Monday, September 13, 2021
The above Quanta article mentions
"Maryna Viazovska’s 2016 discovery of the most efficient
ways of packing spheres in dimensions eight and 24."
From a course to be taught by Viazovska next spring:
The Lovasz reference suggests a review of my own webpage
Cube Space, 19842003.
See as well a review of Log24 posts on Packing.
Comments Off on Cube Space Revisited
Saturday, August 28, 2021
Geometry for Jews continues.
The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , MayJune 1982 —
Related geometric meditation —
Archimedes at Hiroshima
in posts tagged Aitchison.
* As opposed to Solomon's Cube .
Comments Off on Solomon’s Super* Cube…
Sunday, February 21, 2021
“Before time began, there was the Cube.”
— Hassenfeld Brothers merchandising slogan
Comments Off on Cube Woo
Saturday, September 19, 2020
Comments Off on Cube School
Sunday, July 5, 2020
Promotional material —
“Did you buckle up?” — Harlan Kane
The publication date of The Enigma Cube reported above was February 13, 2020.
Related material — Log24 posts around that date now tagged The Reality Bond.
Comments Off on The Enigma Cube
Friday, February 28, 2020
Or: The Newman Prize Continues.
Freeman Dyson reportedly died today. In memoriam ,
some remarks by Dyson from Hiroshima Day 1979 —
(Click to enlarge.)
Comments Off on Up the Trinity Staircase
Monday, February 24, 2020
See also Time Cube elsewhere in this journal.
Comments Off on For “Time Cube” Fans
Sunday, December 22, 2019
Exercise: Use the Guitart 7cycles below to relate the 56 triples
in an 8set (such as the eightfold cube) to the 56 triangles in
a wellknown Kleinquartic hyperbolicplane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M_{24}.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
Comments Off on M_{24} from the Eightfold Cube
Friday, June 21, 2019
See also "SixSet" in this journal
and "Cube Geometry Continues."
Comments Off on Cube Tales for Solstice Day
Comments Off on Cubehenge
Tuesday, May 21, 2019
Comments Off on Inside the White Cube
Monday, May 13, 2019
"Before time began . . . ." — Optimus Prime
Comments Off on Star Cube
Saturday, May 4, 2019
Comments Off on Inside the White Cube
Tuesday, March 5, 2019
For PSL(2,7), this is ((491)(497))/((71)(2))=168.
The group GL(3,2), also of order 168, acts naturally
on the set of seven cubeslicings below —
Another way to picture the seven natural slicings —
Application of the above images to picturing the
isomorphism of PSL(2,7) with GL(3,2) —
For a more detailed proof, see . . .
Comments Off on The Eightfold Cube and PSL(2,7)
Thursday, December 6, 2018
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Related material —
The 56 triangles of the eightfold cube . . .
Image from Christmas Day 2005.
Comments Off on The Mathieu Cube of Iain Aitchison
Sunday, September 30, 2018
Found today in an Internet image search, from the website of
an anonymous amateur mathematics enthusiast —
Forming Gray codes in the eightfold cube with the eight
I Ching trigrams (bagua ) —
This journal on Nov. 7, 2016 —
A different sort of cube, from the makers of the recent
Netflix miniseries "Maniac" —
See also Rubik in this journal.
Comments Off on Iconology of the Eightfold Cube
Tuesday, September 25, 2018
See some posts related to three names
associated with Trinity College, Cambridge —
Atiyah + Shaw + Eddington .
Comments Off on Trinity
Monday, July 23, 2018
Click to enlarge:
Above are the 7 frames of an animated gif from a Wikipedia article.
* For the Furey of the title, see a July 20 Quanta Magazine piece —
See also the eightfold cube in this journal.
"Before time began . . . ." — Optimus Prime
Comments Off on Eightfold Cube for Furey*
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
Thursday, June 28, 2018
Comments Off on Trinity Meditation
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, 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
Saturday, November 18, 2017
Comments Off on Cube Space Continued
Tuesday, August 8, 2017
Comments Off on Cube Quaternions
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
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
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
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
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
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
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
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
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
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
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
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*
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
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
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, August 6, 2017
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 …"
"Would that it were so simple."
Comments Off on Dark Tower Theology
Thursday, August 3, 2017
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 …
Comments Off on Poetic Theology at the New York Times
Friday, March 11, 2016
Thursday, January 8, 2015
"The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being…." — Wallace Stevens
See also Cube Trinity in this journal.
Comments Off on ABC Verlag, Zurich
Saturday, February 25, 2012
(Continued. See previous post and Red and Gray in this journal.)
“Give faith a fighting chance.” —Country song
From a post of June 3, 2007—
Related illustration relevant to theology—
For some background, see Cube Trinity in this journal.
For greater depth, see Levering’s Scripture and Metaphysics:
Aquinas and the Renewal of Trinitarian Theology ,
Blackwell, 2004, page 150.
Comments Off on The Rock
Sunday, August 28, 2011
Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”
A simpler candidate for the “Cube” part of that phrase:
The Eightfold Cube
As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.
“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”
— Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects“
Borovik has a such a diagram—
The planes in Borovik’s figure are those separating the parts of the eightfold cube above.
In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.
In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.
For some related theological remarks, see Cube Trinity in this journal.
Happy St. Augustine’s Day.
* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2element Galois field, these hyperplanes are certain sets of four subcubes.
Comments Off on The Cosmic Part
Monday, September 13, 2021
Comments Off on “What a Swell Party This Is” — Cole Porter
The date of the Rubber Ducky article in the previous post was . . .
November 11, 2019.
Synchronology check:
* A phrase by Woody Allen (NY Times , May 5, 2011).
Comments Off on Using the Dreidel*
Tuesday, August 31, 2021
A passage by Max Jammer quoted in yesterday's post
A Brief Introduction to Ideas suggests further remarks:
There are geometries in which lengths are not invariant
because they are not relevant — for instance, projective
geometry, finite geometry, and of course finite projective
geometry.
See the annus mirabilis introduction to that subject
cited by Jammer in yesterday's Brief Introduction —
Comments Off on Annals of Geometry
Sunday, August 29, 2021
Concepts of Space —
(From the March 2019 post Back to the Annus Mirabilis , 1905 )
Concepts of Space and Time —
Comments Off on “Before Time Began . . .” — Optimus Prime
Monday, August 9, 2021
"Two years ago . . . ." — Synopsis of the August 3 film "Hum"
Two years ago on August 3 . . .
What is going on in this picture?
The above is an image from
the August 3, 2019,
post "Butterfield's Eight."
"Within the week . . . ."
— The above synopsis of "Hum"
This suggests a review of a post
from August 5, 2019, that might
be retitled . . .
"The void she knows,
the tune she hums."
Comments Off on The Tune (Suggested by “Hum: Seek the Void”)
Wednesday, April 7, 2021
Drilling down . . .
My own, more abstract, academic interests are indicated by
a post from this journal on January 20, 2020 —
Dyadic Harmonic Analysis: The Fourfold Square and Eightfold Cube.
Those poetically inclined may regard that post as an instance of the
“intersection of the timeless with time.”
Comments Off on Timeless Capsules
Monday, March 29, 2021
Also on 18 November 2010 —
Comments Off on Graduate School
Sunday, March 21, 2021
“To conquer, three boxes* have to synchronize and join together into the Unity.”
―Wonder Woman in Zack Snyder’s Justice League
See also The Unity of Combinatorics and The Miracle Octad Generator.
* Cf. Aitchison’s Octads —
Comments Off on Mathematics and Narrative: The Unity
Continues from March 17.
See as well some remarks on Chinese perspective
in the Log24 post “Gate” of June 13, 2013.
Comments Off on Mind the Gaps…
Wednesday, March 17, 2021
Comments Off on Mind the Gaps
Friday, March 12, 2021
See Trinity Cube in this journal and . . .
McDonnell’s illustration is from 9 June 1983.
See as well a less official note from later that June.
Comments Off on Grid
Saturday, February 27, 2021
Clue
Here is a midrash on “desmic,” a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …
(The term “desmic,” as noted earlier, is relevant to the structure of
Heidegger’s Sternwürfel .)
“Gadzooks, I’ve done it again!” — Sherlock Hemlock
Comments Off on The Pencil Case
Thursday, February 25, 2021
“… What is your dream—your ideal? What is your News from Nowhere,
or, rather, What is the result of the little shake your hand has given to
the old pasteboard toy with a dozen bits of colored glass for contents?
And, most important of all, can you present it in a narrative or romance
which will enable me to pass an idle hour not disagreeably? How, for instance,
does it compare in this respect with other prophetic books on the shelf?”
— Hudson, W. H.. A Crystal Age (p. 2). Open Road Media. Kindle Edition.
See as well . . .
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
Comments Off on Compare and Contrast
Wednesday, February 24, 2021
“Twentyfour glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —
For those who prefer Borges to Bourne —
Comments Off on Annals of Dim Antiquity
Monday, February 15, 2021
In memory of a Dead Sea Scrolls scholar who
reportedly died on December 29, 2020, here are
links to two Log24 posts from that date:
I Ching Geometry and Raiders of the Lost Coordinates.
Comments Off on Raiders of the Lost Building Blocks
Tuesday, December 29, 2020
“Before time began, there was the Cube.”
— Hassenfeld Brothers cinematic merchandising slogan
Comments Off on I Ching Geometry
Sunday, December 27, 2020
“Knight move” remark from The Eiger Sanction —
“I like to put people on myself by skipping logical steps
in the conversation until they’re dizzy.”
The following logical step — a check of the date Nov. 18, 2017 —
was omitted in the post Futon Dream on this year’s St. Stephen’s Day.
For further context, see James Propp in this journal.
Comments Off on Knight Move for Trevanian
Friday, December 25, 2020
From old posts tagged Change Arises —
From Christmas 2005:
Click on image for details.
For the eightfold cube
as it relates to Klein’s
simple group, see
“A Reflection Group
of Order 168.”
For an rather more
complicated theory of
Klein’s simple group, see
Click on image for details. 
The phrase “change arises” is from ArkaniHamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian —
A related recent illustration from Quanta Magazine —
The above illustration of seven cells is not unrelated to
the eightfoldcube model of the seven projective points in
the Fano plane.
Comments Off on Change Arises: Mathematical Examples
Tuesday, December 15, 2020
Hurt’s dies natalis (date of death, in the saints’ sense) was,
it now seems, 25 January 2017, not 27.
A connection, for fantasy fans, between the Philosopher’s Stone
(represented by the eightfold cube) and the Deathly Hallows
(represented by the usual Fanoplane figure) —
Images from a Log24 search for “Holocron.”
Comments Off on Connection
Sunday, November 22, 2020
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 2329 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
The arXiv versions —
Comments Off on The GaloisFano Plane
Monday, November 16, 2020
Comments Off on The Garland Modelling
Wednesday, September 23, 2020
Various posts here on the geometry underlying the Mathieu group M_{24}
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
Comments Off on Geometry of Even Subsets
Monday, September 21, 2020
“On their way to obscurity, the Simulmatics people
played minor parts in major events, appearing Zeliglike
at crucial moments of 1960s history.”
— James Gleick reviewing a new book by Jill Lepore
Comments Off on ZeligLike?
Saturday, September 19, 2020
“Like Coleridge” . . .
Related material: Bloomsday 2006.
Comments Off on The Summerfield Prize
Thursday, September 17, 2020
Continues in The New York Times :
“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.
Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter
Another such object: the eightfold cube .
Comments Off on Structure and Mutability . . .
Thursday, September 10, 2020
Comments Off on Raiders of . . .
Wednesday, September 9, 2020
For a Jedi holocron of sorts, see this journal on the above YouTube date —
Comments Off on Portrait with Holocron
Tuesday, September 8, 2020
Metaphysical ruminations of Coleridge that might be applied to
the eightfold cube —
See also “Sprechen Sie Neutsch?“.
Comments Off on “The Eight” according to Coleridge
Thursday, July 9, 2020
For those who prefer fiction —
“Twentyfour glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— “Robert Ludlum’s” The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —
Comments Off on The Enigma Glyphs
Sunday, July 5, 2020
“He recounted the story of Adam and Eve, who were banished
from paradise because of their curiosity. Their inability to resist
the temptation of the forbidden fruit. Which itself was a metaphorical
standin for knowledge and power. He urged us to find the restraint
needed to resist the temptation of the cube—the biblical apple
in modern garb. He urged us to remain in Eden until we were able
to work out the knowledge the apple offered, all by ourselves.”
— Richards, Douglas E.. The Enigma Cube (Alien Artifact Book 1)
(pp. 160161). Paragon Press, 2020. Kindle Edition.
The biblical apple also appears in the game, and film, Assassin’s Creed .
Related material —
See the cartoon version of Alfred North Whitehead in the previous post,
and some Whiteheadrelated projective geometry —
Comments Off on It’s Still the Same Old Story …
The previous post reported, perhaps inaccurately, a publication
date of February 13, 2020, for the novel The Enigma Cube .
A variant publication date, Jan. 21, 2020, is reported below.
This journal on that date —
Comments Off on Enigma Variations
Saturday, May 23, 2020
The resemblance to the eightfold cube is, of course,
completely coincidental.
Some background from the literature —
Comments Off on Eightfold Geometry: A Surface Code “Unit Cell”
Friday, May 22, 2020
From a paper cited in the above story:
“Fig. 4 A lattice geometry for a surface code.” —
The above figure suggests a search for “surface code” cube :
Related poetic remarks — “Illumination of a surface.”
Comments Off on Surface Code News
Sunday, May 17, 2020
“Let me say this about that.” — Richard Nixon
Interpenetration in Weyl’s epistemology —
Interpenetration in Mazzola’s music theory —
Interpenetration in the eightfold cube — the three midplanes —
A deeper example of interpenetration:
Aitchison has shown that the Mathieu group M_{24} has a natural
action on the 24 center points of the subsquares on the eightfold
cube’s six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
Comments Off on “The Ultimate Epistemological Fact”
Friday, April 17, 2020
The above title was suggested by the previous post, Explosive Remarks.
Comments Off on A Mechanism of Fission
“Here is the background to Wheeler’s explosive remarks.
John Archibald Wheeler, director of the Center
for Theoretical Physics at the University of Texas,
is one of the world’s top theoretical physicists.
In 1939 he and Niels Bohr published a paper on
‘The Mechanism of Nuclear Fission”
that laid the groundwork for atomic and hydrogen bombs.
Wheeler later played major roles in their development.”
For a rather different explosion of Wheeler’s views, see the previous post.
Comments Off on Explosive Remarks
“In his big book, Gravity [sic ], Wheeler puts our space
into what he calls superspace, and speculates on the
most basic physical laws which operate on superspace.
He comes to the (to me) surprising conclusion that the
rockbottom laws are the laws of the propositional calculus!”
— Martin Gardner, letter to Donald E. Knuth, 8 January 1976,
on cover of Notices of the American Mathematical Society ,
March 2011 issue.
Fact check —
Related reading —
Comments Off on Introduction to Quantum Woo
Sunday, March 22, 2020
A brief summary of the eightfold cube is now at octad.us.
Comments Off on Eightfold Site
Tuesday, March 17, 2020
“Before time began…” — Optimus Prime
See also posts tagged Aitchison.
Comments Off on Geometric Theology
Thursday, March 5, 2020
See the title in this journal.
Such generation occurs both in Euclidean space …
… and in some Galois spaces —
.
In Galois spaces, some care must be taken in defining "reflection."
Comments Off on “Generated by Reflections”
Comments Off on Pythagorean Letter Meets Box of Chocolates
Wednesday, March 4, 2020
On Boston's Hancock Tower:
"I reflect that all art, all beauty, is reflection."
— Fictional character by John Updike (July 1976)
The architect of the tower reportedly died Monday.
See as well "Reflections: Disturbing the Universe I"
by the late Freeman Dyson in The New Yorker
issue dated August 6, 1979.
A reflection I prefer:
Comments Off on Architect’s Elegy
Sunday, March 1, 2020
Freeman Dyson on his staircase at Trinity College
(University of Cambridge) and on Ludwig Wittgenstein:
“I held him in the highest respect and was delighted
to find him living in a room above mine on the same
staircase. I frequently met him walking up or down
the stairs, but I was too shy to start a conversation.”
Frank Close on Ron Shaw:
“Shaw arrived there in 1949 and moved into room K9,
overlooking Jesus Lane. There is nothing particularly
special about this room other than the coincidence that
its previous occupant was Freeman Dyson.”
— Close, Frank. The Infinity Puzzle (p. 78).
Basic Books. Kindle Edition.
See also other posts now tagged Trinity Staircase.
Illuminati enthusiasts may enjoy the following image:
Comments Off on Same Staircase, Different Day
Saturday, February 29, 2020
See as well "Up the Trinity Staircase" (yesterday afternoon)
and "British Pottery" (Log24 , December 22, 2018).
Comments Off on Potter’s Staircase
Roberta Smith on Donald Judd’s
ARTnews Writings:
‘A Great Template for Criticism’
BY ALEX GREENBERGER
February 28, 2020 1:04pm
If Minimalist artist Donald Judd is known as a writer at all, it’s likely for one important text— his 1965 essay “Specific Objects,” in which he observed the rise of a new kind of art that collapsed divisions between painting, sculpture, and other mediums. But Judd was a prolific critic, penning shrewd reviews for various publications throughout his career—including ARTnews . With a Judd retrospective going on view this Sunday at the Museum of Modern Art in New York, ARTnews asked New York Times cochief art critic Roberta Smith— who, early in her career, worked for Judd as his assistant— to comment on a few of Judd’s ARTnews reviews. How would she describe his critical style? “In a word,” she said, “great.” . . . .

And then there is Temple Eight, or Ex Fano Apollinis —
Cicero, In Verrem II. 1. 46 —
He reached Delos. There one night he secretly 46
carried off, from the muchrevered sanctuary of
Apollo, several ancient and beautiful statues, and
had them put on board his own transport. Next
day, when the inhabitants of Delos saw their sanc
tuary stripped of its treasures, they were much
distressed . . . .
Delum venit. Ibi ex fano Apollinis religiosissimo
noctu clam sustulit signa pulcherrima atque anti
quissima, eaque in onerariam navem suam conicienda
curavit. Postridie cum fanum spoliatum viderent ii
qui Delum incolebant, graviter ferebant . . . .
Comments Off on Template
Thursday, February 27, 2020
From the author who in 2001 described "God's fingerprint"
(see the previous post) —
From the same publisher —
From other posts tagged Triskele in this journal —
Other geometry for enthusiasts of the esoteric —
Comments Off on Occult Writings
Sunday, February 23, 2020
"Although art is fundamentally everywhere and always the same,
nevertheless two main human inclinations, diametrically opposed
to each other, appear in its many and varied expressions. ….
The first aims at representing reality objectively, the second subjectively."
— Mondrian, 1936 [Links added.]
An image search today (click to enlarge) —
Comments Off on The Representation of Reality
Friday, February 21, 2020
Also on January 27, 2017 . . .
For other appearances of John Hurt here,
see 1984 Cubes.
Update of 12:45 AM Feb. 22 —
A check of later obituaries reveals that Hurt may well
have died on January 25, 2017, not January 27 as above.
Thus the following remarks may be more appropriate:
Not to mention what, why, who, and how.
Comments Off on To and Fro, Back and …
Wednesday, February 19, 2020
The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.
A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :
The Curtis octads are related to symmetries of the square.
See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —
Aitchison's octads are instead related to symmetries of the cube.
Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .
Image from Christmas Day 2005.
* http://www.math.sci.hiroshimau.ac.jp/branched/files/2018/
presentations/AitchisonHiroshima22018.pdf.
See also Aitchison in this journal.
Comments Off on Aitchison’s Octads
Wednesday, February 12, 2020
The plane at left is modeled naturally by
seven types of “cuts” in the cube at right.
Comments Off on The Reality Bond
Sunday, January 26, 2020
Comments Off on HarmonicAnalysis Building Blocks
Monday, January 20, 2020
The Fourfold Square and Eightfold Cube
Related material: A Google image search for “field dream” + log24.
Comments Off on Dyadic Harmonic Analysis:
Thursday, January 2, 2020
Comments Off on Interality
Saturday, December 14, 2019
(Continued)
The above image is from
"A FourColor Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.
These partitions of an 8set into four 2sets
occur also in Wednesday night's post
Miracle Octad Generator Structure.
This post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers" —
Comments Off on Colorful Tale
Monday, November 11, 2019
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
An Invariance of Symmetry
Comments Off on Time and Chance
Monday, October 7, 2019
Stevens's Omega and Alpha (see previous post) suggest a review.
Omega — The Berlekamp Garden. See Misère Play (April 8, 2019).
Alpha — The Kinder Garten. See Eighfold Cube.
Illustrations —
The sculpture above illustrates Klein's order168 simple group.
So does the sculpture below.
Cube Bricks 1984 —
Comments Off on Berlekamp Garden vs. Kinder Garten
Sunday, September 29, 2019
"The 15 Puzzle and the Magic Cube
are spiritual kin …."
— "Metamagical Themas" column,
Douglas R. Hofstadter, Scientific American ,
Vol. 244, No. 3 (March 1981), pp. 2039
As are the 15 Schoolgirls and the Eightfold Cube.
Comments Off on Spiritual Kin
The previous post dealt with “magic” cubes, so called because of the
analogous “magic” squares. Douglas Hofstadter has written about a
different, physical , object, promoted as “the Magic Cube,” that Hofstadter
felt embodied “a deep invariant”:
Comments Off on Stage Direction: “Comments Off.”
Monday, September 9, 2019
See as well an obituary for Mrs. Wertham from 1987.
Related art —
For further details, search the Web for "Wertham Professor" + Eck.
Comments Off on ART WARS at Harvard: The Wertham Professorship
Monday, July 22, 2019
Lord Peter Wimsey (Balliol 1912) on the BalliolTrinity rivalry at Oxford:
See also Balliol College in the post subtitled Spidey Goes to Church.
Comments Off on The Four Diamonds Meet the Five Red Herrings
Comments Off on Signed, Sealed, Delivered
Good question. See also this journal on the above date —
September 15, 2018.
Comments Off on Visual Languages
Click the image for some remarks on a related novel.
Comments Off on Space, Time, Form
Wednesday, July 10, 2019
… 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 . . .
Comments Off on Artifice* of Eternity …
Tuesday, July 9, 2019
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From John Baez (2018) —
See also this morning's post Perception of Space
and yesterday's Exploring Schoolgirl Space.
Comments Off on Schoolgirl Space: 1984 Revisited
Older Posts »