties of the 4×4 square as
a finite affine 4space — properties that are of use in studying the Mathieu
group M_{24 }with the aid of the MOG.
Sunday, December 2, 2018
Symmetry at Hiroshima
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
http://www.math.sci.hiroshimau.ac.jp/ branched/files/2018/abstract/Aitchison.txt Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein’s quartic curve, respectively), and Bring’s genus 4 curve arises in Klein’s description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the HorrocksMumford bundle. Poincare’s homology 3sphere, and Kummer’s surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay’s binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois’ exceptional finite groups PSL2(p) (for p= 5,7,11), and various other socalled `Arnol’d Trinities’. Motivated originally by the `Eightfold Way’ sculpture at MSRI in Berkeley, we discuss interrelationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential interconnectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato’s concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphicillustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones’ genus 70 Riemann surface previously proposed as a completion of an Arnol’d Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston’s highly symmetric 6 and 8component links, the latter related by Thurston to Klein’s quartic curve. 
See also yesterday morning’s post, “Character.”
Update: For a followup, see the next Log24 post.
Wednesday, October 3, 2018
Sunday, September 23, 2018
Three Times Eight
The New York Times 's Sunday School today —
I prefer the three bricks of the Miracle Octad Generator —
Wednesday, July 11, 2018
Titans
July 10, 2018
The private jets have begun clogging the jetways
in Sun Valley, Idaho, which can only mean one thing:
'Billionaire summer camp’' has begun.
The annual Allen & Company conference, the investment
firm’s inviteonly gathering of some of the world’s most
powerful corporate titans, officially begins on Wednesday."
In other news —
Get ready to see the Titans in training camp."
See also another post now tagged "Clash of the Titans."
Friday, May 4, 2018
Entropy
A more serious note in memory of Anatole Katok:
"Entropy measures the unpredictability
of a system that evolves over time."
— Alex Wright, BULLETIN (New Series)
OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 53, Number 1, January 2016, Pages 41–56
http://dx.doi.org/10.1090/bull/1513
Article electronically published on September 8, 2015:
FROM RATIONAL BILLIARDS
TO DYNAMICS ON MODULI SPACES
Abstract:
"This short expository note gives an elementary
introduction to the study of dynamics on certain
moduli spaces and, in particular, the recent
breakthrough result of Eskin, Mirzakhani,
and Mohammadi. We also discuss the context
and applications of this result, and its connections
to other areas of mathematics, such as algebraic
geometry, Teichmüller theory, and ergodic theory
on homogeneous spaces."
See also the lives of Ratner and Mirzakhani.
Wednesday, April 25, 2018
An Idea
"There was an idea . . ." — Nick Fury in 2012
". . . a calm and objective work that has no special
dance excitement and whips up no vehement
audience reaction. Its beauty, however, is extraordinary.
It’s possible to trace in it terms of arithmetic, geometry,
dualism, epistemology and ontology, and it acts as
a demonstration of art and as a reflection of
life, philosophy and death."
— New York Times dance critic Alastair Macaulay,
quoted here in a post of August 20, 2011.
Illustration from that post —
Thursday, November 16, 2017
Tuesday, September 12, 2017
Think Different
The New York Times online this evening —
"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeplefingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."
Review —
Thursday, September 1, 2011
How It Works

See also 1984 Bricks in this journal.
Saturday, September 9, 2017
How It Works
Del Toro and the History of Mathematics ,
Or: Applied Bullshit Continues
For del Toro —
For the history of mathematics —
Thursday, September 1, 2011
How It Works

Sunday, September 3, 2017
Broomsday Revisited
Ivars Peterson in 2000 on a sort of conceptual art —
" Brill has tried out a variety of gridscrambling transformations
to see what happens. Aesthetic sensibilities govern which
transformation to use, what size the rectangular grid should be,
and which iteration to look at, he says. 'Once a fruitful
transformation, rectangle size, and iteration number have been
found, the artist is in a position to create compelling imagery.' "
— "Scrambled Grids," August 28, 2000
Or not.
If aesthetic sensibilities lead to a 23cycle on a 4×6 grid, the results
may not be pretty —
From "Geometry of the 4×4 Square."
See a Log24 post, Noncontinuous Groups, on Broomsday 2009.
Wednesday, August 23, 2017
Pakanga
("Every Picture Tells a Story," continued from August 15 )
Related material — LaughingAcademy Cartography.
Saturday, January 28, 2017
Cranking It Up
From "Core," a post of St. Lucia's Day, Dec. 13, 2016 —
In related news yesterday —
California yoga mogul’s mysterious death:
Trevor Tice’s drunken last hours detailed
"Police found Tice dead on the floor in his home office,
blood puddled around his head. They also found blood
on walls, furniture, on a sofa and on sheets in a nearby
bedroom, where there was a large bottle of Grey Goose
vodka under several bloodstained pillows on the floor."
See as well an image from "The Stone," a post of March 18, 2016 —
Some backstory —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
Wednesday, January 18, 2017
An Associative Function …
Quoted here on December 16, 2006 —
See also …

"Ryan Reynolds Named Hasty Pudding’s
2017 Man of the Year" 
Reviews of Reynolds' 2015 film "Self/Less" and of
the earlier similar film "Seconds" —4 July 2015 9:00 AM, PDT  The Wrap 
"Ben Kingsley plays a New York real estate mogul who
pays big bucks to have his consciousness microwaved
into Ryan Reynolds' body in 'Self/less,' but the real
reheating of leftovers has already occurred: this new
sciencefiction thriller borrows the foundation of a much
better film — John Frankenheimer’s 1966 'Seconds' —
and strips it of any larger meaning."
The date of the "Seconds" review above, 16 Dec. 2006, was
the reason for the requotation in the first paragraph above.
Sunday, June 19, 2016
In Memoriam
For those who prefer the red pill to the blue pill —
See as well this afternoon's related Vanity Fair piece.
Tuesday, June 7, 2016
Saturday, March 19, 2016
TwobyFour
For an example of "anonymous content" (the title of the
previous post), see a search for "2×4" in this journal.
Monday, February 1, 2016
Historical Note
Possible title:
A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M_{24}
Monday, January 12, 2015
Points Omega*
The previous post displayed a set of
24 unitsquare “points” within a rectangular array.
These are the points of the
Miracle Octad Generator of R. T. Curtis.
The array was labeled Ω
because that is the usual designation for
a set acted upon by a group:
* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.
Sunday, January 11, 2015
Real Beyond Artifice
A professor at Harvard has written about
“the urge to seize and display something
real beyond artifice.”
He reportedly died on January 3, 2015.
An image from this journal on that date:
Another Gitterkrieg image:
The 24set Ω of R. T. Curtis
Click on the images for related material.
Thursday, January 8, 2015
Gitterkrieg
From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —
"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72 I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E_{8} and of the Golay code from
the Hamming code (Turyn 1967)."
On an earlier talk by Nebe at Oberwolfach in 2011 —
"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."
Nebe's Oberwolfach slides include one on
"The history of Turyn's construction" —
Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M_{24}"
by R. T. Curtis, the paper that defined Curtis's
"Miracle Octad Generator."
Turyn's 1967 construction, uncredited by Curtis,
was the basis for Curtis's octadgenerator construction.
See Turyn in this journal.
Tuesday, December 2, 2014
Models
Continued from November 30, 2014
"Number right → Everything right." — Burkard Polster.
See also the six posts of November 30, St. Andrew's Day.
Related material —
Peter J. Cameron today discussing Julia Kristeva on poetry …
"This seems to be saying that the Kolmogorov
complexity of poetry is very low: the entire poem
can be generated from a small amount of information."
… and this journal on St. Andrew's day :
From "A Piece of the Storm,"
by the late poet Mark Strand —
A snowflake, a blizzard of one….
Saturday, October 25, 2014
Foundation Square
In the above illustration of the 345 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
twodimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ_{5}).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…

"The HoffmanSingleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and 
the Web pages "HoffmanSingleton Graph" and "HigmanSims Graph"
of A. E. Brouwer.
Hafner's abstract:
We describe the HoffmanSingleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ_{5}.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a fourdimensional vector space over GF(2).)
Monday, October 6, 2014
Mysterious Correspondences
(Continued from Beautiful Mathematics, Dec. 14, 2013)
“Seemingly unrelated structures turn out to have
mysterious correspondences.” — Jim Holt, opening
paragraph of a book review in the Dec. 5, 2013, issue
of The New York Review of Books
One such correspondence:
For bibliographic information and further details, see
the March 9, 2014, update to “Beautiful Mathematics.”
See as well posts from that same March 9 now tagged “Story Creep.”
Tuesday, September 23, 2014
Meanwhile, Back at Harvard…
"William Deresiewicz argued his claim that students of elite universities
are growingly riskaverse, homogeneous, and careerfocused with a
panel of faculty members and students on Monday evening.
Hosted by Harvard’s Mahindra Humanities Center, the questionand
answerstyle forum involved a panel…. The panel was moderated by
Homi K. Bhabha, director of the Mahindra Center."
— Alexander H. Patel in today's online Harvard Crimson
See also Con Vocation (Sept. 2, 2014).
Sunday, August 31, 2014
Sunday School
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordionpleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”
The geometry of the 4×4 square array is that of the
3dimensional projective Galois space PG(3,2).
This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.
Some history:
Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.
[Rewritten for clarity on Sept. 3, 2014.]
Tuesday, August 26, 2014
Sunday, August 24, 2014
Symplectic Structure…
In the Miracle Octad Generator (MOG):
The above details from a onepage note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:
From R. T. Curtis (1976). A new combinatorial approach to M_{24},
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 2542. doi:10.1017/S0305004100052075.
The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.
Note that the interchange of the two squares in the top row of each
heavy brick induces the diamondtheorem correlation.
Note also that the 20 pictured 3subsets of a 6set in the 1986 note
occur as paired complements in two pictures, each showing 10 of the
3subsets.
This pair of pictures corresponds to the 20 Rosenhain tetrads among
the 35 lines of PG(3,2), while the picture showing the 2subsets
corresponds to the 15 Göpel tetrads among the 35 lines.
See Rosenhain and Göpel tetrads in PG(3,2). Some further background:
Tuesday, June 17, 2014
Finite Relativity
Anyone tackling the Raumproblem described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:
The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper. Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—
This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:
An explanation of the apparent falsity in Curtis's 1989 paper:
By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projectiveline coordinates , in his earlier papers were
mirror images of the octads that resulted later from the Conway coordinates,
as in the images below.
Sunday, June 15, 2014
Aaron Eckhart Strikes Deep
“Even paranoids have real enemies.”
— Attributed to Delmore Schwartz
“There is a difference as to whether you are describing paranoia
or whether you in fact are paranoid yourself.”
— The late Frank Schirrmacher, dw.de , July 2, 2013.
Schirrmacher reportedly died on Thursday, June 12, 2014.
See that date in this journal.
Paranoia is, of course, a fertile field for politicians and filmmakers:
Related material in this journal:
I, Frankenstein (May 15, 2014) and, for the Eckhart film “Erased,”
Hour of the Wolf (Nov. 9, 2006).
Thursday, April 24, 2014
The Inscape of 24
“The more intellectual, less physical, the spell of contemplation
the more complex must be the object, the more close and elaborate
must be the comparison the mind has to keep making between
the whole and the parts, the parts and the whole.”
— The Journals and Papers of Gerard Manley Hopkins ,
edited by Humphry House, 2nd ed. (London: Oxford
University Press, 1959), p. 126, as quoted by Philip A.
Ballinger in The Poem as Sacrament
Related material from All Saints’ Day in 2012:
Friday, March 28, 2014
Blazing Thule
The title is suggested by a new novel (see cover below),
and by an unwritten book by Nabokov —
Related material:
 An artists' book scheduled to be released on March 21, 2014
 A piece by Josefine Lyche in the artists' book
 The original by Borges on which Lyche's piece was based

A solar image from a March 13 post echoing
that on the Blazing World cover above  A Tune for Josefine
 The circular blazing image from last midnight's post Symbol

From March 21, the scheduled date of the Oslo
artists' book release, some remarks on the mathematics of the
Golay code, "Three Constructions of the Miracle Octad Generator"  Backstory: Duelle in this journal.
Friday, March 21, 2014
Three Constructions of the Miracle Octad Generator
See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.
Related material on the TurynCurtis construction
from the University of Cambridge —
— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M_{24},” in slides for lectures 18 from lectures
at Cambridge in 20102011 on “Sporadic and Related Groups.”
See also the Parker lectures of 20122013 on the same topic.
A third construction of Curtis’s 35 4×6 1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4partitions of an 8set with the 35 lines of the projective 3space
over the 2element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22March 23 —
Adding together as (0,1)matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S_{3} on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this “byhand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction, not by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.
* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.
Illustration of array addition from March 23 —
Sunday, March 9, 2014
The Story Creeps Up
For Women’s History Month —
Conclusion of “The Storyteller,” a story
by Cynthia Zarin about author Madeleine L’Engle—
See also the exercise on the Miracle Octad Generator (MOG) at the end of
the previous post, and remarks on the MOG by Emily Jennings (non fiction)
on All Saints’ Day, 2012 (the date the L’Engle quote was posted here).
Friday, February 21, 2014
Raumproblem*
Despite the blocking of Doodles on my Google Search
screen, some messages get through.
Today, for instance —
"Your idea just might change the world.
Enter Google Science Fair 2014"
Clicking the link yields a page with the following image—
Clearly there is a problem here analogous to
the squaretriangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.
I once studied this 24trianglehexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.
* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.
Wednesday, December 25, 2013
Rotating the Facets
“… her mind rotated the facts….”
Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:
“I’ve heard of affairs that are strictly Platonic”
* Footnote added on Dec. 26, 2013 —
See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups,” Trans. Amer. Math. Soc. 31 (1929),
No. 1, 25–42.
On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., threedimensional cubes) to count
rotational symmetries —
See also Diamond Theory in 1937.
Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M_{11} as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M_{24} .
Friday, December 20, 2013
For Emil Artin
(On His Dies Natalis )…
This is asserted in an excerpt from…
"The smallest nonrank 3 strongly regular graphs
which satisfy the 4vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152212—
(Click for clearer image)
Note that Theorem 46 of Klin et al. describes the role
of the Galois tesseract in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric part of the above
exceptional geometriccombinatorial isomorphism.
Saturday, December 14, 2013
Beautiful Mathematics
The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.
Some material relevant to the title adjective:
"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books 
Some relevant links—
 Strangeness and inevitability
 Simply defined abstractions
 Hidden quirks and complexities
 Seemingly unrelated structures
 Mysterious correspondences
 Uncanny patterns
 The rigor of logic
 Beethoven quartet
The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links. See also a post of
Jan. 31, 2014.
Update of March 9, 2014 —
The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).
Saturday, November 23, 2013
Frame Tale (continued)
See The XMen Tree, another tree, and Trinity MOG.
Monday, August 12, 2013
Form
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.
The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—
As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.
Tuesday, July 9, 2013
Vril Chick
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) 
Clearly most of this (the nonhighlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
Tuesday, May 28, 2013
Codes
The hypercube model of the 4space over the 2element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galoistesseract model of the 4space over GF(2).
The thirtyfive 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book coauthored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galoistesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliamsSloane book was first published):
Sunday, April 28, 2013
The Octad Generator
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete npoints in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space fivepoint."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space fivepoint and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The SixSet (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M_{24} —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
Thursday, April 25, 2013
Rosenhain and Göpel Revisited
Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):
The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M_{24}.
For some related material that is more uptodate, search the Web
for Mathieu + Kummer .
Saturday, April 6, 2013
Pascal via Curtis
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M_{24},"
Math. Proc. Camb. Phil. Soc., 79 (1976), 2542.)
The 8subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirtyfive 3subsets of a 7set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 35_{4} Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3sets and all 4sets that can be formed
by the elements of a 7element set; each 'point' is represented
by one of the 3sets, and it is incident with those lines
(represented by 4sets) that contain the 3set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
Wednesday, February 13, 2013
Form:
Story, Structure, and the Galois Tesseract
Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.
The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—
The Klein quadric occurs in the fivedimensional projective space
over a field. If the field is the twoelement Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator (MOG), hence of
the large Mathieu group M_{24}. These properties are also
relevant to the 1976 "Diamond Theory" monograph.
For some background on the quadric, see (for instance)…
See also The Klein Correspondence,
Penrose SpaceTime, and a Finite Model.
Related material:
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) 
Those who prefer story to structure may consult
 today's previous post on the Penrose diamond
 the remarks of Scott Aaronson on August 17, 2012
 the remarks in this journal on that same date
 the geometry of the 4×4 array in the context of M_{24}.
Monday, December 24, 2012
All Over Again
"… the movement of analogy
begins all over once again."
See A Reappearing Number in this journal.
Illustrations:
Figure 1 —
Background: MOG in this journal.
Figure 2 —
Background —
Saturday, November 24, 2012
Reappearing All Over Again
For the title, see the phrase "reappearing number" in this journal.
Some related mathematics—
the Greek labyrinth of Borges, as well as…
Note that "0" here stands for "23," while _{∞} corresponds to today's date.
Monday, November 19, 2012
Sunday, October 14, 2012
Crossroads
"Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself."
— A translated remark by Hermann Weyl, p. 136, "The Current Epistemogical Situation in Mathematics" in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s , Oxford University Press, 1998, pp. 123142, as cited by David Corfield
Corfield once wrote that he would like to know the original German of Weyl's remark. Here it is:
"Die Mathematik ist nicht das starre und Erstarrung bringende Schema, als das der Laie sie so gerne ansieht; sondern wir stehen mit ihr genau in jenem Schnittpunkt von Gebundenheit und Freiheit, welcher das Wesen des Menschen selbst ist."
— Hermann Weyl, page 533 of "Die heutige Erkenntnislage in der Mathematik" (Symposion 1, 132, 1925), reprinted in Gesammelte Abhandlungen, Band II (Springer, 1968), pages 511542
For some context, see a post of January 23, 2006.
Friday, October 5, 2012
The Elegant Fowl
For the late Helen Nicoll—
 A scene from a video starring Nicoll's characters
Meg, Mog, and Owl. The video was uploaded on
November 2, 2008— All Souls' Day.  A different cartoon from All Souls' Day, 2008
 A rhyme from 1871—
The Owl looked up to the stars above, — Edward Lear 
Thursday, October 4, 2012
Kids Grow Up
From an obituary for Helen Nicoll, author
of a popular series of British children's books—
"They feature Meg, a witch whose spells
always seem to go wrong, her cat Mog,
and their friend Owl."
For some (very loosely) related concepts that
have been referred to in this journal, see…
See, too, "Kids grow up" (Feb. 13, 2012).
Sunday, September 9, 2012
Grid Compass
Related material: The Empty Chair Award.
For a different sort of grid compass, see February 3, 2011.
Sunday, July 29, 2012
The Galois Tesseract
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vectorspace structure:
0  c  d  c + d 
a  a + c  a + d  a + c + d 
b  b + c  b + d  b + c + d 
a + b  a + b + c  a + b + d  a + b + c + d 
(This vectorspace a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "SelfDual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 ConwaySloane diagram.
Monday, July 16, 2012
Mapping Problem continued
Another approach to the squaretotriangle
mapping problem (see also previous post)—
For the square model referred to in the above picture, see (for instance)
 Picturing the Smallest Projective 3Space,
 The Relativity Problem in Finite Geometry, and
 Symmetry of Walsh Functions.
Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may be deduced
from the patterns in the projectivehyperplanes array above.
This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.
Update of 9:35 AM ET July 16, 2012:
Note that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —
— share the following vectorspace structure —
0  c  d  c + d 
a  a + c  a + d  a + c + d 
b  b + c  b + d  b + c + d 
a + b  a + b + c  a + b + d  a + b + c + d 
(This vectorspace a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)
Sunday, January 8, 2012
Big Apple
“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”
This suggests that the above structure
be viewed as illustrating not eight parts
but rather 8! = 40,320 parts.
"The Cardinal seemed a little preoccupied today."
The New Yorker , May 13, 2002
See also a note of May 14 , 2002.
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
twothirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79TA37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG’s
4×4 square as the affine 4space over the 2element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four “special tetrads” within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 “special tetrads” rather by the parity
of their intersections with the square’s rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The “35 structures” of the abstract were listed, with an application to
Latinsquare orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M_{24}, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was misnamed as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Thursday, September 1, 2011
How It Works
“Design is how it works.” — Steven Jobs (See Symmetry and Design.)
“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
The name Carmichael is not to be found in Booher’s thesis. A book he does cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937. It should be dated 1931, as the following quotation shows—
From Log24 on Feb. 20, 2010—
“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a fivefold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”
– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217240
Epigraph from Ch. 4 of Design Theory , Vol. I:
“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo XXXIX)
See also “Do you like apples?“
Thursday, August 25, 2011
Design
"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)
Today's American Mathematical Society home page—
Some related material—
The above Rowley paragraph in context (click to enlarge)—
"We employ Curtis's MOG …
both as our main descriptive device and
also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122
And the MOG incorporates the
Geometry of the 4×4 Square.
For this geometry's relation to "design"
in the graphicarts sense, see
Block Designs in Art and Mathematics.
Wednesday, August 24, 2011
Symmetry
An article from cnet.com tonight —
For Jobs, design is about more than aesthetics
By: Jay Greene
… The look of the iPhone, defined by its seamless pane of glass, its chrome border, its perfect symmetry, sparked an avalanche of copycat devices that tried to mimic its aesthetic.
Virtually all of them failed. And the reason is that Jobs understood that design wasn't merely about what a product looks like. In a 2003 interview with the New York Times' Rob Walker detailing the genesis of the iPod, Jobs laid out his vision for product design.
''Most people make the mistake of thinking design is what it looks like,'' Jobs told Walker. "People think it's this veneer— that the designers are handed this box and told, 'Make it look good!' That's not what we think design is. It's not just what it looks like and feels like. Design is how it works.''
Related material: Open, Sesame Street (Aug. 19) continues… Brought to you by the number 24—
"By far the most important structure in design theory is the Steiner system
— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693746) of Handbook of Combinatorics , Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))
Saturday, August 20, 2011
Castle Rock
Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)
"The metaphor for metamorphosis…" —Endgame
Related material:
"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."
— Scholarly paper on "Correlative Cosmologies"
"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.
The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….
'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."
— Alastair Macaulay in today's New York Times
SOCRATES: Let us turn off the road a little….
— Libretto for Mark Morris's 'Socrates'
See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:
The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S_{8}
acting on the above eightpart figure.
See also The Moore Correspondence from last year
on today's date, August 20.
For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—
"At the still point, there the dance is."
— Four Quartets
Saturday, August 6, 2011
Correspondences
Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, “Correspondances ”
From “A FourColor Theorem”—
Figure 1
Note that this illustrates a natural correspondence
between
(A) the seven highly symmetrical fourcolorings
of the 4×2 array at the left of Fig. 1, and
(B) the seven points of the smallest
projective plane at the right of Fig. 1.
To see the correspondence, add, in binary
fashion, the pairs of projective points from the
“points” section that correspond to likecolored
squares in a fourcoloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)
A different correspondence between these 7 fourcoloring
structures and these 7 projectiveline structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—
Figure 2
Here the correspondence between the 7 fourcoloring structures (left section) and the 7 projectiveline structures (center section) is less obvious, but more fruitful. It yields, as shown, all of the 35 partitions of an 8element set (an 8set ) into two 4sets. The 7 fourcolorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.
For some applications of the Curtis MOG, see 
Sunday, July 10, 2011
Wittgenstein’s Diamond
Philosophical Investigations (1953)—
97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest thing there is
(Tractatus LogicoPhilosophicus No. 5.5563).
— Translation by G.E.M. Anscombe
All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.
(Our problems are not abstract but perhaps
the most concrete that there are.)
97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach sein. Sie ist vor aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh. No. 5.5563.)
Related language in Łukasiewicz (1937)—
* Updates of 9:29 PM ET July 10, 2011—
A mnemonic from a course titled “Galois Connections and Modal Logics“—
“Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written (square) for necessarily
and (diamond) for possibly. Then, for example, P can be read as
‘it is possibly the case that P .'”
See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—
“The diamond ⋄ symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬⋄¬. The dual symbol □ was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich ‘possible.’” Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159. Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen. Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century. 
Wednesday, July 6, 2011
NordstromRobinson Automorphisms
A 2008 statement on the order of the automorphism group of the NordstromRobinson code—
"The NordstromRobinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."
— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 122
A statement by Bierbrauer from 2004 has an error that doubles the above figure—
The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order
— Jürgen Bierbrauer, "NordstromRobinson Code and A_{7}Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158170
The error is corrected (though not detected) later in the same 2004 paper—
In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).
For some background, see a wellknown construction of the code from the Miracle Octad Generator of R.T. Curtis—
For some context, see the group of order 322,560 in Geometry of the 4×4 Square.
Sunday, June 19, 2011
Abracadabra (continued)
Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator of R.T. Curtis.
A search tonight for related material on the Web yielded…
Weblog post by Steve Richards titled "The Search for Invariants:
The Diamond Theory of Truth, the Miracle Octad Generator
and Metalibrarianship." The artwork is by Steven H. Cullinane.
Richards has omitted Cullinane's name and retitled the artwork.
The author of the post is an artist who seems to be interested in the occult.
His post continues with photos of pages, some from my own work (as above), some not.
My own work does not deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.
The artist's post concludes with the following (note also the beginning of the preceding post)—
"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.
Sunday, June 5, 2011
Edifice Complex
"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."
— Wallace Stevens, "To an Old Philosopher in Rome"
The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…
"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology
The Kummer 16_{6} Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)
For some background, see Configurations and Squares.
For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes twothirds of the MOG as originally defined .)
Related material— The Schwartz Notes of June 1.
Wednesday, June 1, 2011
The Schwartz Notes
A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signedout search activity was disabled.)
The same search limited to results from only the past month yielded,
in addition, the following—
This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.
Pages 1214 of the document, which is untitled, undated, and
unsigned, discuss the finitegeometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.
See Schwartz’s page 12, page 13, and page 14.
Compare to the web pages from today’s earlier search.
There are no references at the end of the Schwartz document,
but there is this at the beginning—
These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.
It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.
Wednesday, March 2, 2011
Labyrinth of the Line
“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”
“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”
Another singleline labyrinth—
Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—
Related material —
The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.
A search in the latter for miracle octad is updated below.
This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”
Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.
Tuesday, November 16, 2010
Meanwhile, back in 1953…
Monday, October 25, 2010
The Embedding*
A New York Times "The Stone" post from yesterday (5:15 PM, by John Allen Paulos) was titled—
Stories vs. Statistics
Related Google searches—
"How to lie with statistics"— about 148,000 results
"How to lie with stories"— 2 results
What does this tell us?
Consider also Paulos's phrase "imbedding the God character." A less controversial topic might be (with the spelling I prefer) "embedding the miraculous." For an example, see this journal's "Mathematics and Narrative" entry on 5/15 (a date suggested, coincidentally, by the time of Paulos's post)—
* Not directly related to the novel The Embedding discussed at Tenser, said the Tensor on April 23, 2006 ("Quasimodo Sunday"). An academic discussion of that novel furnishes an example of narrative as more than mere entertainment. See Timothy J. Reiss, "How can 'New' Meaning Be Thought? Fictions of Science, Science Fictions," Canadian Review of Comparative Literature , Vol. 12, No. 1, March 1985, pp. 88126. Consider also on this, Picasso's birthday, his saying that "Art is a lie that makes us realize truth…."
Thursday, September 9, 2010
Building a Mystery
Notes on Mathematics and Narrative, continued
Patrick Blackburn, meet Gideon Summerfield…
From a summary of a politically correct 1995 feminist detective novel about quilts, A Piece of Justice—
The story deals with “one Gideon Summerfield, deceased.” Summerfield, a former tutor at (the fictional) St. Agatha’s College, Cambridge University, “is about to become the recipient of the Waymark prize. This prize is awarded in Mathematics and has the same prestige as the Nobel. Summerfield had a rather lackluster career at St. Agatha’s, with the exception of one remarkable result that he obtained. It is for this result that he is being awarded the prize, albeit posthumously.” Someone is apparently trying to prevent a biography of Summerfield from being published.
Compare and contrast with an episode from the resume of a real Gideon Summerfield—
Head of Strategy, Designer City (May 1999 — January 2002)
Secured Web agency business from new and existing clients with compelling digital media strategies and oversaw delivery of creative, production and technical teams…. Clients included… Greenfingers and Lord of the Dance .
For material related to Greenfingers and Lord of the Dance , see Castle Kennedy Gardens at Wicker Man Locations.
Friday, August 20, 2010
The Moore Correspondence
There is a remarkable correspondence between the 35 partitions of an eightelement set H into two fourelement sets and the 35 partitions of the affine 4space L over GF(2) into four parallel fourpoint planes. Under this correspondence, two of the Hpartitions have a common refinement into 2sets if and only if the same is true of the corresponding Lpartitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A_{8} with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M_{24}.
A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.
Edge says that
It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….
Excerpts from the Edge paper—
Excerpts from the Moore paper—
Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439
* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72
** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 41744." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss, Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417444.
Thursday, August 5, 2010
Eightgate
"Eight is a gate."
— This journal, December 2002
Tralfamadorian Structure
in SlaughterhouseFive
includes the following passage:
“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”
This suggests that the above structure be viewed
as illustrating not eight parts but rather
8! = 40,320 parts.
See also April 2, 2003.
Happy birthday to John Huston and
happy dies natalis to Richard Burton.
Saturday, July 24, 2010
Thursday, June 24, 2010
Midsummer Noon
Geometry Simplified
(a projective space)
The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)
The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the verticalandhorizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).
Homogeneous coordinates for the
points of this line —
(1,0), (0,1), (1,1).
Here 0 and 1 stand for the elements
of the twoelement Galois field GF(2).
The 3point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —
(an affine space)
The (Galois) points of this affine plane are
not the single and combined (Euclidean)
line segments that play the role of
points in the 3point projective line,
but rather the four subsquares
that the line segments separate.
For further details, see Galois Geometry.
There are, of course, also the trivial
twopoint affine space and the corresponding
trivial onepoint projective space —
Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affinespace squares.
Saturday, June 19, 2010
Imago Creationis
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
FourPart Tesseract Divisions—
The above figure shows how fourpart partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to fourpart partitions
of the 16 points in a finite Galois space
Euclidean spaces versus Galois spaces in a larger context—
Infinite versus Finite The central aim of Western religion — "Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
 Martha Cooley in The Archivist (1998)
The central aim of Western philosophy — Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy." 
Another picture related to philosophy and religion—
Jung's FourDiamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896). O Paul Valéry, Oeuvres (Paris: Pléiade, 195760) C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 195761) 
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.” If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multidimensionally^{*} whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect. * That is, uses multidimensional symbols beyond our grasp. 
Related material:
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Some context by a British mathematician —
Imago by Wallace Stevens Who can pick up the weight of Britain, Who can move the German load Or say to the French here is France again? Imago. Imago. Imago. It is nothing, no great thing, nor man Of ten brilliancies of battered gold And fortunate stone. It moves its parade Of motions in the mind and heart, A gorgeous fortitude. Medium man In February hears the imagination's hymns And sees its images, its motions And multitudes of motions And feels the imagination's mercies, In a season more than sun and south wind, Something returning from a deeper quarter, A glacier running through delirium, Making this heavy rock a place, Which is not of our lives composed . . . Lightly and lightly, O my land, Move lightly through the air again. 
Tuesday, June 15, 2010
Imago, Imago, Imago
Recommended— an online book—
Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.
See in particular
Valéry and the Discourse On His Method.
Pages 156157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).
O Paul Valéry, Oeuvres (Paris: Pléiade, 195760)
C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 195761)
Compare Jung’s image in Aion of the Self as a fourdiamond figure:
and Cullinane’s purely geometric fourdiamond figure:
For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.
What remains fixed (globally, not pointwise) under these transformations is the system of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009. Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).
Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.
Saturday, May 15, 2010
Mathematics and Narrative continued…
Wednesday, April 28, 2010
Eightfold Geometry
Related web pages:
Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square
Related folklore:
“It is commonly known that there is a bijection between the 35 unordered triples of a 7set [i.e., the 35 partitions of an 8set into two 4sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common.” –“Generalized Polygons and Semipartial Geometries,” by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6
The Miracle Octad Generator may be regarded as illustrating the folklore.
Update of August 20, 2010–
For facts rather than folklore about the above bijection, see The Moore Correspondence.
Wednesday, October 14, 2009
Wednesday October 14, 2009
Singer 7Cycles
Click on images for details.
The 1985 Cullinane version gives some algebraic background for the 1987 Curtis version.
The Singer referred to above is James Singer. See his “A Theorem in Finite Projective Geometry and Some Applications to Number Theory,” Transactions of the American Mathematical Society 43 (1938), 377385.For other singers, see Art Wars and today’s obituaries.
Some background: the Log24 entry of this date seven years ago, and the entries preceding it on Las Vegas and painted ponies.
Thursday, August 6, 2009
Thursday August 6, 2009
Update: The above image was added
at about 11 AM ET Aug. 8, 2009.
Dr. Joe Emerson, April 24, 2005–
— Text: I Peter 2:19
Dr. Emerson falsely claims that the film "On the Waterfront" was based on a book by the late Budd Schulberg (who died yesterday). (Instead, the film's screenplay, written by Schulberg– similar to an earlier screenplay by Arthur Miller, "The Hook"– was based on a series of newspaper articles by Malcolm Johnson.)
"The movie 'On the Waterfront' is once more in rerun. (That’s when Marlon Brando looked like Marlon Brando. That’s the scary part of growing old when you see what he looked like then and when he grew old.) It is based on a book by Budd Schulberg."
Emerson goes on to discuss the book, Waterfront, that Schulberg wrote based on his screenplay–
"In it, you may remember a scene where Runty Nolan, a little guy, runs afoul of the mob and is brutally killed and tossed into the North River. A priest is called to give last rites after they drag him out."
Dr. Emerson's sermon is, as noted above (Text: I Peter 2:19), not mainly about waterfronts, but rather about the "living stones" metaphor of the Big Fisherman.
My own remarks on the date of Dr. Emerson's sermon—
Those who like to mix mathematics with religion may regard the above 4×6 array as a context for the "living stones" metaphor. See, too, the five entries in this journal ending at 12:25 AM ET on November 12 (Grace Kelly's birthday), 2006, and today's previous entry.
Wednesday, May 20, 2009
Wednesday May 20, 2009
Mathieu Group M_{24}
The connection:
 "A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177186.
Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."
 "A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223228.
Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."
Tuesday, May 19, 2009
Tuesday May 19, 2009
"By far the most important structure in design theory is the Steiner system
— "Block Designs," 1995, by Andries E. Brouwer
"The Steiner system S(5, 8, 24) is a set S of 759 eightelement subsets ('octads') of a twentyfourelement set T such that any fiveelement subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M_{24}."
— The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)
"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a littleknown 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."
The 1931 paper of Carmichael is now available online from the publisher for $10.
Saturday, April 4, 2009
Saturday April 4, 2009
"… in some autistic enchantment, pure as one of Bach's inverted canons or Euler's formula for polyhedra."
— George Steiner, "A Death of Kings," in The New Yorker, issue dated Sept. 7, 1968
A correspondence underlying
the Steiner system S(5,8,24)–
The Steiner here is
Jakob, not George.
See "Pope to Pray on
Autism Sunday 2009."
See also Log24 on that
Sunday– February 8:
Sunday, February 15, 2009
Sunday February 15, 2009
From April 28, 2008:
Religious Art
The black monolith of
One artistic shortcoming The following
One approach to "Transformations play See 4/28/08 for examples 
Related material:
From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, pp. 117118:
"… his point of origin is external nature, the fount to which we come seeking inspiration for our fictions. We come, many of Stevens's poems suggest, as initiates, ritualistically celebrating the place through which we will travel to achieve fictive shape. Stevens's 'real' is a bountiful place, continually giving forth life, continually changing. It is fertile enough to meet any imagination, as florid and as multifaceted as the tropical flora about which the poet often writes. It therefore naturally lends itself to rituals of spring rebirth, summer fruition, and fall harvest. But in Stevens's fictive world, these rituals are symbols: they acknowledge the real and thereby enable the initiate to pass beyond it into the realms of his fictions. Two counter rituals help to explain the function of celebration as Stevens envisions it. The first occurs in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer. A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination. For in 'Notes Toward a Supreme Fiction' he tells us that ... the first idea was not to shape the clouds In imitation. The clouds preceded us. There was a muddy centre before we breathed. There was a myth before the myth began, Venerable and articulate and complete. From this the poem springs: that we live in a place That is not our own and, much more, not ourselves And hard it is in spite of blazoned days. We are the mimics. (Collected Poems, 38384) Believing that they are the life and not the mimics thereof, the world and not its fictionforming imitators, these young men cannot find the savage transparence for which they are looking. In its place they find the pediment, a scowling rock that, far from being life's source, is symbol of the human delusion that there exists a 'form alone,' apart from 'chains of circumstance.' A far more productive ritual occurs in 'Sunday Morning.'…." 
For transformations of a more
specifically religious nature,
see the remarks on
Richard Strauss,
"Death and Transfiguration,"
(Tod und Verklärung, Opus 24)
in Mathematics and Metaphor
on July 31, 2008, and the entries
of August 3, 2008, related to the
death of Alexander Solzhenitsyn.
Tuesday, January 6, 2009
Tuesday January 6, 2009
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:
“… we do not need to accept Jung’s theory as true in order to find it illuminating.”
The same is true of Jung’s remarks on synchronicity.
For example —
Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s fourdiamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 19842003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
“What, then, is a frame, and what is frame work?”
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.
windows of knowledge.”
— Vladimir Nabokov
Monday, January 5, 2009
Monday January 5, 2009
A Wealth of
Algebraic Structure
A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M_{24}, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was misnamed as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
— Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345353, doi:10.1017/S0013091500004600.
(Published online by Cambridge University Press 19 Dec 2008.)
In the above article, Curtis explains how twothirds of his 4×6 MOG array may be viewed as the 4×4 model of the fourdimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)
For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):
A new combinatorial approach to M_{24}, by R. T. Curtis. Abstract:
“In this paper, we define M_{24} from scratch as the subgroup of S_{24} preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”
(Received June 15 1974)
— Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 2542, doi:10.1017/S0305004100052075.
(Published online by Cambridge University Press 24 Oct 2008.)
Saturday, December 27, 2008
Saturday, December 13, 2008
Saturday December 13, 2008
The Shining
of Dec. 13
continued from
Dec. 13, 2003
“There is a place for a hint
somewhere of a big agent
to complete the picture.”
— Notes for an unfinished novel,
The Last Tycoon,
by F. Scott Fitzgerald
Filmography:William Grady
The Good Earth (1937)
casting: Chinese extras
(uncredited)
See also
yesterday’s entries
as well as…
Serpent’s Eyes Shine,
Alice’s Tea Party,
Janet’s Tea Party,
Hollywood Memory,
and
Hope of Heaven.
“… it’s going to be
accomplished in steps,
this establishment of
the Talented
in the scheme of things.”
Monday, November 24, 2008
Monday November 24, 2008
Sunday, October 12, 2008
Sunday October 12, 2008
— Today’s New York Times
review of the Very Rev.
Francis Bowes Sayre Jr.
Related material:
Log24 entries from
the anniversary this
year of Sayre’s birth
and from the date
of his death:
A link from the former
suggests the following
graphic meditation–
(Click on figure for details.)
A link from the latter
suggests another
graphic meditation–
(Click on figure for details.)
Although less specifically
American than the late
Reverend, who was
born in the White House,
hence perhaps irrelevant
to his political views,
these figures are not
without relevance to
his religion, which is
more about metanoia
than about paranoia.
Thursday, July 31, 2008
Thursday July 31, 2008
“Put bluntly, who is kidding whom?”
— Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.
Good question.
Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —

1. The performance of a work by
Richard Strauss,
“Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008
2. Headline of a music review
in today’s New York Times:
Welcoming a Fresh Season of
Transformation and Death
3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:
4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”
5. Symmetry from Plato to
the FourColor Conjecture
7. Yesterday’s entry,
“Theories of Everything“
Coda:
as a tesseract.“
— Madeleine L’Engle
For a profile of
L’Engle, click on
the Easter eggs.
Monday, April 28, 2008
Monday April 28, 2008
The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.
One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.
The following
figure does
allow such
an epiphany.
One approach to
the epiphany:
"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein
The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:
For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose SpaceTime, and
a Finite Model."
For a related structure–
not rectangle but cube–
see Epiphany 2008.
Saturday, April 19, 2008
Saturday April 19, 2008
On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see GianCarlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)
From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–
“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into
____________________
m(m – 1) … (m – k + 1)
This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.
For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.
Thursday, March 6, 2008
Thursday March 6, 2008
“The historical road
from the Platonic solids
to the finite simple groups
is well known.”
— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the FourColor Conjecture
“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
This Steiner system is closely connected to M_{24} and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of M_{24}):
“Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix
— Op. cit., p. 719
Finite Geometry of
the Square and Cube
and
Jewel in the Crown
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?'”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
“story theory” of truth
Those who prefer stories to truth
may consult the Log24 entries
of March 1, 2, 3, 4, and 5.
They may also consult
the poet Rubén Darío:
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.