Tuesday, May 21, 2019
Inside the White Cube
Monday, May 13, 2019
Thursday, March 22, 2018
The Diamond Cube
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.
Tuesday, April 5, 2016
“Puzzle Cube of a Novel”
Sunday, December 28, 2014
Cube of Ultron
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)
Monday, May 19, 2014
Thursday, January 24, 2013
Cube Space
For the late Cardinal Glemp of Poland,
who died yesterday, some links:
Friday, December 28, 2012
Cube Koan
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…"
Sunday, August 5, 2012
Cube Partitions
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.
Sunday, February 5, 2012
Wednesday, January 11, 2012
Cuber
"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.
Friday, December 30, 2011
Quaternions on a Cube
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…
 The 1985 note from which the above figures were drawn
 Visualizing GL(2,p)
 Quaternions in an Affine Galois Plane
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)
Saturday, August 27, 2011
Cosmic Cube*
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
Friday, June 24, 2011
The Cube
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.
Friday, May 13, 2011
Apollo’s 13
Continued … See related previous posts.
Those who prefer narrative to mathematics
may consult Wikipedia on The Cosmic Cube.
Wednesday, July 10, 2019
Artifice* of Eternity …
… and Schoolgirl Space
"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."
— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 1420.
See also Schoolgirl Space in this journal.
* See, for instance, Lewis Hyde on the word "artifice" and . . .
Tuesday, July 9, 2019
Perception of Space
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
Dreamtimes
“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”
— John Malkovich at squarespace.com, January 10, 2017
Also on that date . . .
Monday, July 8, 2019
Exploring Schoolgirl Space
See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."
Sunday, July 7, 2019
Schoolgirl Problem
Anonymous remarks on the schoolgirl problem at Wikipedia —
"This solution has a geometric interpretation in connection with
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing."
See also Polster + Tetrahedron in this journal.
There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square model rather
than a tetrahedral model. The square model of PG(3,2) last
appeared in the schoolgirlproblem article on Feb. 11, 2017, just
before a revision that removed it.
Saturday, June 8, 2019
Art Object, continued and continued
See as well posts mentioning "An Object of Beauty."
Update of 12 AM June 11 — A screenshot of this post
is now available at http://dx.doi.org/10.17613/hqk7nx97 .
Monday, May 13, 2019
Doris Day at the Hudson Rock
" 'My public image is unshakably that of
America’s wholesome virgin, the girl next door,
carefree and brimming with happiness,'
she said in Doris Day: Her Own Story ,
a 1976 book . . . ."
From "Angels & Demons Meet Hudson Hawk" (March 19, 2013) —
From the March 1 post "Solomon and the Image," a related figure —
Friday, March 1, 2019
Solomon and the Image
"Maybe an image is too strong
Or maybe is not strong enough."
— "Solomon and the Witch,"
by William Butler Yeats
Saturday, March 24, 2018
Sure, Whatever.
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.
Friday, March 23, 2018
Reciprocity
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.
From the Personal to the Platonic
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 —
Compare and contrast the shifting hexagon logo in the Rushkoff novel above
with the hexagoninsideacube in my "Diamonds and Whirls" note (1984).
Thursday, March 22, 2018
Wednesday, March 7, 2018
Unite the Seven.
Related material —
The seven points of the Fano plane within
"Before time began . . . ."
— Optimus Prime
Monday, January 22, 2018
Hollywood Moment
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."
Friday, October 13, 2017
Wednesday, September 13, 2017
Summer of 1984
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
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.
Chin Music
Tuesday, June 20, 2017
AllSpark Notes
"For years, the AllSpark rested, sitting dormant
like a giant, useless art installation."
— Vinnie Mancuso at Collider.com yesterday
Related material —
Giant, useless art installation —
Sol LeWitt at MASS MoCA. See also LeWitt in this journal.
Wednesday, April 12, 2017
Contracting the Spielraum
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 112, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
Monday, April 3, 2017
Friday, January 13, 2017
Thursday, January 12, 2017
The Cherished Gift
From "Solomon's Cube" —
Related material —
"Is this a dagger I see before me?"
"No." (A line suggested by Polanski's 2010 "The Ghost Writer")
Sunday, January 1, 2017
Like the Horizon
(Continued from a remark by art critic Peter Schjeldahl quoted here
last year on New Year's Day in the post "Art as Religion.")
"The unhurried curve got me.
It was like the horizon of a world
that made a nonworld of
all of the space outside it."
— Peter Schjeldahl, "Postscript: Ellsworth Kelly,"
The New Yorker , December 30, 2015
This suggests some further material from the paper
that was quoted here yesterday on New Year's Eve —
"In teaching a course on combinatorics I have found
students doubting the existence of a finite projective
plane geometry with thirteen points on the grounds
that they could not draw it (with 'straight' lines)
on paper although they had tried to do so. Such a
lack of appreciation of the spirit of the subject is but
a consequence of the elements of formal geometry
no longer being taught in undergraduate courses.
Yet these students were demanding the best proof of
existence, namely, production of the object described."
— Derrick Breach (See his obituary from 1996.)
A related illustration of the 13point projective plane
from the University of Western Australia:
Projective plane of order 3
(The four points on the curve
at the right of the image are
the points on the line at infinity .)
The above image is from a post of August 7, 2012,
"The Space of Horizons." A related image —
Click on the above image for further remarks.
Saturday, November 12, 2016
Friday, September 30, 2016
Desmic Midrash
The author of the review in the previous post, Dara Horn, supplies
below a midrash on "desmic," a term derived from the Greek desme
( δεσμή , bundle, sheaf, or, in the mathematical sense, pencil —
French faisceau ), which is apparently related to the term desmos , bond …
(The term "desmic," as noted earlier, is relevant to the structure of
Heidegger's Sternwürfel .)
The Horn midrash —
(The "medieval philosopher" here is not the remembered preChristian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:
Guide for the Perplexed , 3:51.)
Here of course "that bond" may be interpreted as corresponding to the
Greek desmos above, thus also to the desmic structure of the
stellated octahedron, a sort of threedimensional Star of David.
See "desmic" in this journal.
Thursday, September 29, 2016
Articulation
Cassirer vs. Heidegger at Harvard —
A remembrance for Michaelmas —
A version of Heidegger's "Sternwürfel " —
From Log24 on the upload date for the above figure —
Wednesday, September 28, 2016
Star Wars
See also in this journal "desmic," a term related
to the structure of Heidegger's Sternwürfel .
Saturday, September 17, 2016
Friday, September 2, 2016
Raiders of the Lost Birthday
Some images from the posts of last July 13
(Harrison Ford's birthday) may serve as funeral
ornaments for the late Prof. David Lavery.
See as well posts on "Silent Snow" and "Starlight Like Intuition."
Wednesday, July 27, 2016
Deathly Hallows
The previous post, on the July 13 death of computer scientist Robert Fano,
suggests a review of "Deathly Hallows" posts in this journal. From that review —
Mathematics
The Fano plane block design 
Magic
The Deathly Hallows symbol— 
For further information, click the image below —
Tuesday, July 26, 2016
In Nomine Patris
"Robert Fano, an electrical engineer who was instrumental
in creating a world of instantly responsive computers, died
on July 13 in Naples, Fla. He was 98."
— John Markoff in this evening's online New York Times
"Fano's father was the mathematician Gino Fano . . . ."
A mnemonic I associate with the Fano plane — "Seven is Heaven . . . ."
Log24 on the date of Robert Fano's death —
Wednesday, July 13, 2016
Luminosity
"At CERN the LHC has reached design luminosity,
— Peter Woit, Thursday, June 30, 2016, 
Another sort of design luminosity —
Art Wars
Wil S. Hylton today in the online New York Times —
"It seems to me now, with greater reflection,
that the value of experiencing another person’s art
is not merely the work itself, but the opportunity
it presents to connect with the interior impulse of another.
The arts occupy a vanishing space in modern life:
They offer one of the last lingering places to seek out
empathy for its own sake, and to the extent that
an artist’s work is frustrating or difficult or awful,
you could say this allows greater opportunity to try to
meet it. I am not saying there is no room for discriminating
taste and judgment, just that there is also, I think,
this other portal through which to experience creative work
and to access a different kind of beauty, which might be
called communion."
Or damnation.
Always Nice to See You
Thursday, December 17, 2015
Hint of Reality
From an article* in Proceedings of Bridges 2014 —
As artists, we are particularly interested in the symmetries of real world physical objects. Three natural questions arise: 1. Which groups can be represented as the group of symmetries of some realworld physical object? 2. Which groups have actually been represented as the group of symmetries of some realworld physical object? 3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not? 
The article was cited by Evelyn Lamb in her Scientific American
weblog on May 19, 2014.
The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:
"… finite projective planes [in particular, the 7point Fano plane,
about which Lamb is writing] seem like a triumph of purely
axiomatic thinking over any hint of reality…."
For related hints of reality, see Eightfold Cube in this journal.
* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman
Saturday, October 10, 2015
Epiphany in Paris
Friday, August 7, 2015
Parts
Spielerei —
"On the most recent visit, Arthur had given him
a brightly colored cube, with sides you could twist
in all directions, a new toy that had just come onto
the market."
— Daniel Kehlmann, F: A Novel (2014),
translated from the German by
Carol Brown Janeway
Nicht Spielerei —
A figure from this journal at 2 AM ET
on Monday, August 3, 2015
Also on August 3 —
FRANKFURT — "Johanna Quandt, the matriarch of the family
that controls the automaker BMW and one of the wealthiest
people in Germany, died on Monday in Bad Homburg, Germany.
She was 89."
MANHATTAN — "Carol Brown Janeway, a Scottishborn
publishing executive, editor and awardwinning translator who
introduced American readers to dozens of international authors,
died on Monday in Manhattan. She was 71."
Related material — Heisenberg on beauty, Munich, 1970
Wednesday, May 13, 2015
Space
Notes on space for day 13 of May, 2015 —
The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."
Related poetic material:
The ninefold square and Apollo, as well as …
Thursday, May 7, 2015
Paradigm for Pedagogues
Illustrations from a post of Feb. 17, 2011:
Plato’s paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
Ultron: By the Book
If The New York Times interviewed Ultron for its
Sunday Book Review "By the Book" column —
What books are currently on your night stand?
Steve Fuller's Thomas Kuhn: A Philosophical History for Our Times
Gerald Holton's Thematic Origins of Scientific Thought
John Gray's The Soul of the Marionette
Wednesday, May 6, 2015
Soul
Nonsense…
See Gary Zukav, Harvard '64, in this journal.
and damned nonsense —
"Every institution has a soul."
— Gerald Holton in Harvard Gazette today
Commentary —
"The Ferris wheel came into view again…."
— Malcom Lowry, Under the Volcano
See also Holton in a Jan. 1977 interview:
"If people have souls, and I think a few have, it shows…."
Wednesday, April 1, 2015
WürfelMärchen
Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —
For Tschira in this journal, see Stiftung .
For some Würfel illustrations, see this morning's post
Manifest O. A related webpage —
Manifest O
The title was suggested by
http://benmarcus.com/smallwork/manifesto/.
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —

An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001  Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. 
831016  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915  Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. 
Thursday, December 18, 2014
Platonic Analogy
(Five by Five continued)
As the 3×3 grid underlies the order3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged GaloisPlane Models.
Wednesday, November 26, 2014
Class Act
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.
A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:
The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former. [9] I am aware only of a series of inhouse publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie IX.
— Stephen Eberhart, Dept. of Mathematics, 
Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…
… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. nonEuclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful. Steve 
See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.
Tuesday, November 25, 2014
EuclideanGalois Interplay
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Thursday, November 13, 2014
Mort de Grothendieck
“Alexandre Grothendieck est mort jeudi matin
à l’hôpital de SaintGirons (Ariège), à l’âge de 86 ans.”
Update of 6: 16 PM ET: A memorial of sorts, from May 27 this year:
Wednesday, September 17, 2014
Raiders of the Lost Articulation
Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :
An unarticulated (but colored) cube:
A 2x2x2 articulated cube:
A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:
Friday, August 29, 2014
Raum
A possible answer to the 1923 question of Walter Gropius, "Was ist Raum?"—
See also yesterday's Source of the Finite and the image search
on the Gropius question in last night's post.
Thursday, August 28, 2014
Brutalism Revisited
Yesterday's 11 AM post was a requiem for a brutalist architect.
Today's LA Times has a related obituary:
"Architectural historian Alan Hess, who has written several books on
MidCentury Modern design, said Meyer didn't have a signature style,
'which is one reason he is not as wellknown as some other architects
of the period. But whatever style he was working in, he brought a real
sense of quality to his buildings.'
A notable example is another bank building, at South Beverly Drive
and Pico Boulevard, with massive concrete columns, a hallmark of
the New Brutalism style. 'This is a really good example of it,' Hess said."
— David Colker, 5:43 PM LA time, Aug. 28, 2014
A related search, suggested by this morning's post Source of the Finite:
(Click to enlarge.)
Source of the Finite
"Die Unendlichkeit ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche stamme…."
— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: AubierFlammarion, 1969), fragment 120, p. 118
Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p. 181.
This quotation was suggested by the Bauhausrelated phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 19842003.
For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.
Wednesday, August 27, 2014
Altar
"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"
— Macaulay, quoted in the April 2013 film "Oblivion"
"Leave a space." — Tom Stoppard, "Jumpers"
Related material: The August 16, 2014, sudden death in Scotland
of an architect of the above Cardross seminary, and a Log24 post,
Plato's Logos, from the date of the above photo: June 26, 2010.
See also…
Here “eidolon” should instead be “eidos .”
An example of eidos — Plato's diamond (from the Meno ) —
Schau der Gestalt
(Continued from Aug. 19, 2014)
"Christian contemplation is the opposite
of distanced consideration of an image:
as Paul says, it is the metamorphosis of
the beholder into the image he beholds
(2 Cor 3.18), the 'realisation' of what the
image expresses (Newman). This is
possible only by giving up one's own
standards and being assimilated to the
dimensions of the image."
— Hans Urs von Balthasar,
The Glory of the Lord:
A Theological Aesthetics,
Vol. I: Seeing the Form
[ Schau der Gestalt ],
Ignatius Press, 1982, p. 485
A Bauhaus approach to Schau der Gestalt :
I prefer the I Ching 's approach to the laws of cubical space.
Saturday, July 12, 2014
Sequel
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
Verbum theme:
A search today for Verbum in this journal yielded
a Georgetown University Chomskyite, Professor
David W. Lightfoot.
"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."
Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.
See Tecumseh Fitch in this journal.
For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.
That birthday was also the upload date for the following:
See esp. the comments section.
Wednesday, June 4, 2014
Monkey Business
The title refers to a Scientific American weblog item
discussed here on May 31, 2014:
Some closely related material appeared here on
Dec. 30, 2011:
A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:
"Is there a geometric realization of Quaternion group?" —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).
Saturday, May 31, 2014
Quaternion Group Models:
The ninefold square, the eightfold cube, and monkeys.
For posts on the models above, see quaternion
in this journal. For the monkeys, see
"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American weblog, May 19, 2014:
The Scientific American item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):
See also Finite Geometry and Physical Space.
Thursday, March 27, 2014
Diamond Space
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the sixdimensional affine space
over the twoelement Galois field.
The reason for the name:
Click to enlarge.
Thursday, February 13, 2014
Plan 9
The final link in today's previous post leads to
a post whose own final link leads to…
Thursday, December 13, 2012

Fashion Week — The Conclusion
Winter’s Game*
Part I: Continued from January 20 — "Arising Heaven" —
Part II: The Stars My Destination in this journal
Part III: Ender's Game —
* The title refers to a character, Rogue Winter, in Alfred Bester's
1981 novel The Deceivers .
Monday, January 13, 2014
A Prime for Marissa
Wednesday, December 25, 2013
Tuesday, December 24, 2013
Through a Mirror, Darkly
Review of a book first published in 1989—
Reality's Mirror: Exploring the Mathematics of Symmetry —
"Here is a book that explains in laymen language
what symmetry is all about, from the lowliest snowflake
and flounder to the lofty group structures whose
astonishing applications to the Old One are winning
Nobel prizes. Bunch's book is a marvel of clear, witty
science writing, as delightful to read as it is informative
and uptodate. The author is to be congratulated on
a job well done." — Martin Gardner
A completely different person whose name
mirrors that of the Mathematics of Symmetry author —
See also this journal on the date mentioned in the Princetonian .
"Always with a little humor." — Yen Lo
Wednesday, December 18, 2013
Bing Bang Theory
Microsoft in 2009 on its new search engine name—
"We like Bing because it sounds off in our heads
when we think about that moment of discovery
and decision making— when you resolve those
important tasks."
A search on Bing today —
A colorful tale —
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).
American Beauty
Thursday, December 12, 2013
Outsider Art
"… Galois was a mathematical outsider…."
— Tony Mann, "head of the department of mathematical sciences,
University of Greenwich, and president, British Society for the
History of Mathematics," in a May 6, 2010, review of Duel at Dawn
in Times Higher Education.
Related art:
(Click for a larger image.)
For a less outside version of the central image
above, see Kunstkritikk on Oct. 15, 2013.
Monday, December 9, 2013
Heaven Descending
An I Ching study quoted in Waiting for Ogdoad (St. Andrew’s Day, 2013)—
(Click for clearer image.)
The author of the above I Ching study calls his lattice “Arising Heaven.”
The following lattice might, therefore, be called “Heaven Descending.”
Click for the source, mentioned in Anatomy of a Cube (Sept. 18, 2011).
Thursday, December 5, 2013
Tuesday, December 3, 2013
Diamond Space
A new website illustrates its URL.
See DiamondSpace.net.
Thursday, November 21, 2013
ART WARS:
The Mitgang Menu
Related material: This morning's 6 AM post and Wiener News.
Update of 3:29 PM:
From Herbert Mitgang's New York Times
obituary of Cleanth Brooks—
"The New Critics advocated close reading of literary texts
and detailed analysis, concentrating on semantics, meter,
imagery, metaphor and symbol as well as references to
history, biography and cultural background."
But It Rings…
"The shaving razor's cold and it stings."
The above image is from Ulysses “Seen,” adapted
by Robert Berry from the novel by James Joyce.
Quad Rants
Continued from 24 hours ago.
"AA had no rules but many traditions (that were, in fact, rules).
One of the most ironclad was that you never made a Twelfth Step
call on an active alcoholic by yourself, unless the alkie in question
was safely incarcerated in a hospital, detox, or the local bughouse.
If you did, you were apt to end up matching him drink for drink and
line for line."
— King, Stephen (20130924). Doctor Sleep: A Novel
(p. 272). Scribner. Kindle Edition.
" Aus 'It' wurde 'Es', und King sprach es so aus,
dass man sich alleine vom Klang des Titels
gruselte: 'Essssss!' "
— Last night's online
Hamburger Abendblatt
Tuesday, November 12, 2013
The XMen Tree
Related material:
The comments on a Log24 post of Nov. 6, 2013,
remarks by Michael Worton on the tree in
"Waiting for Godot," images from the film
"The Tree of Life," and, in memory of Robert
de Marrais, an image search from this evening:
"Spelling the Tree" + "de Marrais," 2 MB.
Funeral Canticle
For and by composer Sir John Tavener, 69,
who reportedly died today.
Update of 8:28 PM ET Nov. 12—
The obituary link above is to The Telegraph.
Here is a link to the version in The New York Times—
Soundtrack
"DEVIL – MUSIC
20 pages of incidental music written at school
for G. K. Chesterton's play MAGIC
by D. Coxeter."
See also…
Related material — Chesterton + Magic in this journal.
Wednesday, October 9, 2013
Sign in
According to Hoyle
To Apollo*
Friday, October 4, 2013
Christian’s Question
— Christian Bale in "American Hustle," a film
scheduled for limited release on St. Lucy's Day
and wide release on Christmas Day, 2013
See also this journal on November 26, 2012.
Walter’s Wake
(Continued from October First)
"It gets to the end
We get to run it again"
— James Taylor,
"One More Go Round" from
New Moon Shine album
For the Feast of St. Francis
"According to Vladimir Nabokov, Salvador Dalí
was 'really Norman Rockwell’s twin brother
kidnapped by gypsies in babyhood.'
But actually there were triplets: the third one is
Stephen King."
— Margaret Atwood, "Shine On,"
online Sept. 19, 2013
"The metaphor for metamorphosis
no keys unlock."
Roll Credits
Inceptions
Sunday, September 22, 2013
Incarnation, Part 2
"… a list of group theoretic invariants
and their geometric incarnation…"
— David Lehavi on the Kummer 16_{6} configuration in 2007
Related material —
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
"This is not theology; this is mathematics."
— Steven H. Cullinane on four quartets
To wit:
Saturday, September 21, 2013
Geometric Incarnation
The Kummer 16_{6} configuration is the configuration of sixteen
6sets within a 4×4 square array of points in which each 6set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
Mathematics and Narrative (continued)
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013

Narrative:
Aooo.
Happy birthday to Stephen King.