Friday, June 29, 2018
From a post of July 25, 2008, "56 Triangles," on the Klein quartic
and the eightfold cube —
"Baez's discussion says that the Klein quartic's 56 triangles
can be partitioned into 7 eighttriangle Egan 'cubes' that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eighttriangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane."
Related material from 1975 —
More recently …
Comments Off on Triangles in the Eightfold Cube
Monday, April 2, 2018
From a Toronto Star video pictured here on April 1 three years ago:
The three connected cubes are labeled "Harmonic Analysis," 'Number Theory,"
and "Geometry."
Related cultural commentary from a review of the recent film "Justice League" —
"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."
The nineteenthcentury German mathematician Felix Christian Klein
as Steppenwolf —
Volume I of a treatise by Klein is subtitled
"Arithmetic, Algebra, Analysis." This covers
two of the above three Toronto Star cubes.
Klein's Volume II is subtitled "Geometry."
An excerpt from that volume —
Further cultural commentary: "Glitch" in this journal.
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Sunday, April 1, 2018
Or: Hector and the Horse
"How many roads . . . . ?" — Bob Dylan
Comments Off on The Truth Cube
Saturday, March 31, 2018
For Greta Gerwig and Saoirse Ronan —
See also a Log24 post from the above Cube Theory date —
April 12, 2016 — Lyrics for a Cartoon Graveyard — as well as . . .
Comments Off on Cube Theory
Thursday, March 22, 2018
The Java applets at the webpage "Diamonds and Whirls"
that illustrate Cullinane cubes may be difficult to display.
Here instead is an animated GIF that shows the basic unit
for the "design cube" pages at finitegeometry.org.
Comments Off on The Diamond Cube
Wednesday, January 17, 2018
Comments Off on “Before Time Began, There Was the Cube”
Saturday, November 18, 2017
Comments Off on Cube Space Continued
Thursday, August 17, 2017
Operation Blockhead continues …
See also Weyl + Palermo in this journal.
Comments Off on For Time Cube Fans
Tuesday, August 8, 2017
Comments Off on Cube Quaternions
Wednesday, July 26, 2017
See 4x4x4 in this journal. See also …
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Sunday, June 4, 2017
From this journal on August 18, 2015, "A Wrinkle in Terms" —
For two misuses by John Baez of the phrase “permutation group”
at the nCategory Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —
“There is such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle
* See RIP, Time Cube at gizmodo.com (September 1, 2015).
Comments Off on In Memory of the Time Cube Page*
Tuesday, April 4, 2017
"Inside the White Cube" —
"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20thcentury art."
"Space: what you
damn well have to see."
— James Joyce, Ulysses
Comments Off on White Cube
Thursday, January 19, 2017
The Silvia of the title is from the previous post.
For the Time Cube, see …
Comments Off on A Time Cube for Silvia
Friday, January 6, 2017
The assignments page for a graduate algebra course at Cornell
last fall had a link to the eightfold cube:
Comments Off on Eightfold Cube at Cornell
Thursday, October 6, 2016
Tuesday, August 30, 2016
A KUNSTforum.as article online today (translation by Google) —
Update of Sept. 7, 2016: The corrections have been made,
except for the misspelling "Cullinan," which was caused by
Google translation, not by KUNSTforum.
Comments Off on The Eightfold Cube in Oslo
Monday, August 1, 2016
From this journal —
See (for instance) Sacred Order, July 18, 2006 —
From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —
The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelvefoot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralowfrequency vibration of a massive engine. I drift toward the box, mesmerized.
— Crouch, Blake. Dark Matter: A Novel
(Kindle Locations 20042010).
Crown/Archetype. Kindle Edition.

See also Log24 on the publication date of Dark Matter .
Comments Off on Cube
Sunday, April 24, 2016
Comments Off on Prima Materia as Cubes
Tuesday, April 5, 2016
Comments Off on “Puzzle Cube of a Novel”
Monday, April 4, 2016
Foreword by Sir Michael Atiyah —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
Judy Bass, Los Angeles Times , March 12, 1989 —
"Like Rubik's Cube, The Eight demands to be pondered."
As does a figure from 1984, Cullinane's Cube —
For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
See also the recent post Cube Bricks 1984 —
Related remark from the literature —
Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 19842003."
(From Anatomy of a Cube, Sept. 18, 2011.)
Comments Off on Cube for Berlin
Thursday, March 17, 2016
The following page quotes "Raiders of the Lost Crucible,"
a Log24 post from Halloween 2015.
From KUNSTforum.as, a Norwegian art quarterly, issue no. 1 of 2016.
Related posts — See Lyche Eightfold.
Comments Off on On the Eightfold Cube
Friday, March 4, 2016
Related aesthetics —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
Comments Off on Cube Bricks 1984
Sunday, November 15, 2015
Anyone who clicked on the Dirac search at the end of
the previous post, "Dirac's Diamond," may wonder why the
"Solomon's Cube" post of 11 AM Sunday, March 1, 2009,
appeared in the Dirac search results, since there is no
apparent mention of Dirac in that Sunday post.
Use the source —
<!– See also "a linear transformation of V6… which preserves
the Klein quadric; in this way we arrive at the isomorphism of
Sym(8) withthe full orthogonal group O+(6; 2)." in "The
Classification of Flats in PG(9,2) which are External to the
Grassmannian G1,4,2 Authors: Shaw, Ron;
 Maks, Johannes; Gordon, Neil; Source: Designs,
Codes and Cryptography, Volume 34, Numbers 23, February
2005 , pp. 203227; Publisher: Springer.  For more details,
see "Finite Geometry, Dirac Groups and the Table of Real
Clifford Algebras," by R. Shaw (U. of Hull), pp. 5999 in
Clifford Algebras and Spinor Structures, by By Albert
Crumeyrolle, Rafał Abłamowicz, Pertti Lounesto,
published by Springer, 1995. –>
Comments Off on The Diamond and the Cube
Friday, October 9, 2015
An eightfold cube appears in this detail
of a photo by Josefine Lyche of her
installation "4D Ambassador" at the
Norwegian Sculpture Biennial 2015 —
(Detail from private Instagram photo.)
Catalog description of installation —
Google Translate version —
In a small bedroom to Foredragssalen populate
Josefine Lyche exhibition with a group sculptures
that are part of the work group 4D Ambassador
(20142015). Together they form an installation
where she uses light to amplify the feeling of
stepping into a new dimension, for which the title
suggests, this "ambassadors" for a dimension we
normally do not have access to. "Ambassadors"
physical forms presents nonphysical phenomena.
Lyches works have in recent years been placed
in something one might call an "esoteric direction"
in contemporary art, and defines itself this
sculpture group humorous as "glamminimalist."
She has in many of his works returned to basic
geometric shapes, with hints to the occult,
"new spaceage", mathematics and where
everything in between.
See also Lyche + "4D Ambassador" in this journal and
her website page with a 2012 version of that title.
Comments Off on Eightfold Cube in Oslo
Comments Off on Cube Design
Monday, September 28, 2015
Click to enlarge:
For the hypercube as a vector space over the twoelement field GF(2),
see a search in this journal for Hypercube + Vector + Space .
For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 19781986.
For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 57.
Comments Off on Hypercube Structure
Thursday, July 23, 2015
Comments Off on Design Cube
Monday, July 13, 2015
Why "Omega?"
Omega is a Greek letter, Ω , used in
mathematics to denote a set on which
a group acts.
Comments Off on The Omega Cube
Thursday, March 26, 2015
The incidences of points and planes in the
Möbius 8_{4 } configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and faceplanes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the pointplane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 8_{4}," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x_{4}, x_{3}, x_{2}, x_{1}) over the twoelement
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "SelfDual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413455
^{†} The subscripts' usual 1234 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Comments Off on The Möbius Hypercube
Sunday, December 28, 2014
The Blacklist “Pilot” Review
"There is an element of camp to this series though. Spader is
quite gleefully channeling Anthony Hopkins, complete with being
a well educated, elegant man locked away in a supercell.
Speaking of that supercell, it’s kind of ridiculous. They’ve got him
locked up in an abandoned post office warehouse on a little
platform with a chair inside a giant metal cube that looks like
it could have been built by Tony Stark. And as Liz approaches
to talk to him, the entire front of the cube opens and the whole
thing slides back to leave just the platform and chair. Really?
FUCKING REALLY ? "
— Kate Reilly at Geekenstein.com (Sept. 27, 2013)
Comments Off on Cube of Ultron
Wednesday, December 17, 2014
Continued from Nobel Note (Jan. 29, 2014).
From Tradition in Action , "The Missal Crisis of '62,"
remarks on the revision of the Catholic missal in that year—
"Neither can the claim that none of these changes
is heretical in content be used as an argument
in favor of its use, for neither is the employment of
hula girls, fireworks, and mariachis strictly speaking
heretical in itself, but they belong to that class of novel
and profane things that do not belong in the Mass."
— Fr. Patrick Perez, posted Sept. 11, 2007
See also this journal on November 22, 2014…
… and on Bruce Springsteen's birthday this year —
Tuesday, September 23, 2014

Comments Off on Launched from Cuber
Monday, May 19, 2014
A sequel to this afternoon's Rubik Quote:
"The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos."
— Professor Ernő Rubik at Chrome Cube Lab
(Click image below to enlarge.)
Comments Off on Cube Space
See also Cube Symmetry Planes in this journal.
Comments Off on UnRubik Cube
Saturday, January 25, 2014
The archived Java rotatable hypercube of
Harry J. Smith is no longer working.
For an excellent JavaScript replacement,
see Pete Michaud's
http://petemichaud.github.io/4dhypercube/.
This JavaScript version can easily be saved.
Comments Off on Rotatable Hypercube
Friday, June 7, 2013
From Night of Lunacy (Sunday, May 5, 2013):
Related posts: Rubric, Cuber, and Pound Sign.
Click image for some background.
See also Story Theory and Princeton Apocalypse.
Comments Off on Rubric’s Cuber
Thursday, January 24, 2013
For the late Cardinal Glemp of Poland,
who died yesterday, some links:
Comments Off on Cube Space
Friday, December 28, 2012
From Don DeLillo's novel Point Omega —
I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."
"What reality?"
"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreedupon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become threedimensional. The reality stands, it walks, it squats. Except when it doesn't."
He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."
I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.
"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—
This view of a die's faces 3, 6, and 5, in counter
clockwise order (see previous post) suggests a way
of labeling the eight corners of a die (or cube):
123, 135, 142, 154, 246, 263, 365, 456.
Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counterclockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertexlabeling may be used in describing
the automorphisms of the order8 quaternion group.
For a more literary approach to quaternions, see
Pynchon's novel Against the Day .
* From Peter J. Cameron's weblog:
"The big name associated with this is Major MacMahon,
an associate of Hardy, Littlewood and Ramanujan,
of whom Robert Kanigel said,
His expertise lay in combinatorics, a sort of
glorified dicethrowing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.
Glorified dicethrowing, indeed…"
Comments Off on Cube Koan
Monday, November 5, 2012
Continued from April 2, 2012.
Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties—
Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz cubes (see links below).
Horowitz Design Cubes Package
Horowitz Design Cubes (1971)
1973 Horowitz Design Cubes Patent
Horowitz Biography
Comments Off on Design Cubes
Tuesday, October 16, 2012
Last Wednesday's 11 PM post mentioned the
adjacencyisomorphism relating the 4dimensional
hypercube over the 2element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.
A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).
In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6dimensional hypercube over GF(2)
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.
The above cube may be used to illustrate some properties
of the 64point Galois 6space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.
See
Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."
Comments Off on Cube Review
Thursday, September 27, 2012
Denote the ddimensional hypercube by γ_{d} .
"… after coloring the sixtyfour vertices of γ_{6}
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."
— From "Kummer's 16_{6 }," section 12 of Coxeter's 1950
"Selfdual Configurations and Regular Graphs"
Just as the 4×4 square represents the 4dimensional
hypercube γ_{4 }over the twoelement Galois field GF(2),
so the 4x4x4 cube represents the 6dimensional
hypercube γ_{6} over GF(2).
For religious interpretations, see
Nanavira Thera (Indian) and
I Ching geometry (Chinese).
See also two professors in The New York Times
discussing images of the sacred in an oped piece
dated Sept. 26 (Yom Kippur).
Comments Off on Kummer and the Cube
Sunday, August 5, 2012
The second Logos figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—
Click for further details.
Comments Off on Cube Partitions
Wednesday, July 11, 2012
Monday, June 4, 2012
Yesterday's post Child's Play displayed a cube formed
by a Hasse diagram of the 8 subsets of a 3set.*
This suggests a review of a post from last January—
* See a comment on yesterday's post relating it to earlier,
very similar, remarks by Margaret Masterman.
I was unaware yesterday that those remarks exist.
Comments Off on Cube to Tesseract
Saturday, May 26, 2012
See also Finite Geometry and Physical Space.
Related material from MacTutor—
Harriot and binary numbers
The paper by J. W. Shirley, Binary numeration before Leibniz, Amer. J. Physics 19 (8) (1951), 452454, contains an interesting look at some mathematics which appears in the hand written papers of Thomas Harriot [15601621]. Using the photographs of the two original Harriot manuscript pages reproduced in Shirley’s paper, we explain how Harriot was doing arithmetic with binary numbers.
Leibniz [16461716] is credited with the invention [16791703] of binary arithmetic, that is arithmetic using base 2. Laplace wrote:
Leibniz saw in his binary arithmetic the image of Creation. … He imagined the Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration. This conception was so pleasing to Leibniz that he communicated it to the Jesuit, Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, who was very fond of the sciences …
However, Leibniz was certainly not the first person to think of doing arithmetic using numbers to base 2. Many years earlier Harriot had experimented with the idea of different number bases….

For a discussion of Harriot on the discretevs.continuous question,
see Katherine Neal, From Discrete to Continuous: The Broadening
of Number Concepts in Early Modern England (Springer, 2002),
pages 6971.
Comments Off on Harriot’s Cubes
Monday, April 9, 2012
A search today (Élie Cartan's birthday) for material related to triality*
yielded references to something that has been called a Bhargava cube .
Two pages from a 2006 paper by Bhargava—
Bhargava's reference [4] above for "the story of the cube" is to…
Higher Composition Laws I:
A New View on Gauss Composition,
and Quadratic Generalizations
Manjul Bhargava
The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217250
Published by: Annals of Mathematics
Article Stable URL: http://www.jstor.org/stable/3597249
A brief account in the context of embedding problems (click to enlarge)—
For more ways of slicing a cube,
see The Eightfold Cube —
* Note (1) some remarks by Tony Smith
related to the above Dynkin diagram
and (2) another colorful variation on the diagram.
Comments Off on Eightfold Cube Revisited
Saturday, February 25, 2012
Click images for further details.
See also Crimson Tide, Rubik, and Cuber.
For another monochromatic enigma without
guaranteed equality of results, see
Finite Geometry of the Square and Cube.
Comments Off on Cuber (continued)
Sunday, February 5, 2012
(Continued from January 11, 2012)
Comments Off on Cuber
Wednesday, January 11, 2012
"Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing. And once you have made or acquired a new 'cube'… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube. What is the essence of each operator? One senses a deep invariant lying somehow 'down underneath' it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment. In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….
… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube. It is the answer; it simply has the right spirit ."
— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern (Kindle edition, locations 1155711572)
See also Many Dimensions in this journal and Solomon's Cube.
Comments Off on Cuber
Friday, December 30, 2011
The following picture provides a new visual approach to
the order8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she cofounded—
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
Comments Off on Quaternions on a Cube
Friday, November 18, 2011
The hypercube has 192 rotational symmetries.
Its full symmetry group, including reflections,
is of order 384.
See (for instance) Coxeter—
Related material—
The rotational symmetry groups of the Platonic solids
(from April 25, 2011)—
— and the figure in yesterday evening's post on the hypercube—
(Animation source: MIQEL.com)
Clearly hypercube rotations of this sort carry any
of the eight 3D subcubes to the central subcube
of a central projection of the hypercube—
The 24 rotational symmeties of that subcube induce
24 rigid rotations of the entire hypercube. Hence,
as in the logic of the Platonic symmetry groups
illustrated above, the hypercube has 8 × 24 = 192
rotational symmetries.
Comments Off on Hypercube Rotations
Sunday, September 18, 2011
R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott's remarks.
For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 19842003."
Comments Off on Anatomy of a Cube
Saturday, August 27, 2011
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
Comments Off on Cosmic Cube*
Saturday, July 9, 2011
The New York Times has a skateboarder obit with a URL date of July 9.
Here is an earlier version from the LA Times—
July 4, 2011
By Keith Thursby, Los Angeles Times
Chris Cahill, one of the original Dogtown ZBoys
who brought seismic changes to skateboarding
with their style and attitude, has died. He was 54.
Cahill was found June 24 at his Los Angeles home,
said Larry Dietz of the Los Angeles County
coroner's office. A cause of death has not been
determined and tests are ongoing, Dietz said.
More…
Related material from Midsummer Day, June 24, the day Cahill was found dead—
The Gleaming and The Cube.
An illustration from the latter—
The above was adapted from a 1996 cover—
Vintage Books, July 1996. Cover: Evan Gaffney.
For the significance of the flames,
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.
Comments Off on Gleaming the Cube (continued)
Monday, June 27, 2011
The 3×3×3 Galois Cube
See Unity and Multiplicity.
This cube, unlike Rubik's, is a
purely mathematical structure.
Its properties may be compared
with those of the order2 Galois
cube (of eight subcubes, or
elements ) and the order4 Galois
cube (of 64 elements). The
order3 cube (of 27 elements)
lacks, because it is based on
an odd prime, the remarkable
symmetry properties of its smaller
and larger cube neighbors.
Comments Off on Galois Cube Revisited
Friday, June 24, 2011
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.
Comments Off on The Cube
Thursday, May 26, 2011
The title refers not to numbers of the form p^{ 3}, p prime, but to geometric cubes with p ^{3} subcubes.
Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p^{ 3}.
For the case p =2, see The Eightfold Cube.
For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")
For symmetries of group actions for larger primes, see the related 1985 remark* on two dimensional linear groups—
"Actions of GL(2,p ) on a p ×p coordinatearray
have the same sorts of symmetries,
where p is any odd prime."
* Group Actions, 19842009
Comments Off on Prime Cubes
Wednesday, April 20, 2011
It was a dark and stormy night…
— Page 180, Logicomix
“… the class of reﬂections is larger in some sense over an arbitrary ﬁeld than over a characteristic zero ﬁeld.”
– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups”
For some context, see the small cube in “A Simple Reflection Group of Order 168.”
See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).
That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix page above—
A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night—
“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”
— GianCarlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.
Comments Off on Romancing the Cube
Monday, June 21, 2010
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
groupactions theorem has not yet been ripped off.
Comments Off on Cube Spaces
Saturday, October 24, 2009
A search for “Chinese Cube” (based on the the previous entry’s title) reveals the existence of a most interesting character, who…
“… has attempted in his books to produce a Science and Art of Reasoning using the simplest of the Platonic solids, the Cube. [His] model also parallels, in some ways, the Cube of Space constructed from the Sepher Yetzirah’s attributions for the Hebrew letters and their direction. [He] elucidated his theories at great length….”
— More…
For related remarks, see the link to Solomon’s Cube from the previous entry.
Then of course there is…
Click on figure for details.
Comments Off on Chinese Cubes Continued
Thursday, October 22, 2009
From the Bulletin of the American Mathematical Society, Jan. 26, 2005:
What is known about unit cubes
by Chuanming Zong, Peking University
Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in ndimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….
From Log24, now:
What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated
Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in ndimensional binary space. In fact, as one will see from the links below, it is not simple at all.
Solomon’s Cube
The Klein Correspondence, Penrose SpaceTime, and a Finite Model
NonEuclidean Blocks
Geometry of the I Ching
Related material:
Monday’s entry Just Say NO and a poem by Stevens,
“The Well Dressed Man with a Beard.”
Comments Off on Chinese Cubes
Sunday, July 22, 2018
Saturday, July 21, 2018
(A sequel to yesterday's Geometry for Jews)
From Dr/ Yau's own website —
From this journal on the above UCI posting date — April 6, 2018 —
From this journal on the above lecture date — April 26, 2018 —
illustrations in a post titled Defining Form —
For the relevance of the above material to building blocks,
see Eightfold Cube in this journal.
Wednesday, July 18, 2018
Epiphany 2012: An Exercise
in Bulk Apperception
This post was suggested by a Marrific photo posted today,
by a New Orleans song as played by Bix Beiderbecke,
by a trailer for a new Zemeckis film that appeared at YouTube
on the way to the New Orleans song, and by
the longing for Bix by Mira Sorvino in "Intruders."
The YouTube Bix date, Jan. 6, 2012, suggested a trip back
to that date via a Zemeckis Cube (see "Ready Player One.")
From Log24 on Epiphany 2012 —
A version of the Zemeckis Cube —
From the Zemeckis trailer —
* See a Log24 search for Mira + Intruders.
Sunday, July 15, 2018
"… Lincoln Plaza Cinemas, the Juilliard String Quartet,
and the Strand Book Store remained oases
for cultural and intellectual stimulation."
— John S. Friedman in The Forward , Jan. 21, 2018
Read more:
https://forward.com/culture/392483/
howfredbassdantalbotrobertmann
shapednewyorkculture/
From the Oasis in Steven Spielberg's "Ready Player One" (2018) —
I prefer, from a Log24 search for Flux Capacitor …
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
Saturday, July 14, 2018
(Continued)
The walkerart.org passage above is from Feb. 17, 2011.
See also this journal on Feb. 17, 2011 —
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
Friday, July 13, 2018
Related drama —
Thursday, July 12, 2018
Three hidden keys open three secret gates
Wherein the errant will be tested for worthy traits
And those with the skill to survive these straits
Will reach The End where the prize awaits
— Ready Player One , by Ernest Cline
"Look, my favorite expression is,
'When you go up to the bell, ring it,
or don’t go up to the bell.' …
We’ve gone too far. We have to ring the bell."
— Mel Brooks on "The Producers"
in The New York Times today.
A 2016 Scribner edition of Stephen King's IT —
Related material —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
Wednesday, July 11, 2018
"The whole meaning of the word is
looking into something with clarity and precision,
seeing each component as distinct,
and piercing all the way through
so as to perceive the most fundamental reality
of that thing."
For the word itself, try a Web search on
noteworthy phrases above.
“. . . the utterly real thing in writing is
the only thing that counts . . . ."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"168"
— Page number in a 2016 Scribner edition
of Stephen King's IT
Sunday, July 8, 2018
Monday, July 2, 2018
This post is in memory of dancerchoreographer Gillian Lynne,
who reportedly died at 92 on Sunday, July 1, 2018.
For a scene from her younger days, click on Errol Flynn above.
The cube contemplated by Flynn is from Log24 on Sunday.
"This is how we enter heaven, enter dancing."
— Paraphrase of Lorrie Moore (See Oct. 18, 2003.)
Comments Off on In Memoriam
For Cady Heron
"Why you gotta be so mean?" — Taylor Swift
* See references to that Greek island in this journal.
Comments Off on A Mykonos* Narrative …
Sunday, July 1, 2018
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film "Triple Cross."
Related structures —
Greg Egan's animated image of the Klein quartic —
For a tetrahedral key to the arrangement of the 56 triangles within the above
structure, see a book chapter by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
Abstract for a talk at the City University of New York:
The Experience of Meaning
Jan Zwicky, University of Victoria
09:0009:40 Friday, April 5, 2013
Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to 'being simple': consider Eliot's Four Quartets or Mozart's late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it's often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we've seen into the heart of things. I'll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them.

For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
Comments Off on Deutsche Ordnung
Comments Off on The Perpetual Motion of T. S. Eliot
Friday, June 29, 2018
The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis
(birth into heaven) of St. Stanley Kubrick.
Related material from March 7, 2014,
in this journal —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
Comments Off on For St. Stanley
Thursday, June 28, 2018
Comments Off on Trinity Meditation
Wednesday, June 27, 2018
A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :
From Mere Christianity , by C. S. Lewis (1952) —
"Book IV – Beyond Personality:
or First Steps in the Doctrine of the Trinity"
. . . .
I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully.
You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares.
Do you see the point? A world of one dimension would be a straight line. In a twodimensional world, you still get straight lines, but many lines make one figure. In a threedimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels.
Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine.
In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something superpersonal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.
You may ask, "If we cannot imagine a threepersonal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that threepersonal life, and that may begin any time —tonight, if you like.
. . . .

But beware of being drawn into the personal life of the Happy Family .
https://www.jstor.org/stable/24966339 —
"The colorful story of this undertaking begins with a bang."
And ends with …
Martin Gardner on Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
Comments Off on Taken In
Monday, June 11, 2018
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "UhOh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computereye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
Comments Off on Arty Fact
Sunday, June 10, 2018
The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.
Propp's remarks included the following:
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
Comments Off on Number Concept
Thursday, June 7, 2018
See also Eightfold Trinity in this journal.
Comments Off on For Dan Brown
Wednesday, June 6, 2018
Mystery box merchandise from the 2011 J. J. Abrams film Super 8 —
A mystery box that I prefer —
Click image for some background.
See also Nicht Spielerei .
Comments Off on Geometry for Goyim
Monday, June 4, 2018
"Unsheathe your dagger definitions." — James Joyce, Ulysses
The "triple cross" link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .
Comments Off on The Trinity Stone Defined
Thursday, May 31, 2018
A New, Improved Version of Quantum Suffering !
Background for group actions on the eightfold cube —
See also other posts now tagged Quantum Suffering
as well as — related to the image above of the Great Wall —
Myspace China.
Comments Off on Eightfold Suffering:
2011 —
2014 —
See also other Log24 posts on quaternion group models.
Comments Off on Quaternion Group Models
Sunday, May 20, 2018
Dialogue from the 1984 fourth draft of the script, as found on the Web,
for "Back to the Future" (1985) (apparently some changes were made
in the filming) —
A sort of "flux capacitor" (see previous post) —
The RollsRoyce Cullinan
… plus "e" for Einstein …
Comments Off on Some Style
Saturday, May 19, 2018
For Tom Hanks and Dan Brown —
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
See as well a Google search for flux philosophy —
https://www.google.com/search?q=flux+philosophy.
Comments Off on Flux Capacitor
Thursday, May 17, 2018
On the film "Anna" in the previous post —
See also the above world premiere date in the posts of October 2013 —
esp. the post Conundrum.
Related material — An early scene in "Mindscape" . . .
. . . and "The Abacus Conundrum" in this journal.
Comments Off on Speak, Memory
Friday, May 4, 2018
A star figure and the Galois quaternion.
The square root of the former is the latter.
See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —
Comments Off on Art & Design
Sunday, April 29, 2018
From a search in this journal for Desmic —
"As the chaos grew . . . ."
"We have, in fact, the corners of a cube . . . ."
Comments Off on Sunday School
Monday, April 23, 2018
Mike Hale in The New York Times online today —
Review: ‘Genius’ Paints Picasso by the Numbers
"… the production’s tinselly soul.
For instance, it’s on the record that Picasso’s lovers
Dora Maar and MarieThérèse Walter had
a wrestling match in his studio while he was
painting 'Guernica.' 'Genius' includes that
scene, naturally, but adds its own detail:
The altercation helps Picasso overcome a creative block
and gleefully set to work on the gigantic painting.
It may be news to scholars that one of art’s
greatest testaments to the horror of war was
inspired, in part, by the excitement of being
fought over by a pair of jealous women."
Related Art —
A Creative Block —
Comments Off on Blockbuster Exhibition
Sunday, April 8, 2018
From a Log24 post of Feb. 5, 2009 —
An online logo today —
See also Harry Potter and the Lightning Bolt.
Comments Off on Design
Saturday, April 7, 2018
The FBI holding cube in "The Blacklist" —
" 'The Front' is not the whole story . . . ."
— Vincent Canby, New York Times film review, 1976,
as quoted in Wikipedia.
See also Solomon's Cube in this journal.
Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was allmeaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”
A less poetic meditation on the above 4x4x4 design cube —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
See also a related remark by LéviStrauss in 1955:
"…three different readings become possible:
left to right, top to bottom, front to back."
Comments Off on Sides
Monday, April 2, 2018
See the posts of April 1 three years ago.
Some context from a personal Kindle library —
"Elementary Mathematics from an Ad" suggests . . .
Comments Off on Review
Sunday, April 1, 2018
Saturday, March 31, 2018
“The greatest obstacle to discovery
is not ignorance —
it is the illusion of knowledge.”
— Daniel J. Boorstin,
Librarian of Congress,
quoted here in 2006.
Related material —
Remarks on Rubik's Cube from June 13, 2014 and . . .
See as well a different Gresham, author of Nightmare Alley ,
and Log24 posts on that book and the film of the same name .
Comments Off on Illusion
Friday, March 30, 2018
The online New York Times this evening has an obituary
for "an unorthodox … drama scholar" who reportedly died
on Thursday, March 22, 2018.
Some drama in this journal from around that date — in posts
tagged "The Cubes" — includes the following excerpt from
a graphic novel:
"Program or be programmed."
— A saying by the author of the above graphic novel.
Comments Off on Unorthodox Drama
Thursday, March 29, 2018
The title reverses a phrase of Fano —
“costruire (o, dirò meglio immaginare).”
Illustrations of imagining (the Fano plane) and of constructing (the eightfold cube) —
The Fano plane and the eightfold cube
Comments Off on To Imagine (or, Better, to Construct)
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
"Is this Time Cube?"
Notes toward an answer —
And from SixSet Geometry in this journal . . .
Comments Off on “Before Creation Itself . . .”
Wednesday, March 28, 2018
A comment on the the Diamond Theorem Facebook page —
Those who enjoy asymmetry may consult the "Expert's Cube" —
For further details see the previous post.
Comments Off on On Unfairly Excluding Asymmetry
Tuesday, March 27, 2018
Related material on automorphism groups —
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order168 simple group.
Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eightcube mathematical structure above
than they are an eightcube mechanical puzzle like the one below.
Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."
Comments Off on Compare and Contrast
Saturday, March 24, 2018
The search for Langlands in the previous post
yields the following Toronto Star illustration —
From a review of the recent film "Justice League" —
"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."
For other cubic adventures, see yesterday's post on A Piece of Justice
and the block patterns in posts tagged Design Cube.
Comments Off on Sure, Whatever.
Friday, March 23, 2018
Copy editing — From Wikipedia
"Copy editing (also copyediting or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."
An example of the need for copy editing:
Related material: Langlands and Reciprocity in this journal.
Comments Off on Reciprocity
Comments Off on Piece Prize
On the Oslo artist Josefine Lyche —
"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."
— Ann Cathrin Andersen,
http://bryggmagasin.no/2017/behindtheglitter/
Personal —
The Rushkoff Logo
— From a 2016 graphic novel by Douglas Rushkoff.
See also Rushkoff and Talisman in this journal.
Platonic —
The Diamond Cube.
Compare and contrast the shifting hexagon logo in the Rushkoff novel above
with the hexagoninsideacube in my "Diamonds and Whirls" note (1984).
Comments Off on From the Personal to the Platonic
Thursday, March 22, 2018
Also on March 18, 2015 . . .
Comments Off on In Memoriam
Wednesday, March 21, 2018
WISC = Wechsler Intelligence Scale for Children
RISC = Reduced Instruction Set Computer or
Rust Inventory of Schizotypal Cognitions
See related material in earlier WISC RISC posts.
See also . . .
"Many parents ask us about the Block Design section
on the WISC and hope to purchase blocks and exercises
like those used on the WISC test. We explain that doing that
has the potential to invalidate their child's test results.
These Froebel Color Cubes will give you a tool to work with
your child on the skills tested for in the Block Design section
of the WISC in an ethical and appropriate way. These same
skills are applicable to any test of nonverbal reasoning like
the NNAT, Raven's or nonverbal sections of the CogAT or OLSAT. "
— An online marketing webpage
For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.
Comments Off on WISC RISC
Monday, March 12, 2018
Stein reportedly died at 100 last Friday (March 9).
Related material —
Textiles by Stein arranged on the six faces of a cube —
Ethel Stein, "Circus & Slapstick," 1996
See also a less amusing approach to
patterns on the faces of a cube.
Comments Off on Stein
Wednesday, March 7, 2018
Related material —
The seven points of the Fano plane within
The Eightfold Cube.
"Before time began . . . ."
— Optimus Prime
Comments Off on Unite the Seven.
Tuesday, February 27, 2018
On the recent film "Justice League" —
From DC Extended Universe Wiki, "Mother Box" —
"However, during World War I, the British rediscovered
mankind's lost Mother Box. They conducted numerous studies
but were unable to date it due to its age. The Box was then
shelved in an archive, up until the night Superman died,
where it was then sent to Doctor Silas Stone, who
recognized it as a perpetual energy matrix. . . ." [Link added.]
The cube shape of the lost Mother Box, also known as the
Change Engine, is shared by the Stone in a novel by Charles Williams,
Many Dimensions . See the Solomon's Cube webpage.
See too the matrix of Claude LéviStrauss in posts tagged
Verwandlungslehre .
Some literary background:
Who speaks in primordial images speaks to us
as with a thousand trumpets, he grips and overpowers,
and at the same time he elevates that which he treats
out of the individual and transitory into the sphere of
the eternal. — C. G. JUNG
"In the conscious use of primordial images—
the archetypes of thought—
one modern novelist stands out as adept and
grand master: Charles Williams.
In The Place of the Lion he incarnates Plato’s
celestial archetypes with hairraising plausibility.
In Many Dimensions he brings a flock of ordinary
mortals face to face with the stone bearing
the Tetragrammaton, the Divine Name, the sign of Four.
Whether we understand every line of a Williams novel
or not, we feel something deep inside us quicken
as Williams tells the tale.
Here, in The Greater Trumps , he has turned to
one of the prime mysteries of earth . . . ."
— William Lindsay Gresham, Preface (1950) to
Charles Williams's The Greater Trumps (1932)
For fans of what the recent series Westworld called "bulk apperception" —
Comments Off on Raiders of the Lost Images
Saturday, February 17, 2018
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the twoelement Galois field GF(2) —

"The digital revolution based on the 2 symbols (0,1)"

"The algebra of George Boole"

"Binary codes"

"Dirac's spinors, with their up/down dichotomy"
These phrases are from the yearend review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —

The 2×2 square

The 2x2x2 cube

The 4×4 square

The 4x4x4 cube
See Finite Geometry of the Square and Cube.
See also today's earlier post God's Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on The Binary Revolution
Tuesday, February 13, 2018
For the late Anne M. Treisman, who reportedly died Friday, Feb. 9:
From "A FeatureIntegration Theory of Attention" —
"The controversy between analytic and synthetic theories
of perception goes back many years: the Associationists
asserted that the experience of complex wholes is built
by combining more elementary sensations, while the
Gestalt psychologists claimed that the whole precedes
its parts, that we initially register unitary objects and
relationships, and only later, if necessary, analyze these
objects into their component parts or properties. This view
is still active now . . . ."
— Anne M. Treisman, University of British Columbia,
and Garry Gelade, Oxford University, in
Cognitive Psychology 12, 97136 (1980)
"Before time began, there was the Cube." — Optimus Prime
Comments Off on Attention Must Be Paid
Tuesday, February 6, 2018
The metaphor for metamorphosis no keys unlock.
— Steven H. Cullinane, "Endgame"
* See Times Square Church in this journal and
the posts of July 2010. Related material:
A Monday night death —
Comments Off on For Times Square Church*
Monday, January 22, 2018
A death on the date of the above symmetry chat,
Wednesday, August 17, 2016 —
An Hispanic Hollywood moment:
Ojo de Dios —
Click for related material.
For further Hispanic entertainment,
see Ben Affleck sing
"Aquellos Ojos Verdes "
in "Hollywoodland."
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Wednesday, January 17, 2018
(Continued from September 12, 2005)
The previous post contrasted the numbertriple 1178 below
with number triples 1295 and 1259.
A perhaps more logical counterpart of the triple 1178, based
on opposite locations of starpoints or cubeedges, is
the triple 9125. For a theological interpretation, see 9/12/05.
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Tuesday, January 16, 2018
Lines from characters played in the film by Tom Hanks and Halle Berry —
— Cloud Atlas , by David Mitchell (2004).
An orison of sorts from a post on Martin Scorsese's
birthday, Sunday, Nov. 18, 2007 —
Displayed on the BlackBerry are parts
of Log24 posts from October 25, 2007,
and October 24, 2007.
Related pattern geometry
From a Log24 search for Angleton + Brotherhood:
A photo of Angleton in a post from 12/9/5 —
From a post of 11/7/8 —
A cryptic note for Dan Brown:
The above dates 11/7/8 and 12/9/5 correspond to the cornerlabels
(read clockwise and counterclockwise) of the two large triangles
in the Finkelstein Talisman —
Above: More symbology for Tom Hanks from
this morning's post The Pentagram Papers.
The above symbology is perhaps better suited to Hanks in his
role as Forrest Gump than in his current role as Ben Bradlee.
For Hanks as Dan Brown's Harvard symbologist
Robert Langdon, see the interpretation 12/5/9, rather
than 12/9/5, of the above triangle/cubecorner label.
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Other intersectionpointscounting material —
The Finkelstein Talisman:
See also Hanks + Cube in this journal —
.
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Saturday, January 6, 2018
Tom Wolfe in The Painted Word (1975):
"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde understood, they
were not only theories but … hot news,
straight from the studios, from the scene."
Harold Rosenberg in The New Yorker (click to enlarge)—
See also Interality and the Eightfold Cube .
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Friday, January 5, 2018
* See the term interality in this journal.
For many synonyms, see
"The Human Seriousness of Interality,"
by Peter Zhang, Grand Valley State University,
China Media Research 11(2), 2015, 93103.
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Wednesday, December 27, 2017
See the 27part structure of
the 3x3x3 Galois cube
as well as Autism Sunday 2015.
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Tuesday, December 26, 2017
(Continued)
Two Students of Structure
A comment on Sean Kelly's Christmas Morning column on "aliveness"
in the New York Times philosophy series The Stone —
Diana Senechal's 1999 doctoral thesis at Yale was titled
"Diabolical Structures in the Poetics of Nikolai Gogol."
Her mother, Marjorie Senechal, has written extensively on symmetry
and served as editorinchief of The Mathematical Intelligencer .
From a 2013 memoir by Marjorie Senechal —
"While I was in Holland my enterprising student assistant at Smith had found, in Soviet Physics – Crystallography, an article by N. N. Sheftal' on tetrahedral penetration twins. She gave it to me on my return. It was just what I was looking for. The twins Sheftal' described had evidently begun as (111) contact twins, with the two crystallites rotated 60^{o} with respect to one another. As they grew, he suggested, each crystal overgrew the edges of the other and proceeded to spread across the adjacent facet. When all was said and done, they looked like they'd grown through each other, but the reality was overandaround. Brilliant! I thought. Could I apply this to cubes? No, evidently not. Cube facets are all (100) planes. But . . . these crystals might not have been cubes in their earliest stages, when twinning occurred! I wrote a paper on "The mechanism of certain growth twins of the penetration type" and sent it to Martin Buerger, editor of Neues Jarbuch für Mineralogie. This was before the Wrinch symposium; I had never met him. Buerger rejected it by return mail, mostly on the grounds that I hadn't quoted any of Buerger's many papers on twinning. And so I learned about turf wars in twin domains. In fact I hadn't read his papers but I quickly did. I added a reference to one of them, the paper was published, and we became friends.[5]
After reading Professor Sheftal's paper I wrote to him in Moscow; a warm and encouraging correspondence ensued, and we wrote a paper together long distance.[6] Then I heard about the scientific exchanges between the Academies of Science of the USSR and USA. I applied to spend a year at the Shubnikov Institute for Crystallography, where Sheftal' worked. I would, I proposed, study crystal growth with him, and color symmetry with Koptsik. To my delight, I was accepted for an 11month stay. Of course the children, now 11 and 14, would come too and attend Russian schools and learn Russian; they'd managed in Holland, hadn't they? Diana, my older daughter, was as delighted as I was. We had gone to Holland on a Russian boat, and she had fallen in love with the language. (Today she holds a Ph.D. in Slavic Languages and Literature from Yale.) . . . .
. . . we spent the academic year 197879 in Moscow.

Philosophy professors and those whose only interest in mathematics
is as a path to the occult may consult the Log24 posts tagged Tsimtsum.
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