See Eightfold Froebel.
See Eightfold Froebel.
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 19842003 (Oct. 24, 2008).
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).
"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
The Silvia of the title is from the previous post.
For the Time Cube, see …
The assignments page for a graduate algebra course at Cornell
last fall had a link to the eightfold cube:
Nobel Flashback:
Wednesday, January 29, 2014

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.
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 
See also Log24 on the publication date of Dark Matter .
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.)
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.
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
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.
<!– 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. –>
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.
For Aaron Sorkin and Walter Isaacson —
Related material —
Bauhaus Cube, Design Cube, and
Nabokov's Transparent Things .
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.
Omega is a Greek letter, Ω , used in
mathematics to denote a set on which
a group acts.
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).
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)
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

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.)
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.
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.
For the late Cardinal Glemp of Poland,
who died yesterday, some links:
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…"
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).
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."
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).
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.
For Pete Rustan, space recon expert, who died on June 28—
See also Galois vs. Rubik and Group Theory Template.
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.
See also Finite Geometry and Physical Space.
Related material from MacTutor—
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:
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.
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.
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.
"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.
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—
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
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
rotational symmetries.
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."
The New York Times has a skateboarder obit with a URL date of July 9.
Here is an earlier version from the LA Times—
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.
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.
The 3×3×3 Galois Cube
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.
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.
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."
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.
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.
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.
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.
The Klein Correspondence, Penrose SpaceTime, and a Finite Model
Related material:
Monday’s entry Just Say NO and a poem by Stevens,
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. "
For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.
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.
Related material —
The seven points of the Fano plane within
"Before time began . . . ."
— Optimus Prime
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 cubic 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" —
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the twoelement Galois field GF(2) —
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) —
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:
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
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 —
(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.
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.
Other intersectionpointscounting material —
See also Hanks + Cube in this journal —
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 .
* 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.
See the 27part structure of
the 3x3x3 Galois cube
as well as Autism Sunday 2015.
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.) . . . . 
Philosophy professors and those whose only interest in mathematics
is as a path to the occult may consult the Log24 posts tagged Tsimtsum.
The previous post, "Mind," suggests a search for "n+1" in this journal.
From that search —
The above psychoanalytic remarks suggest . . .
See also "Transformers" (2007).
"Before time began, there was the Cube."
— Optimus Prime
David E. Wellbery on Goethe
From an interview published on 2 November 2017 at
http://literaturwissenschaftberlin.de/interviewwithdavidwellbery/
as later republished in
The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .
This is a sequel to yesterday's post Cube Space Continued.
Logo from the above webpage —
See also the similar structure of the eightfold cube, and …
Related dialogue from the new film "Unlocked" —
1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.
1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.
The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017 —
A visually related concept — See Solomon's Cube in this journal.
Chronologically related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem —
The title is from this morning's online New York Times review
of a new Jackie Chan film.
Click the image below for some related posts.
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.
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.
From the Log24 post "A Point of Identity" (August 8, 2016) —
A logo that may be interpreted as oneeighth of a 2x2x2 array
of cubes —
The figure in white above may be viewed as a subcube representing,
when the eightcube array is coordinatized, the identity (i.e., (0, 0, 0)).
An image in memory of a publisher* who reportedly died
on Saturday, August 26, 2017.
He and his wife wrote a novel, The Twelve , that has been compared to
the classic film "Village of the Damned." (See a sequel in this journal.)
For more on the image, see posts now tagged The Finkelstein Talisman.
In memory of a TV gunslinger who reportedly died Thursday, August 3, 2017 . . .
From this journal on that day (posts now tagged Dark Tower Theology) —
"The concept under review is that of the Holy Trinity.
See also, in this journal, Cube Trinity.
For a simpler Trinity model, see the threepoint line …"
Or: Trinity Test Site
From the New York Times Book Review of
next Sunday, August 6, 2017 —
"In a more conventional narrative sequence,
even a sequence of poems,
this interpenetration would acquire
sequence and evolution." [Link added.]
The concept under review is that of the Holy Trinity.
See also, in this journal, Cube Trinity.
For a simpler Trinity model, see the threepoint line …
From Solomon's Cube —
"Here MSRI, an acronym for Mathematical Sciences Research Institute,
is pronounced 'Misery.' See Stephen King [and] K.C. Cole . . . ."
From a manuscript by Mikhail Gromov cited yesterday in MSRI Program —
"The field of geometric group theory emerged from Gromov’s insight
that even mathematical objects such as groups, which are defined
completely in algebraic terms, can be profitably viewed as geometric
objects and studied with geometric techniques."
— Mathematical Sciences Research Institute, 2016:
See also some writings of Gromov from 201516:
For a simpler example than those discussed at MSRI
of both algebraic and geometric techniques applied to
the same group, see a post of May 19, 2017,
"From Algebra to Geometry." That post reviews
an earlier illustration —
For greater depth, see "Eightfold Cube" in this journal.
In memory of a Disney "imagineer" who reportedly died yesterday.
From the opening scene of a 2017 film, "Gifted":
Frank calls his niece Mary to breakfast on the morning she is
to enter first grade. She is dressed, for the first time, for school —
 Hey! Come on. Let's move!  No!  Let me see.  No.  Come on, I made you special breakfast.  You can't cook.  Hey, Mary, open up. (She opens her door and walks out.)  You look beautiful.  I look like a Disney character. Where's the special?  What?  You said you made me special breakfast. Read more: http://www.springfieldspringfield.co.uk/ movie_script.php?movie=gifted 
Michiko Kakutani in The New York Times —
"The detective story genre concerns the finding of clues
and the search for hidden designs, and its very form
underscores Mr. Pynchon’s obsession with conspiracies
and the existence of systems too complicated to understand."
— Review of Pynchon's Bleeding Edge , Sept. 10, 2013
Background: "Moss on the Wall," this journal on that date.
A less complicated system —
"Plan 9 deals with the resurrection of the dead."
— Bill Murray in "Ed Wood"
(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)
The above title was suggested by a film trailer quoted here Saturday —
" Jeremy Irons' dry Alfred Pennyworth:
'One misses the days when one's biggest concerns
were exploding windup penguins.' "
"Penguin Classics Deluxe Edition" describes, among other books,
an edition of the I Ching published on December 1, 2015.
Excerpt from this journal on that date —
Tuesday, December 1, 2015
Verhexung

Related material —
Why was the Cosmic Cube named the Tesseract
in the Marvel movie series? Is there any specific reason
for the name change? According to me, Cosmic Cube
seems a nice and cooler name.
— Asked March 14, 2013, by Dhwaneet Bhatt
At least it wasn't called 'The AllSpark.'
It's not out of the realm of possibility.
— Solemnity, March 14, 2013
Publishers Weekly on a Nov. 1, 2011, book, Under Blue Cup —
"Krauss’s core argument (what she deems a 'crusade')
is that the 'white cube,' which conceptual and installation
artists have deemed obsolete, actually thrives."
For other "core arguments," see Satuday's post "Common Core"
and the Art Space posts "Odd Core" and "Even Core."
Paul Krugman:
Asimov's Foundation novels grounded my economics
In the Foundation novels of Isaac Asimov …
"The Prime Radiant can be adjusted to your mind, and all
corrections and additions can be made through mental rapport.
There will be nothing to indicate that the correction or addition
is yours. In all the history of the Plan there has been no
personalization. It is rather a creation of all of us together.
Do you understand?"
"Yes, Speaker!"
— Isaac Asimov, Second Foundation , Ch. 8: Seldon's Plan
"Before time began, there was the Cube."
See also Transformers in this journal.
The 4x4x4 cube is the natural setting
for the finite version of the Klein quadric
and the eight "heptads" discussed by
Conwell in 1910.
As R. Shaw remarked in 1995,
"The situation is indeed quite pleasing."
From an Anthony Lane movie review in the April 8, 2013,
issue of The New Yorker —
"When the Lord God forbade his worshippers to bow down
before any graven image, [Rosario] Dawson’s face was
exactly the kind of thing He had in mind. No other star can
boast such sculptured features—except Vincent Cassel,
who is pretty damn graven himself. When the two of them
make love, in 'Trance,' one strong bone structure pressed
against another, it’s like a clash of major religions. What if
they had a family? The kids would be practically Cubist."
As for the other film Lane reviewed in that issue, "Blancanieves" —
See Snow White + Cube in this journal.
See as well a related cartoon graveyard, also from April 8, 2013.
Detail from the previous post —
See Space Cross in this journal.
See also Anthony Hopkins' new film
"Transformers: The Last Knight" and …
Remark on conceptual art quoted in the previous post —
"…he’s giving the concept but not the realization."
A concept — See a note from this date in 1983:
A realization —
Not the best possible realization, but enough for proof of concept .
"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.
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
See the previous three posts… and the Nobel flashback titled Cuber.
Cambridge University Press in 1999 —
See also Cube Bricks.
From a post of last Friday, June 2 —
See also Transformers in this journal.
"Before time began, there was the Cube."
— Transformers (2007)
The Cube
CodePen logo, pictured here on May 28, 2017 —
From YouTube, "The Cube," published on April 6, 2016 —
Meanwhile, also on April 6, 2016, at 2:01 AM ET …
* See The Pinterest Directive and Expanding the Spielraum.
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
See also "Cornerstone" in this journal and …
A sidebar from a Google search today —
This suggests a review of posts now tagged Obelisk,
which include …
See also "Romancing the Omega" —
Related mathematics — Guitart in this journal —
See also Weyl + Palermo in this journal —
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).
"Plan 9 deals with the resurrection of the dead."
— Bill Murray in "Ed Wood"
(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)
Prequel —
Note that Yale's die design and use of the phrase "rigid motions"
differ from those in the webpage "Solomon's Cube."
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
From Log24, "Cube Bricks 1984" —
Also on March 9, 2017 —
For those who prefer graphic art —
Powered by WordPress