From the current New Yorker,
a link for Katherine Neville —
Saturday, January 24, 2026
Eightfold Cube AI Overview
Comments Off on Eightfold Cube AI Overview
Sunday, September 4, 2022
Dice and the Eightfold Cube
At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, as
in the Gell-Mann picture above, but rather at the ends of one of
a cube's four diagonal axes of symmetry.
See some related illustrations below.
Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.
The exceptional role of 0 and ∞ in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —
Transposition of 0 and ∞ in the knight coordinatization
induces the symplectic polarity of PG(3,2) discussed by
(for instance) Anne Duncan in 1968.
Comments Off on Dice and the Eightfold Cube
Sunday, December 22, 2019
M24 from the Eightfold Cube
Exercise: Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
Comments Off on M24 from the Eightfold Cube
Tuesday, March 5, 2019
The Eightfold Cube and PSL(2,7)
For PSL(2,7), this is ((49-1)(49-7))/((7-1)(2))=168.
The group GL(3,2), also of order 168, acts naturally
on the set of seven cube-slicings 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) —
.gif)
For a more detailed proof, see . . .
Comments Off on The Eightfold Cube and PSL(2,7)
Sunday, September 30, 2018
Iconology of the Eightfold Cube
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
Monday, July 23, 2018
Eightfold Cube for Furey*
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
Triangles in the Eightfold Cube
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 eight-triangle 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 eight-triangle 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
Friday, January 6, 2017
Eightfold Cube at Cornell
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
The Eightfold Cube in Oslo
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
Thursday, March 17, 2016
On the Eightfold Cube
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, October 9, 2015
Eightfold Cube in Oslo
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
(2014-2015). 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 "glam-minimalist."
She has in many of his works returned to basic
geometric shapes, with hints to the occult,
"new space-age", 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
Monday, April 9, 2012
Eightfold Cube Revisited
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. 217-250
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
Monday, December 29, 2025
Octad Art — Bricks, Cubes, Flowers
For the bricks of the title, see other posts tagged Brick Space.
For some cubes* and flowers, see below.
Combining features of the above two images, one might picture the 24
cells of the 4×6 array underlying the Curtis Miracle Octad Generator
(MOG) as each containing an eightfold cube, pictured as above with seven
of its subcubes showing and an eighth subcube hidden behind them.
The seven visible subcubes may be colored, as in the Curtis image of
the Klein map, with seven distinct colors… corresponding to the seven
edge-colors used in the Curtis-Klein map. Each of the seven visible
subcubes in a cell may also be labeled, on its visible faces, with a symbol
denoting one of the 24 points of the projective line over GF(23), just as the
faces in the Curtis-Klein map are labeled. The hidden subcube in each cell
may be regarded as also so labeled, by the MOG label of the cell's position.
There is then enough information in the array's eightfold cubes' colors and
labels to construct the seven generating permutations of M24 described by
Curtis, and the 24 array cells may be regarded as now containing 24 distinct
entities — which perhaps might be called "octoids."
Those desiring a more decorative approach may replace the 24 labeled cubes
with 24 labeled "flowers." Each flower — like the map's symmetric seven
"petals" and the central "infinity heptagon" they surround — forms an octad.
Related Illustrations . . .
* See as well posts tagged Mathieu Cube . . .
Related material —
The 56 triangles of the eightfold cube . . .
- in Aitchison's March 9, 2018, talk (slides 32-34), and
- in this journal on July 25, 2008, and later.
Image from Christmas Day 2005.
Post last revised: December 30, 2025 @ 21:30 E.S.T.
Comments Off on Octad Art — Bricks, Cubes, Flowers
Wednesday, August 6, 2025
Cubes
From a post on the Feast of St. Nicholas, 2018,
"The Mathieu Cube of Iain Aitchison" —
Compare and contrast . . .
The Supercube of Solomon Golomb.
Comments Off on Cubes
Thursday, April 24, 2025
Uniting the Three Cubes
Note that the number 8, a cube, may be represented as
either a literal "eightfold cube" — a 2x2x2 array — or as,
in the manner of R. T. Curtis, a 4-row 2-column "brick."
Related art . . .
Some will prefer a more dramatic approach to uniting three cubes . . .
Comments Off on Uniting the Three Cubes
Monday, December 16, 2024
Whitecaps for Beethoven
"Each of the 64 subcubes is supposed to be marked identically,
with white caps on two opposite vertices and a black band
around the subcube that separates the two white caps."
Related whitecap reading . . .
Comments Off on Whitecaps for Beethoven
Thursday, June 13, 2024
Of London Bondage … continues.
From a bondage search . . .
“Loitering in Lara’s dressing room, she tries on
the faux-bondage harness she picked up in London….”
From Geometry for Belgium —
Comments Off on Of London Bondage … continues.
Tuesday, May 21, 2024
Geometry for Belgium
Other matching patterns . . .
Tuesday Weld in the 1972 film of Didion's Play It As It Lays :
Note the making of a matching pattern.
Comments Off on Geometry for Belgium
Saturday, May 4, 2024
What Lies Between
"I perceived . . . cinema is that which is between things,
not things [themselves] but between one and another."
— Jean-Luc Godard, "Introduction à une véritable histoire
du cinéma," Albatros , Paris, 1980, p. 145
|
Log24 on 10 Dec. 2008 —
|
Log24 on 12 Dec. 2008 —
|
Between the two image-dates above . . .
" 'The jury is still out on how long – and whether – people are actually
going to understand this.' It took the world 150 years to realize
the true power of the printing press . . . ." — Cade Metz
Comments Off on What Lies Between
Thursday, March 21, 2024
The Cross Section
Addendum for Christopher Nolan — Dice and the Eightfold Cube.
Comments Off on The Cross Section
Tuesday, February 20, 2024
Backlight
The epigraph of the previous post —
"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig
Related reading and art for academic nihilists — See . . .
Reading and art I prefer —
Love in the Ruins , by Walker Percy, and . . .
Van Gogh (by Ed Arno) and an image and
a passage from The Paradise of Childhood
(by Edward Wiebé):
Comments Off on Backlight
Tuesday, October 24, 2023
A Bond with Reality: The Geometry of Cuts
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation of the above
seven partitions of the eightfold cube:
|
|
Related mathematics:
|
The use of binary coordinate systems
Natural physical transformations of square or cubical arrays See "The Thing and I." |
and . . .
Related entertainment:
Or Matt Helm by way of a Jedi cube.
Comments Off on A Bond with Reality: The Geometry of Cuts
Saturday, July 1, 2023
Mechanical Plaything (Hinged) vs. Conceptual Art (Unhinged)
"Infinity Cube" … hinged plaything, for sale —
"Eightfold Cube" … un hinged concept, not for sale—
See as well yesterday's Trickster Fuge ,
and a 1906 discussion of the eightfold cube:
Comments Off on Mechanical Plaything (Hinged) vs. Conceptual Art (Unhinged)
Wednesday, June 21, 2023
Annals of Magical Thinking: The Rushmore Embedding
Comments Off on Annals of Magical Thinking: The Rushmore Embedding
Saturday, June 10, 2023
Green, Orange, Black
The colors surrounding Watson's body in the above
"bandeau" photo suggest a review. A search in this journal
for Green+Orange+Black yields . . .
In the above image, the "hard core of objectivity" is represented
by the green-and-white eightfold cube. The orange and black are,
of course, the Princeton colors.
Comments Off on Green, Orange, Black
Friday, June 2, 2023
Reichenbach’s Fell Swoop
See The Eightfold Cube and . . .
Art is magic delivered from
the lie of being truth.
— Theodor Adorno, Minima moralia,
London, New Left Books, 1974, p. 222
(First published in German in 1951.)
The director, Carol Reed, makes…
impeccable use of the beauty of black….
— V. B. Daniel on The Third Man
I see your ironical smile.
— Hans Reichenbach
Adorno, The Third Man, and Reichenbach
are illustrated below (l. to r.) above the names of
cities with which they are associated.
Comments Off on Reichenbach’s Fell Swoop
Friday, January 27, 2023
The Stone
Here stands the mean, uncomely stone,
’Tis very cheap in price!
The more it is despised by fools,
The more loved by the wise.
— https://jungcurrents.com/
the-story-of-the-stone-at-bollingen
Not so cheap:
Identical copies of the above image are being offered for sale
on three websites as representing a Masonic "cubic stone."
None of the three sites say where, exactly, the image originated.
Image searches for "Masonic stone," "Masonic cube," etc.,
fail to yield any other pictures that look like the above image —
that of a 2x2x2 array of eight identical subcubes.
For purely mathematical — not Masonic — properties of such
an array, see "eightfold cube" in this journal.
The websites offering to sell the questionable image —
|
Getty —
|
|
Alamy —
https://www.alamy.com/
|
|
Photo12 —
https://www.photo12.com/en/image/
No price quoted on public page:
|
Comments Off on The Stone
Saturday, January 14, 2023
Châtelet on Weil — A “Space of Gestures”
|
From Gilles Châtelet, Introduction to Figuring Space Metaphysics does have a catalytic effect, which has been described in a very beautiful text by the mathematician André Weil: Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another.4 André Weil cuts to the quick here: he conjures these 'murky reflections', these 'furtive caresses', the 'theory of Galois that Lagrange touches … with his finger through a screen that he does not manage to pierce.' He is a connoisseur of these metaphysical 'fogs' whose dissipation at one point heralds their reforming at another. It would be better to talk here of a horizon that tilts thereby revealing a new space of gestures which has not as yet been elucidated and cut out as structure. 4 A. Weil, 'De la métaphysique aux mathématiques', (Oeuvres, vol. II, p. 408.) |
For gestures as fogs, see the oeuvre of Guerino Mazzola.
For some clearer remarks, see . . .
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation
of the above seven partitions
of the eightfold cube:
|
|
Related material: Galois.space .
Comments Off on Châtelet on Weil — A “Space of Gestures”
Monday, December 26, 2022
Super-8 Box
Comments Off on Super-8 Box
Friday, December 23, 2022
“Was ist Raum?” — Bauhaus Founder Walter Gropius
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
The Theory and
Organization of the
Bauhaus (1923)
A relevant illustration:
At math.stackexchange.com on March 1-12, 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).
These references will not appeal to those who enjoy modernism as a religion.
(For such a view, see Rosalind Krauss on grids and another writer's remarks
on the religion's 100th anniversary this year.)
Some related nihilist philosophy from Cormac McCarthy —
"Forms turning in a nameless void."
Comments Off on “Was ist Raum?” — Bauhaus Founder Walter Gropius
Saturday, November 12, 2022
Inside a White Cube
For the late Brian O'Doherty, from posts now tagged "Pless Birthday 2022" —
This post was suggested by an obituary of O'Doherty and by
"The Life and Work of Vera Stepen Pless" in
Notices of the American Mathematical Society , December 2022.
Comments Off on Inside a White Cube
Tuesday, November 1, 2022
From “Goethe on All Souls’ Day”
The above title is that of a Log24 post on St. Cecilia's Day in 2017
that quoted some earlier All Souls' Day remarks from Berlin.
From that post —
Exercise: Explain why the lead article in the November issue of
Notices of the American Mathematical Society misquotes Weyl
and gives the misleading impression that the example above,
the eightfold cube , might be part of the mathematical pursuit
known as geometric group theory .
Background: Earlier instances here of the phrase "geometric group theory."
Comments Off on From “Goethe on All Souls’ Day”
Saturday, September 3, 2022
1984 Revisited
Note the three quadruplets of parallel edges in the 1984 figure above.
The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —
Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.
Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —
And then, more simply, there is the Galois tesseract —
For parts of my own world in June 2010, see this journal for that month.
The above Galois tesseract appears there as follows:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
Comments Off on 1984 Revisited
Monday, August 1, 2022
Enowning
Related material — The Eightfold Cube.
See also . . .
"… 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."
— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010
Comments Off on Enowning
Tuesday, April 12, 2022
Dealing with Cubism
"It’s important, as art historian Reinhard Spieler has noted,
that after a brief, unproductive stay in Paris, circa 1907,
Kandinsky chose to paint in Munich. That’s where he formed
the Expressionist art group Der Blaue Reiter (The Blue Rider) —
and where he avoided having to deal with cubism."
— David Carrier,
Images from an earlier Christmas Day, in 2005 —
Comments Off on Dealing with Cubism
Saturday, February 12, 2022
Das Geheimnis der Einheit
Thomas Mann on "the mystery of the unity" —
"Denn um zu wiederholen, was ich anfangs sagte:
in dem Geheimnis der Einheit von Ich und Welt,
Sein und Geschehen, in der Durchschauung des
scheinbar Objectiven und Akzidentellen als
Veranstaltung der Seele glaube ich den innersten Kern
der analytischen Lehre zu erkennen." (GW IX 488)
An Einheit-Geheimnis that is perhaps* more closely related
to pure mathematics** —
"What is the nature of the original unity
that throws itself apart in this separation,
and in what sense are the separated ones
here as the essence of the abyss?
Here it cannot be a question of any kind of 'dialectic,'
but only of the essence of the ground
(that is, of truth) itself." [Tr. by Google]
" Welcher Art ist die ursprüngliche Einheit,
daß sie sich in diese Scheidung auseinanderwirft,
und in welchem Sinn sind die Geschiedenen
hier als Wesung der Ab-gründigkeit gerade einig?
Hier kann es sich nicht um irgend eine »Dialektik«
handeln, sondern nur um die Wesung des Grundes
(der Wahrheit also) selbst."
* Or perhaps not .
** For a relevant Scheidung , see Eightfold Cube.
Comments Off on Das Geheimnis der Einheit
Friday, February 11, 2022
For Space Groupies
A followup to Wednesday's post Deep Space —
Related material from this journal on July 9, 2019 —
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From other posts tagged 1984 Cubes —
Comments Off on For Space Groupies
Saturday, February 5, 2022
Mathieu Cube Labeling
Shown below is an illustration from "The Puzzle Layout Problem" —
- September 2003
-
Lecture Notes in Computer Science 2912:500-501
DOI:10.1007/978-3-540-24595-7_50 - Source: DBLP
-
Conference: Graph Drawing, 11th International Symposium,
GD 2003, Perugia, Italy, September 21-24, 2003, Revised Papers - Authors: Kozo Sugiyama, Seok-Hee Hong, Atsuhiko Maeda
Exercise: Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points
∞, 0, 1, 2, 3 … 22 of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on
cube-part permutations by the Mathieu group M24.
A note for Northrop Frye —
Interpenetration in the eightfold cube — the three midplanes —
A deeper example of interpenetration:
Aitchison has shown that the Mathieu group M24 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.
* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.
Comments Off on Mathieu Cube Labeling
Friday, December 31, 2021
Aesthetics in Academia
Related art — The non-Rubik 3x3x3 cube —
The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.
I prefer the radiance (in the sense of Aquinas) of the central, structuring
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).
Comments Off on Aesthetics in Academia
Thursday, December 30, 2021
Antidote to Chaos?
Some formal symmetry —
|
"… each 2×4 "brick" in the 1974 Miracle Octad Generator
Folding a 2×4 Curtis array yet again yields — Steven H. Cullinane on April 19, 2016 — The Folding. |
Related art-historical remarks:
The Shape of Time (Kubler, Yale U.P., 1962).
See yesterday's post The Thing .
Comments Off on Antidote to Chaos?
Tuesday, October 5, 2021
Wednesday, September 15, 2021
The Razor and the Touchstone
It is often good to remember that writers of headlines (and subheadlines)
are usually not the same people as the authors of the following texts.
In particular, in the above example, neither the word "touchstone" nor
the use of "enquires" to mean "enquiries" appears in the text proper.
Still, the mixed metaphor of "razor" as "touchstone" is not without interest.
See The Eightfold Cube and Modernist Cuts.
Comments Off on The Razor and the Touchstone
Monday, August 9, 2021
The Tune (Suggested by “Hum: Seek the Void”)
"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
Timeless Capsules
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
Friday, March 12, 2021
Grid
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
Friday, December 25, 2020
Change Arises: Mathematical Examples
From old posts tagged Change Arises —
|
From Christmas 2005:
For the eightfold cube
For an rather more
Click on image for details. |
The phrase "change arises" is from Arkani-Hamed 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 eightfold-cube model of the seven projective points in
the Fano plane.
Comments Off on Change Arises: Mathematical Examples
Tuesday, December 15, 2020
Connection
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 Fano-plane figure) —
Images from a Log24 search for “Holocron.”
Comments Off on Connection
Sunday, November 22, 2020
The Galois-Fano Plane
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. 23-29 (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.
Comments Off on The Galois-Fano Plane
Thursday, September 17, 2020
Structure and Mutability . . .
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 . . .
Tuesday, September 8, 2020
“The Eight” according to Coleridge
Metaphysical ruminations of Coleridge that might be applied to
the eightfold cube —
See also "Sprechen Sie Neutsch?".
Update of December 29, 2022 —
Comments Off on “The Eight” according to Coleridge
Saturday, May 23, 2020
Eightfold Geometry: A Surface Code “Unit Cell”
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”
Sunday, May 17, 2020
“The Ultimate Epistemological Fact”
"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 M24 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”
Sunday, March 22, 2020
Eightfold Site
A brief summary of the eightfold cube is now at octad.us.
Comments Off on Eightfold Site
Thursday, March 5, 2020
“Generated by Reflections”
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”
Sunday, March 1, 2020
Same Staircase, Different Day
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
Template
|
Roberta Smith on Donald Judd’s 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 co-chief 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 much-revered 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
Occult Writings
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 —
|
Monday, November 4, 2019
As Above, So Below*
|
|
|
Comments Off on Occult Writings
Sunday, February 23, 2020
The Representation of Reality
"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
Wednesday, February 19, 2020
Aitchison’s Octads
The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.
A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :
The Curtis octads are related to symmetries of the square.
See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —
Aitchison's octads are instead related to symmetries of the cube.
Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .
Image from Christmas Day 2005.
* http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/
presentations/Aitchison-Hiroshima-2-2018.pdf.
See also Aitchison in this journal.
[addtoany]
Comments Off on Aitchison’s Octads
Wednesday, February 12, 2020
The Reality Bond
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
Harmonic-Analysis Building Blocks
Comments Off on Harmonic-Analysis Building Blocks
Monday, January 20, 2020
Dyadic Harmonic Analysis:
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
Saturday, December 14, 2019
Colorful Tale
The above image is from
"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.
These partitions of an 8-set into four 2-sets
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, October 7, 2019
Berlekamp Garden vs. Kinder Garten
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 order-168 simple group.
So does the sculpture below.
Cube Bricks 1984 —
Comments Off on Berlekamp Garden vs. Kinder Garten
Sunday, September 29, 2019
Spiritual Kin
"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. 20-39
As are the 15 Schoolgirls and the Eightfold Cube.
Comments Off on Spiritual Kin
Tuesday, July 9, 2019
Schoolgirl Space: 1984 Revisited
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
Thursday, June 20, 2019
Sunday, May 26, 2019
Sunday, May 19, 2019
The Building Blocks of Geometry
From "On the life and scientific work of Gino Fano"
by Alberto Collino, Alberto Conte, and Alessandro Verra,
ICCM Notices , July 2014, Vol. 2 No. 1, pp. 43-57 —
|
" Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view: ' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' " 13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010). |
Restoring the severed link —
See also Espacement and The Thing and I.
Related material —
[addtoany]
Comments Off on The Building Blocks of Geometry
Monday, May 6, 2019
One Stuff
Comments Off on One Stuff
Saturday, May 4, 2019
Inside the White Cube
See also Espacement and The Thing and I.
Comments Off on Inside the White Cube
Monday, March 25, 2019
Espacement
(Continued from the previous post.)
|
In-Between "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' 1), writing (of which 'spacing' is said to be 'the fundamental property' 2) and deconstruction (with one of Derrida's last major texts, Le Toucher: Jean-Luc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' 3)." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, Jean-Luc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,16 we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207-227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'in-between' more 'originary' than the play of differences it in-forms. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a pre-ontological 'opening' – or, shall we say, 'in-between.' |
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
Comments Off on Espacement
Saturday, March 16, 2019
Thursday, February 21, 2019
A Tale of Eight Building Blocks*
* For another such tale, see Eightfold Cube in this journal.
Comments Off on A Tale of Eight Building Blocks*
Thursday, December 6, 2018
The Mathieu Cube of Iain Aitchison
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Search Results
|
Related material —
The 56 triangles of the eightfold cube . . .
- in Aitchison's March 9, 2018, talk (slides 32-34), and
- in this journal on July 25, 2008, and later.
Image from Christmas Day 2005.
Comments Off on The Mathieu Cube of Iain Aitchison
Wednesday, November 28, 2018
Geometry and Experience
Einstein, "Geometry and Experience," lecture before the
Prussian Academy of Sciences, January 27, 1921–
|
… This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. The very word geometry, which, of course, means earth-measuring, proves this. For earth-measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. …. |
Later in the same lecture, Einstein discusses "the theory of a finite
universe." Of course he is not using "finite" in the sense of the field
of mathematics known as "finite geometry " — geometry with only finitely
many points.
Nevertheless, his remarks seem relevant to the Fano plane , an
axiomatically defined entity from finite geometry, and the eightfold cube ,
a physical object embodying the properties of the Fano plane.
|
I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed. First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe. |
Comments Off on Geometry and Experience
Thursday, November 8, 2018
Reality vs. Axiomatic Thinking
From https://blogs.scientificamerican.com/…
|
A Few of My Favorite Spaces:
The intuition-challenging Fano plane may be By Evelyn Lamb on October 24, 2015
"…finite projective planes seem like |
For Fano's axiomatic approach, see the Nov. 3 Log24 post
"Foundations of Geometry."
For the Fano plane's basis in reality , see the eightfold cube
at finitegeometry.org/sc/ and in this journal.
See as well "Two Views of Finite Space" (in this journal on the date
of Lamb's remarks — Oct. 24, 2015).
Some context: Gödel's Platonic realism vs. Hilbert's axiomatics
in remarks by Manuel Alfonseca on June 7, 2018. (See too remarks
in this journal on that date, in posts tagged "Road to Hell.")
Comments Off on Reality vs. Axiomatic Thinking
Saturday, September 15, 2018
Eidetic Reduction in Geometry
|
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by Gian-Carlo Rota Some results from a Google search — Eidetic reduction | philosophy | Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. |
For example —
The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
Comments Off on Eidetic Reduction in Geometry
Friday, August 31, 2018
Perception of Number
Review of yesterday's post Perception of Space —
From Harry Potter and the Philosopher's Stone (1997),
republished as "… and the Sorcerer's Stone ," Kindle edition:
In a print edition from Bloomsbury (2004), and perhaps in the
earliest editions, the above word "movements" is the first word
on page 168:
Click the above ellipse for some Log24 posts on the eightfold cube,
the source of the 168 automorphisms ("movements") of the Fano plane.
"Refined interpretation requires that you know that
someone once said the offspring of reality and illusion
is only a staggering confusion."
— Poem, "The Game of Roles," by Mary Jo Bang
Related material on reality and illusion —
an ad on the back cover of the current New Yorker —
"Hey, the stars might lie, but the numbers never do." — Song lyric
Comments Off on Perception of Number
Thursday, August 30, 2018
Perception* of Space
* A footnote in memory of a dancer who reportedly died
yesterday, August 29 — See posts tagged Paradigm Shift.
"Birthday, death-day — what day is not both?" — John Updike
Comments Off on Perception* of Space
Saturday, August 25, 2018
“Waugh, Orwell. Orwell, Waugh.”
Suggested by a review of Curl on Modernism —
Related material —
Waugh + Orwell in this journal and …
Comments Off on “Waugh, Orwell. Orwell, Waugh.”
Sunday, July 29, 2018
The Materialization
McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is being, after all,
beyond the arbitrary flux of existence."
Neither does the above materialization of 281 as the page
number of her philosophical remark.
See also the materialization of 281 as a page number in
the book Witchcraft by Charles Williams —
The materialization of 168 as a page number in a
Stephen King novel is somewhat more convincing,
but less convincing than the materialization of Klein's
simple group of of 168 elements in the eightfold cube.
Comments Off on The Materialization
Sunday, July 22, 2018
Saturday, July 21, 2018
Building-Block Theory
(A sequel to yesterday’s Geometry for Jews)
From this journal on the above UCI posting date — April 6, 2018 —
From this journal on the above lecture date — April 26, 2018 —
illustrations in a post titled Defining Form —
For the relevance of the above material to building blocks,
see Eightfold Cube in this journal.
Comments Off on Building-Block Theory
Sunday, July 1, 2018
Deutsche Ordnung
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film "Triple Cross."
Related structures —
Greg Egan's animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
|
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to 'being simple': consider Eliot's Four Quartets or Mozart's late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it's often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we've seen into the heart of things. I'll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
Comments Off on Deutsche Ordnung
Sunday, June 10, 2018
Number Concept
The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.
Propp's remarks included the following:
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
Comments Off on Number Concept
Wednesday, June 6, 2018
Geometry for Goyim
Mystery box merchandise from the 2011 J. J. Abrams film Super 8 —
A mystery box that I prefer —
Click image for some background.
See also Nicht Spielerei .
Comments Off on Geometry for Goyim
Monday, June 4, 2018
The Trinity Stone Defined
“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
Sunday, April 1, 2018
Thursday, March 29, 2018
To Imagine (or, Better, to Construct)
The title reverses a phrase of Fano —
“costruire (o, dirò meglio immaginare).”
Illustrations of imagining (the Fano plane) and of constructing (the eightfold cube) —
Comments Off on To Imagine (or, Better, to Construct)
Tuesday, March 27, 2018
Compare and Contrast
Related material on automorphism groups —
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.
Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure above
than they are an eight-cube mechanical puzzle like the one below.
Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."
Comments Off on Compare and Contrast
Wednesday, March 7, 2018
Unite the Seven.
Related material —
The seven points of the Fano plane within
|
|
"Before time began . . . ."
— Optimus Prime
Comments Off on Unite the Seven.
Saturday, January 6, 2018
Report from Red Mountain
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 .
Comments Off on Report from Red Mountain
Friday, January 5, 2018
Seven Types of Interality*
* 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, 93-103.
Comments Off on Seven Types of Interality*
Wednesday, November 22, 2017
Goethe on All Souls’ Day
David E. Wellbery on Goethe
From an interview published on 2 November 2017 at
http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/
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 .
Comments Off on Goethe on All Souls’ Day
Saturday, November 18, 2017
Cube Space Continued
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).
Comments Off on Cube Space Continued
Sunday, October 29, 2017
File System… Unlocked
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.
Comments Off on File System… Unlocked
Thursday, October 19, 2017
Graphic Design: Fast Forward
Typographical: »
Eightfold Cube:
Comments Off on Graphic Design: Fast Forward
Saturday, October 7, 2017
Wednesday, September 13, 2017
Summer of 1984
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
Comments Off on Summer of 1984
Tuesday, August 8, 2017
Saturday, July 29, 2017
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 2015-16:
- Memorandum Ergo (October 29, 2015)
- Great Circle of Mysteries (November 15, 2015)
- Quotations and Ideas (April 15, 2016)
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.
Comments Off on MSRI Program
Tuesday, June 20, 2017
Epic
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
Comments Off on Epic
Tuesday, May 2, 2017
Image Albums
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
Comments Off on Image Albums
Wednesday, April 12, 2017
Contracting the Spielraum
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 1-12, 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).
Comments Off on Contracting the Spielraum
Thursday, March 9, 2017
One Eighth
From Wikipedia's Iceberg Theory —
Related material:
The Eightfold Cube and The Quantum Identity —
See also the previous post.
Comments Off on One Eighth
Monday, January 9, 2017
Analogical Extension at Cornell
Click to enlarge the following (from Cornell U. Press in 1962) —
For a more recent analogical extension at Cornell, see the
Epiphany 2017 post on the eightfold cube and yesterday
evening's post "A Theory of Everything."
Comments Off on Analogical Extension at Cornell
Sunday, January 8, 2017
A Theory of Everything
The title refers to the Chinese book the I Ching ,
the Classic of Changes .
The 64 hexagrams of the I Ching may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.
A related post —
Comments Off on A Theory of Everything
Saturday, January 7, 2017
Conceptualist Minimalism
"Clearly, there is a spirit of openhandedness in post-conceptual art
uses of the term 'Conceptualism.' We can now endow it with a
capital letter because it has grown in scale from its initial designation
of an avant-garde grouping, or various groups in various places, and
has evolved in two further phases. It became something like a movement,
on par with and evolving at the same time as Minimalism. Thus the sense
it has in a book such as Tony Godfrey’s Conceptual Art. … Beyond that,
it has in recent years spread to become a tendency, a resonance within
art practice that is nearly ubiquitous." — Terry Smith, 2011
See also the eightfold cube —
Comments Off on Conceptualist Minimalism
Sunday, November 27, 2016
A Machine That Will Fit
Or: Notes for the Metaphysical Club
Northrop Frye on Wallace Stevens:
"He… stands in contrast to the the dualistic
approach of Eliot, who so often speaks of poetry
as though it were an emotional and sensational
soul looking for a 'correlative' skeleton of
thought to be provided by a philosopher, a
Cartesian ghost trying to find a machine that
will fit."
Ralph Waldo Emerson on "vacant and vain" knowledge:
"The new position of the advancing man has all
the powers of the old, yet has them all new. It
carries in its bosom all the energies of the past,
yet is itself an exhalation of the morning. I cast
away in this new moment all my once hoarded
knowledge, as vacant and vain."
Harold Bloom on Emerson:
"Emerson may not have invented the American
Sublime, yet he took eternal possession of it."
Wallace Stevens on the American Sublime:
"And the sublime comes down
To the spirit itself,
The spirit and space,
The empty spirit
In vacant space."
A founding member of the Metaphysical Club:
See also the eightfold cube.
Comments Off on A Machine That Will Fit
Thursday, November 3, 2016
Triple Cross
(Continued … See the title in this journal, as well as Cube Bricks.)
Cube Bricks 1984 —
Related material —
Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge and Nanavira Thera.
See as well Bill Murray's 1984 film "The Razor's Edge" …
Movie poster from 1984 —
"A thin line separates
love from hate,
success from failure,
life from death."
Three other dualities, from Nanavira Thera in 1959 —
"I find that there are, in every situation,
three independent dualities…."
(Click to enlarge.)
Comments Off on Triple Cross
Sunday, October 23, 2016
Quartet
“The man who lives in contact with what he believes to be a living Church
is a man always expecting to meet Plato and Shakespeare to-morrow
at breakfast.”
— G. K. Chesterton
Or Sunday dinner.
|
Platonic |
Shakespearean |
| Not to mention Euclid and Picasso. | |
|
|
|
|
In the above pictures, Euclid is represented by |
|
Comments Off on Quartet
Saturday, September 24, 2016
Core Structure
For the director of "Interstellar" and "Inception" —
At the core of the 4x4x4 cube is …
Cover modified.
Comments Off on Core Structure
Thursday, September 22, 2016
Binary Opposition Illustrated
Click the above image for remarks on
"deep structure" and binary opposition.
See also the eightfold cube.
Comments Off on Binary Opposition Illustrated
Thursday, September 15, 2016
Metaphysics at Notre Dame
Recommended reading —
"When Analogies Fail," by Alexander Stern,
a doctoral candidate in philosophy at Notre Dame, in
The Chronicle of Higher Education online September 11, 2016.
Related material —
That same Alexander Stern in this journal on April 17, 2016:
See also the eightfold cube in the previous post,
Metaphysics at Scientific American:
Comments Off on Metaphysics at Notre Dame
Wednesday, August 31, 2016
The Lost Crucible
Yesterday's post The Eightfold Cube in Oslo suggests a review of
posts that mention The Lost Crucible.
(The crucible in question is from a book by Katherine Neville,
The Eight . Any connection with Arthur Miller's play "The Crucible"
is purely coincidental.)
Comments Off on The Lost Crucible
Saturday, August 27, 2016
Incarnation
See a search for the title in this journal.
Related material:
The incarnation of three permutations,
named A, B, and C,
on the 7-set of digits {1, 2, 3, 4, 5, 6, 7}
as permutations on the eightfold cube.
See Minimal ABC Art, a post of August 22, 2016.
Comments Off on Incarnation
Monday, April 25, 2016
Peirce’s Accounts of the Universe
Compare and contrast Peirce's seven systems of metaphysics with
the seven projective points in a post of March 1, 2010 —
From my commentary on Carter's question —
Comments Off on Peirce’s Accounts of the Universe
Wednesday, April 20, 2016
Symmetric Generation of a Simple Group
The reference in the previous post to the work of Guitart and
The Road to Universal Logic suggests a fiction involving
the symmetric generation of the simple group of order 168.
See The Diamond Archetype and a fictional account of the road to Hell …
The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.
For related symmetric generation of a much larger group, see Solomon's Cube.
Comments Off on Symmetric Generation of a Simple Group