^{Typographical:} »
Eightfold Cube:
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
"The field of geometric group theory emerged from Gromov’s insight
that even mathematical objects such as groups, which are defined
completely in algebraic terms, can be profitably viewed as geometric
objects and studied with geometric techniques."
— Mathematical Sciences Research Institute, 2016:
See also some writings of Gromov from 201516:
For a simpler example than those discussed at MSRI
of both algebraic and geometric techniques applied to
the same group, see a post of May 19, 2017,
"From Algebra to Geometry." That post reviews
an earlier illustration —
For greater depth, see "Eightfold Cube" in this journal.
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
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
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 112, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
From Wikipedia's Iceberg Theory —
Related material:
The Eightfold Cube and The Quantum Identity —
See also the previous post.
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."
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 —
"Clearly, there is a spirit of openhandedness in postconceptual 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 avantgarde 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 —
The assignments page for a graduate algebra course at Cornell
last fall had a link to the eightfold cube:
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.
(Continued … See the title in this journal, as well as Cube Bricks.)
Cube Bricks 1984 —
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.)
“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 tomorrow
at breakfast.”
— G. K. Chesterton
Or Sunday dinner.
Platonic 
Shakespearean 
Not to mention Euclid and Picasso.  


In the above pictures, Euclid is represented by 
For the director of "Interstellar" and "Inception" —
At the core of the 4x4x4 cube is …
Cover modified.
Click the above image for remarks on
"deep structure" and binary opposition.
See also the eightfold cube.
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.)
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.
See a search for the title in this journal.
Related material:
The incarnation of three permutations,
named A, B, and C,
on the 7set 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.
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 —
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.
A recent post about the eightfold cube suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :
As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding a 1×8 array from Turyn's
1967 construction of the Golay code.
Folding a 2×4 Curtis array yet again yields the 2x2x2 eightfold cube .
Those who prefer an entertainment approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space."
The New York Times philosophy column yesterday —
The Times's philosophy column "The Stone" is named after the legendary
"philosophers' stone." The column's name, and the title of its essay yesterday
"Is that even a thing?" suggest a review of the eightfold cube as "The object
most closely resembling a 'philosophers' stone' that I know of" (Page 51 of
the current issue of a Norwegian art quarterly, KUNSTforum.as).
The eightfold cube —
Definition of Epiphany
From James Joyce’s Stephen Hero , first published posthumously in 1944. The excerpt below is from a version edited by John J. Slocum and Herbert Cahoon (New York: New Directions Press, 1959). Three Times: … By an epiphany he meant a sudden spiritual manifestation, whether in the vulgarity of speech or of gesture or in a memorable phase of the mind itself. He believed that it was for the man of letters to record these epiphanies with extreme care, seeing that they themselves are the most delicate and evanescent of moments. He told Cranly that the clock of the Ballast Office was capable of an epiphany. Cranly questioned the inscrutable dial of the Ballast Office with his no less inscrutable countenance: — Yes, said Stephen. I will pass it time after time, allude to it, refer to it, catch a glimpse of it. It is only an item in the catalogue of Dublin’s street furniture. Then all at once I see it and I know at once what it is: epiphany. — What? — Imagine my glimpses at that clock as the gropings of a spiritual eye which seeks to adjust its vision to an exact focus. The moment the focus is reached the object is epiphanised. It is just in this epiphany that I find the third, the supreme quality of beauty. — Yes? said Cranly absently. — No esthetic theory, pursued Stephen relentlessly, is of any value which investigates with the aid of the lantern of tradition. What we symbolise in black the Chinaman may symbolise in yellow: each has his own tradition. Greek beauty laughs at Coptic beauty and the American Indian derides them both. It is almost impossible to reconcile all tradition whereas it is by no means impossible to find the justification of every form of beauty which has ever been adored on the earth by an examination into the mechanism of esthetic apprehension whether it be dressed in red, white, yellow or black. We have no reason for thinking that the Chinaman has a different system of digestion from that which we have though our diets are quite dissimilar. The apprehensive faculty must be scrutinised in action. — Yes … — You know what Aquinas says: The three things requisite for beauty are, integrity, a wholeness, symmetry and radiance. Some day I will expand that sentence into a treatise. Consider the performance of your own mind when confronted with any object, hypothetically beautiful. Your mind to apprehend that object divides the entire universe into two parts, the object, and the void which is not the object. To apprehend it you must lift it away from everything else: and then you perceive that it is one integral thing, that is a thing. You recognise its integrity. Isn’t that so? — And then? — That is the first quality of beauty: it is declared in a simple sudden synthesis of the faculty which apprehends. What then? Analysis then. The mind considers the object in whole and in part, in relation to itself and to other objects, examines the balance of its parts, contemplates the form of the object, traverses every cranny of the structure. So the mind receives the impression of the symmetry of the object. The mind recognises that the object is in the strict sense of the word, a thing , a definitely constituted entity. You see? — Let us turn back, said Cranly. They had reached the corner of Grafton St and as the footpath was overcrowded they turned back northwards. Cranly had an inclination to watch the antics of a drunkard who had been ejected from a bar in Suffolk St but Stephen took his arm summarily and led him away. — Now for the third quality. For a long time I couldn’t make out what Aquinas meant. He uses a figurative word (a very unusual thing for him) but I have solved it. Claritas is quidditas . After the analysis which discovers the second quality the mind makes the only logically possible synthesis and discovers the third quality. This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany. Having finished his argument Stephen walked on in silence. He felt Cranly’s hostility and he accused himself of having cheapened the eternal images of beauty. For the first time, too, he felt slightly awkward in his friend’s company and to restore a mood of flippant familiarity he glanced up at the clock of the Ballast Office and smiled: — It has not epiphanised yet, he said. 
Foreword by Sir Michael Atiyah —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
Judy Bass, Los Angeles Times , March 12, 1989 —
"Like Rubik's Cube, The Eight demands to be pondered."
As does a figure from 1984, Cullinane's Cube —
For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
See also the recent post Cube Bricks 1984 —
Related remark from the literature —
Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 19842003."
(From Anatomy of a Cube, Sept. 18, 2011.)
The following page quotes "Raiders of the Lost Crucible,"
a Log24 post from Halloween 2015.
From KUNSTforum.as, a Norwegian art quarterly, issue no. 1 of 2016.
Related posts — See Lyche Eightfold.
A March 10, 2016, Facebook post from KUNSTforum.as,
a Norwegian art quarterly —
Click image above for a view of pages 5051 of a new KUNSTforum
article showing two photos relevant to my own work — those labeled
"after S. H. Cullinane."
(The phrase "den pensjonerte Oxfordprofessoren Stephen H. Cullinane"
on page 51 is almost completely wrong. I have never been a professor,
I was never at Oxford, and my first name is Steven, not Stephen.)
For some background on the 15 projective points at the lower left of
the above March 10 Facebook post, see "The Smallest Projective Space."
From an article* in Proceedings of Bridges 2014 —
As artists, we are particularly interested in the symmetries of real world physical objects. Three natural questions arise: 1. Which groups can be represented as the group of symmetries of some realworld physical object? 2. Which groups have actually been represented as the group of symmetries of some realworld physical object? 3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not? 
The article was cited by Evelyn Lamb in her Scientific American
weblog on May 19, 2014.
The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:
"… finite projective planes [in particular, the 7point Fano plane,
about which Lamb is writing] seem like a triumph of purely
axiomatic thinking over any hint of reality…."
For related hints of reality, see Eightfold Cube in this journal.
* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman
"… if your requirement for success is to be like Steve Jobs,
good luck to you."
— "Transformation at Yahoo Foiled by Marissa Mayer’s
Inability to Bet the Farm," New York Times online yesterday
"Design is how it works." — Steve Jobs
Related material: Posts tagged Ambassadors.
"… the A B C of being…." — Wallace Stevens
Scholia —
Compare to my own later note, from March 4, 2010 —
"It seems that Guitart discovered these 'A, B, C' generators first,
though he did not display them in their natural setting,
the eightfold cube." — Borromean Generators (Log24, Oct. 19)
See also Raiders of the Lost Crucible (Halloween 2015)
and "Guitar Solo" from the 2015 CMA Awards on ABC.
Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —
"First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013"
This journal on the date Friday, April 5, 2013 —
The object most closely resembling a "philosophers' stone"
that I know of is the eightfold cube .
For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —
Related material by Schöter —
A 20page PDF, "Boolean Algebra and the Yi Jing."
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)
I differ with Schöter's emphasis on Boolean algebra.
The appropriate mathematics for I Ching studies is,
I maintain, not Boolean algebra but rather Galois geometry.
See last Saturday's post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter's work, not
suitable for a general Internet audience.
This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —
"Hexagonal Logic of the Field F_{8} as a Boolean Logic
with Three Involutive Modalities," pp. 191220 in
The Road to Universal Logic:
Festschrift for 50th Birthday of
JeanYves Béziau, Volume I,
Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.
See also the eightfold cube in this journal.
From slides dated June 28, 2008 —
Compare to my own later note, from March 4, 2010 —
It seems that Guitart discovered these "A, B, C" generators first,
though he did not display them in their natural setting,
the eightfold cube.
Some context: The epigraph to my webpage
"A Simple Reflection Group of Order 168" —
"Let G be a finite, primitive subgroup of GL(V) = GL(n,D) ,
where V is an ndimensional vector space over the
division ring D . Assume that G is generated by 'nice'
transformations. The problem is then to try to determine
(up to GL(V) conjugacy) all possibilities for G . Of course,
this problem is very vague. But it is a classical one,
going back 150 years, and yet very much alive today."
— William M. Kantor, "Generation of Linear Groups,"
pp. 497509 in The Geometric Vein: The Coxeter Festschrift ,
published by Springer, 1981
Norwegian Sculpture Biennial 2015 catalog, p. 70 —
" 'Ambassadørene' er fysiske former som presenterer
ikkfysiske fenomener. "
Translation by Google —
" 'Ambassadors' physical forms presents
nonphysical phenomena. "
Related definition —
Are the "line diagrams" of the diamond theorem and
the analogous "plane diagrams" of the eightfold cube
nonphysical entities? Discuss.
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.
The Fano Plane —
"A balanced incomplete block design , or BIBD
with parameters b , v , r , k , and λ is an arrangement
of b blocks, taken from a set of v objects (known
for historical reasons as varieties ), such that every
variety appears in exactly r blocks, every block
contains exactly k varieties, and every pair of
varieties appears together in exactly λ blocks.
Such an arrangement is also called a
(b , v , r , k , λ ) design. Thus, (7, 3, 1) [the Fano plane]
is a (7, 7, 3, 3, 1) design."
— Ezra Brown, "The Many Names of (7, 3, 1),"
Mathematics Magazine , Vol. 75, No. 2, April 2002
W. Cherowitzo uses the notation (v, b, r, k, λ) instead of
Brown's (b , v , r , k , λ ). Cherowitzo has described,
without mentioning its close connection with the
Fanoplane design, the following —
"the (8,14,7,4,3)design on the set
X = {1,2,3,4,5,6,7,8} with blocks:
{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}
{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}
{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."
We can arrange these 14 blocks in complementary pairs:
{1,2,3,6} {4,5,7,8}
{1,2,4,8} {3,5,6,7}
{1,2,5,7} {3,4,6,8}
{1,3,4,5} {2,6,7,8}
{1,3,7,8} {2,4,5,6}
{1,4,6,7} {2,3,5,8}
{1,5,6,8} {2,3,4,7}.
These pairs correspond to the seven natural slicings
of the following eightfold cube —
Another representation of these seven natural slicings —
These seven slicings represent the seven
planes through the origin in the vector
3space over the twoelement field GF(2).
In a standard construction, these seven
planes provide one way of defining the
seven projective lines of the Fano plane.
A more colorful illustration —
"It is as if one were to condense
all trends of present day mathematics
onto a single finite structure…."
— GianCarlo Rota, foreword to
A Source Book in Matroid Theory ,
Joseph P.S. Kung, Birkhäuser, 1986
"There is such a thing as a matroid."
— Saying adapted from a novel by Madeleine L'Engle
Related remarks from Mathematics Magazine in 2009 —
See also the eightfold cube —
Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts.
For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3space over the
twoelement Galois field GF(2), or, if you prefer, on the Galois
field Ω = GF(8).
Related fiction: The Eight , by Katherine Neville.
Related nonfiction: A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —
Mathematics
The Fano plane block design 
Magic
The Deathly Hallows symbol— 
… industrial designer Kenji Ekuan —
The adjective "eightfold," intrinsic to Buddhist
thought, was hijacked by GellMann and later
by the Mathematical Sciences Research Institute
(MSRI, pronounced "misery"). The adjective's
application to a 2x2x2 cube consisting of eight
subcubes, "the eightfold cube," is not intended to
have either Buddhist or Semitic overtones.
It is pure mathematics.
The seven symmetry axes of the regular tetrahedron
are of two types: vertextoface and edgetoedge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertextoface axes and one edgetoedge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edgetoedge axes.
(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book , pp. 1617.)
There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetricdifference sum of the
other two members.
(This is the eightfold cube discussed at finitegeometry.org.)
Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :
An unarticulated (but colored) cube:
A 2x2x2 articulated cube:
A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:
An image related to the recent posts Sense and Sensibility:
A quote from yesterday's post The Eight:
A possible source for the above phrase about phenomena "carved at their joints":
See also the carving at the joints of Plato's diamond from the Meno :
Related material: Phaedrus on Kant as a diamond cutter
in Zen and the Art of Motorcycle Maintenance .
"Die Unendlichkeit ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche stamme…."
— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: AubierFlammarion, 1969), fragment 120, p. 118
Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p. 181.
This quotation was suggested by the Bauhausrelated phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 19842003.
For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.
The title refers to a Scientific American weblog item
discussed here on May 31, 2014:
Some closely related material appeared here on
Dec. 30, 2011:
A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:
"Is there a geometric realization of Quaternion group?" —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).
The ninefold square, the eightfold cube, and monkeys.
For posts on the models above, see quaternion
in this journal. For the monkeys, see
"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American weblog, May 19, 2014:
The Scientific American item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):
See also Finite Geometry and Physical Space.
From a Huffington Post discussion of aesthetics:
“The image below on the left… is… overly simplistic, and lacks reality:
It’s all a matter of perspective: the problem here is that opposite sides
of the cube, which are parallel in real life, actually look parallel in the
left image! The image on the right is better….”
A related discussion: Eight is a Gate.
A screenshot of the new page on the eightfold cube at Froebel Decade:
Click screenshot to enlarge.
Continued from previous post and from Sept. 8, 2009.
Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the nonEuclidean geometry of Galois space.
In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.
Continued from October 30 (Devil's Night), 2013.
“In a sense, we would see that change
arises from the structure of the object.”
— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013
This suggests a review of mathematics and the
"Classic of Change ," the I Ching .
The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching as the corners of a cube.
See also…
(Click for clearer image.)
The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching trigrams and as
seven natural ways of slicing the cube.
For a different approach to the mathematics of cube slices,
related to Gauss's composition law for binary quadratic forms,
see the Bhargava cube in a post of April 9, 2012.
This journal on July 5, 2007 —
“It is not clear why MySpace China will be successful."
— The Chinese magazine Caijing in 2007, quoted in
Asia Sentinel on July 12, 2011
This journal on that same date, July 12, 2011 —
See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.
Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":
"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the everrecurring dwarfs…."
This journal, Group Actions, Nov. 14, 2012:
"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of Snow White (representing (0,0,0))
and the Seven Dwarfs—
Promotional description of a new book:
"Like Gödel, Escher, Bach before it, Surfaces and Essences will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."
"Like Gödel, Escher, Bach before it…."
Or like Metamagical Themas .
Rubik core:
Non Rubik cores:
Of the odd nxnxn cube: 
Of the even nxnxn cube: 
Related material: The Eightfold Cube and…
"A core component in the construction
is a 3dimensional vector space V over F_{2 }."
— Page 29 of "A twist in the M_{24} moonshine story,"
by Anne Taormina and Katrin Wendland.
(Submitted to the arXiv on 13 Mar 2013.)
"It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is reassembled into a linear array, are physical transformations not symbolic transformations. …"
— Storrs McCall, Department of Philosophy, McGill University, "The Consistency of Arithmetic"
"It should be emphasized…."
OK:
Storrs McCall at a 2008 philosophy conference .
His blocks talk was at 2:50 PM July 21, 2008.
See also this journal at noon that same day:
Froebel's Third Gift and the Eightfold Cube
Robert A. Wilson, in an inaugural lecture in April 2008—
Representation theory
A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).
But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .
This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.
Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.
We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.
As Poincaré said,
Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.
Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:
* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on "Lowdimensional
Topology, Geometry, and Dynamics"—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eightcube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel's Third Gift—
Photo by Norman Brosterman fom the Inventing Kindergarten
exhibit at The Institute for Figuring (cofounded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops' topics—
lowdimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
Last night's post on The Trinity of Max Black and the use of
the term "eightfold" by the Mathematical Sciences Research Institute
at Berkeley suggest a review of an image from Sept. 22, 2011—
The triskele detail above echoes a Buddhist symbol found,
for instance, on the Internet in an ad for meditation supplies—
Related remarks—
http://www.spencerart.ku.edu/about/dialogue/fdpt.shtml—
Mary Dusenbury (Radcliffe '64)—
"… I think a textile, like any work of art, holds a tremendous amount of information— technical, material, historical, social, philosophical— but beyond that, many works of art are very beautiful and they speak to us on many layers— our intellect, our heart, our emotions. I've been going to museums since I was a very small child, thinking about what I saw, and going back to discover new things, to see pieces that spoke very deeply to me, to look at them again, and to find more and more meaning relevant to me in different ways and at different times of my life. …
… I think I would suggest to people that first of all they just look. Linger by pieces they find intriguing and beautiful, and look deeply. Then, if something interests them, we have tried to put a little information around the galleries to give a bit of history, a bit of context, for each piece. But the most important is just to look very deeply."
http://en.wikipedia.org/wiki/Nikaya_Buddhism—
According to Robert Thurman, the term "Nikāya Buddhism" was coined by Professor Masatoshi Nagatomi of Harvard University, as a way to avoid the usage of the term Hinayana.^{[12]} "Nikaya Buddhism" is thus an attempt to find a more neutral way of referring to Buddhists who follow one of the early Buddhist schools, and their practice.
12. The Emptiness That is Compassion:
An Essay on Buddhist Ethics, Robert A. F. Thurman, 1980
[Religious Traditions , Vol. 4 No. 2, Oct.Nov. 1981, pp. 1134]
http://dsal.uchicago.edu/cgibin/philologic/getobject.pl?c.2:1:6.pali—
Nikāya [Sk. nikāya, ni+kāya]
collection ("body") assemblage, class, group
http://en.wiktionary.org/wiki/नि—
Sanskrit etymology for नि (ni)
नि (ni)
http://www.rigpawiki.org/index.php?title=Kaya—
Kaya (Skt. kāya ; སྐུ་, Tib. ku ; Wyl. sku ) —
the Sanskrit word kaya literally means ‘body’
but can also signify dimension, field or basis.
• structure, existentiality, founding stratum ▷HVG KBEU
Note that The Trinity of Max Black is a picture of a set—
i.e., of an "assemblage, class, group."
Note also the reference above to the word "gestalt."
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—
Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—
"ANDOVER–HARVARD THEOLOGICAL LIBRARY"
Detail from Knowles's obituary in this morning's New York Times—
William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.
Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
From Pilate Goes to Kindergarten—
The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—
Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?
A paper written in a somewhat similar spirit—
"Chiral Tetrahedrons as Unitary Quaternions"—
ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, noncommutativity of quaternions is a feature directly related to the chirality of molecules….
“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”
– Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975
Related material—
"The group of 8" is a phrase from politics, not mathematics.
Of the five groups of order 8 (see today's noon post),
the one pictured* in the center, Z_{2} × Z_{2} × Z_{2} , is of particular
interest. See The Eightfold Cube. For a connection of this
group of 8 to the last of the five pictured at noon, the
quaternion group, see Finite Geometry and Physical Space.
* The picture is of the group's cycle graph.
John Baez wrote in 1996 ("Week 91") that
"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."
Baez seems to come close to doing this with a
somewhat different i , j , and k — Hurwitz
quaternions— in his 2005 book review
quoted here yesterday.
See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.
…. and John Golding, an authority on Cubism who "courted abstraction"—
"Adam in Eden was the father of Descartes." — Wallace Stevens
Fictional symbologist Robert Langdon and a cube—
From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—
A related quotation—
"… quaternions provide a useful paradigm
for studying the phenomenon of 'triality.'"
— David A. Richter's webpage Zometool Triality
See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.
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.
The Cube Model and Peano Arithmetic
The eightfold cube model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—
The Consistency of Arithmetic, by Storrs McCall
"Is Peano arithmetic (PA) consistent? This paper contains a proof that it is. …
Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules. 'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a nonsymbolic, nonlinguistic component, a domain of objects. If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PAconsistency. But in addition to axiomatic proofs there is another kind of proof. In this paper I give a proof of PA's consistency based on a formal semantics for PA. To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.
The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :
"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory. This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models. In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations. The models occupy centre stage." (1980, p. 44)
Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a nonlinguistic model. Suppose we wish to prove the consistency of the following geometric axioms:
A1. For any two lines, there is at most one point that lies on both.
A2. For any two points, there is exactly one line that lies on both.
A3. On every line there lie at least two points.
The following diagram shows the axioms to be consistent:
The consistency proof is not a 'syntactic' one, in which the consistency of A1A3 is derived as a theorem of a deductive system, but is based on a nonlinguistic structure. It is a semantic as opposed to a syntactic proof. The proof constructed in this paper, like van Fraassen's, is based on a nonlinguistic component, not a diagram in this case but a physical domain of threedimensional cubeshaped blocks. ….
… The semantics presented in this paper I call 'block semantics', for reasons that will become clear…. Block semantics is based on domains consisting of cubeshaped objects of the same size, e.g. children's wooden building blocks. These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns. A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable. Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be rearranged or transformed into a linear array by successive concatenation of its rows. The result is called the 'linear transformation' of the rectangle. An essential characteristic of block semantics is that every domain of every block model is finite. In this respect it differs from Tarski’s semantics for firstorder logic, which permits infinite domains. But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.
It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is reassembled into a linear array, are physical transformations not symbolic transformations. …"
— Storrs McCall, Department of Philosophy, McGill University
See also…
In memory of artist Ronald Searle—
Searle reportedly died at 91 on December 30th.
From Log24 on that date—
Click the above image for some context.
Update of 9:29 PM EST Jan. 3, 2012—
Theorum
Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution. Problems with "theory" Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just a theory" – and the often cited response of "but gravity is also just a theory". To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives. This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

Some backgound— In this journal, "Diamond Theory of Truth."
The following picture provides a new visual approach to
the order8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she cofounded—
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—
From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—
"… we are saying much more than that
some set of seven involutions, which would be a very weak
requirement. We are asserting that M_{ 24} is generated by a set
of seven involutions which possesses all the symmetries of
acting on the points of the 7point projective plane…."
— Symmetric Generation , p. 41
"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42
See also (click to enlarge)—
Cassirer's remarks connect the concept of objectivity with that of object .
The above quotations perhaps indicate how the Mathieu group
"This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."
— James Joyce, Stephen Hero
For a simpler object "which possesses all the symmetries of
For symmetric generation of
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."
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.
An image that may be viewed as
a cube with a "+" on each face—
The eightfold cube
Underlying structure
For the Pope and others on St. Benedict's Day
who prefer narrative to mathematics—
The New York Times at 9 PM ET June 23, 2011—
ROBERT FANO: I’m trying to think briefly how to put it.
GINO FANO: "On the Fundamental Postulates"—
"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."
"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."
Also from the Times article quoted above…
"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”
— Tom Van Vleck
Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—
The art is also related to the mathematics of Gino Fano.
For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."
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."
The following is a new illustration for Cubist Geometries—
(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
For advanced, see Solomon's Cube and Geometry of the I Ching .)
Excerpt from a post of 8 AM May 26, 2006 —
A Living Church "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 tomorrow at breakfast." – G. K. Chesterton

A related scene from the opening of Blake Edwards's "S.O.B." —
The above is the result of a (fruitless) image search today for a current version of Giovanni Sambin's "Basic Picture: A Structure for Topology."
That search was suggested by the title of today's New York Times oped essay "Found in Translation" and an occurrence of that phrase in this journal on January 5, 2007.
Further information on one of the images above—
A search in this journal on the publication date of Giaquinto's Visual Thinking in Mathematics yields the following—
In defense of Plato’s realism (vs. sophists’ nominalism– see recent entries.) Plato cited geometry, notably in the Meno , in defense of his realism. 
For the Meno 's diamond figure in Giaquinto, see a review—
— Review by Jeremy Avigad (preprint)
Finite geometry supplies a rather different context for Plato's "basic picture."
In that context, the Klein fourgroup often cited by art theorist Rosalind Krauss appears as a group of translations in the mathematical sense. (See Kernel of Eternity and Sacerdotal Jargon at Harvard.)
The Times oped essay today notes that linguistic translation "… is not merely a job assigned to a translator expert in a foreign language, but a long, complex and even profound series of transformations that involve the writer and reader as well."
The list of fourgroup transformations in the mathematical sense is neither long nor complex, but is apparently profound enough to enjoy the close attention of thinkers like Krauss.
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.
URBI
(Toronto)–
Click on image for some background.
ORBI
(Globe and Mail)–
See also Baaad Blake and
Fearful Symmetry.
Harvard Crimson headline today–
"Deconstructing Design"
Reconstructing Design
The phrase "eightfold way" in today's
previous entry has a certain
graphic resonance…
For instance, an illustration from the
Wikipedia article "Noble Eightfold Path" —
Adapted detail–
See also, from
St. Joseph's Day—
Harvard students who view Christian symbols
with fear and loathing may meditate
on the above as a representation of
the Gankyil rather than of the Trinity.
It is well known that the seven
Similarly, recent posts* have noted that the thirteen
These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finitegeometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4space over the twoelement Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."
Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).
The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…
See also Geometry of the I Ching and a search in this journal for
* February 27 and March 13
** G_{20160} in Mitchell 1910, LF(3,2^{2}) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,2^{2}),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21Point Plane," Math. Zeitschr. 87, 348362 (1965)
The current article on group theory at Wikipedia has a Rubik's Cube as its logo–
The article quotes Nathan C. Carter on the question "What is symmetry?"
This naturally suggests the question "Who is Nathan C. Carter?"
A search for the answer yields the following set of images…
Click image for some historical background.
Carter turns out to be a mathematics professor at Bentley University. His logo– an eightfoldcube labeling (in the guise of a Cayley graph)– is in much better taste than Wikipedia's.
"The cube has…13 axes of symmetry:
6 C_{2} (axes joining midpoints of opposite edges),
4 C_{3} (space diagonals), and
3C_{4} (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube
These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13point Galois plane.
The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubiklike mechanism–
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
A closely related structure–
the finite projective plane
with 13 points and 13 lines–
A later version of the 13point plane
by Ed Pegg Jr.–
A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–
The above images tell a story of sorts.
The moral of the story–
Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.
The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3space converted to a vector 3space.
If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.
The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.
The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.
(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)
See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.
From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005— as revised on Nov. 25, 2009—
Historically, (Coxeter 1934) proved that every reflection group [Euclidean, by the current Wikipedia definition] is a Coxeter group (i.e., has a presentation where all relations are of the form r_{i}^{2} or (r_{i}r_{j})^{k}), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group [again, Euclidean], and classified finite Coxeter groups. Finite fields
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as −1=1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981). 
Related material:
"A Simple Reflection Group of Order 168," by Steven H. Cullinane, and
by Ascher Wagner, U. of Birmingham, received 27 July 1977
Journal  Geometriae Dedicata 
Publisher  Springer Netherlands 
Issue  Volume 9, Number 2 / June, 1980 
[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]
Clearly the eightfold cube is a counterexample.
Deep Play:
Mimzy vs. Mimsy
From a 2007 film, "The Last Mimzy," based on
the classic 1943 story by Lewis Padgett
"Mimsy Were the Borogoves"–
As the above mandala pictures show,
the film incorporates many New Age fashions.
The original story does not.
A more realistic version of the story
might replace the mandalas with
the following illustrations–
For a commentary, see "NonEuclidean Blocks."
(Here "nonEuclidean" means simply
other than Euclidean. It does not imply any
violation of Euclid's parallel postulate.)
Truth, Geometry, Algebra
The following notes are related to A Simple Reflection Group of Order 168.
1. According to H.S.M. Coxeter and Richard J. Trudeau
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
— Coxeter, 1987, introduction to Trudeau’s The NonEuclidean Revolution
1.1 Trudeau’s Diamond Theory of Truth
1.2 Trudeau’s Story Theory of Truth
2. According to Alexandre Borovik and Steven H. Cullinane
2.1 Coxeter Theory according to Borovik
2.1.1 The Geometry–
Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
Coxeter Languages in Coxeter Theory
2.2 Diamond Theory according to Cullinane
2.2.1 The Geometry–
Examples: Eightfold Cube and Solomon’s Cube
2.2.2 The Algebra–
Examples: Cullinane and (rather indirectly related) Gerhard Grams
Summary of the story thus far:
Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side. Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)
From today's NY Times—
Obituaries for mystery authors
Ralph McInerny and Dick Francis
From the date (Jan. 29) of McInerny's death–
"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"
– Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson
From the date (Feb. 14) of Francis's death–
The EIghtfold Cube
The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.
This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order168 simple group of Felix Christian Klein.
For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
and the death of Coxeter–
Putting Descartes Before Dehors
For a more Protestant meditation,
see The Cross of Descartes—
"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke
For further details, click on
the image below–
Notre Dame Philosophical Reviews
Today I revised the illustrations
in Finite Geometry of the
Square and Cube
for consistency in labeling
the eightfold cube.
Related material:
The Sept. 8 entry on nonEuclidean* blocks ended with the phrase “Go figure.” This suggested a MAGMA calculation that demonstrates how Klein’s simple group of order 168 (cf. Jeremy Gray in The Eightfold Way) can be visualized as generated by reflections in a finite geometry.
* i.e., other than Euclidean. The phrase “nonEuclidean” is usually applied to only some of the geometries that are not Euclidean. The geometry illustrated by the blocks in question is not Euclidean, but is also, in the jargon used by most mathematicians, not “nonEuclidean.”
NonEuclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads…. Tesseract "Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Hardening of the thoughtarteries," Jane interjected. Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–" "Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–" "Poor kid," Jane said. Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–" "Blocks? What kind?" Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid." — "Mimsy Were the Borogoves," Lewis Padgett, 1943 
For the intuitive basis of one type of nonEuclidean* geometry– finite geometry over the twoelement Galois field– see the work of…
Friedrich Froebel
(17821852), who
invented kindergarten.
His "third gift" —
Pilate Goes
to Kindergarten
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The NonEuclidean Revolution
Consider the following question in a paper cited by V. S. Varadarajan:
E. G. Beltrametti, “Can a finite geometry describe physical spacetime?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.
Simplifying:
“Can a finite geometry describe physical space?”
Simplifying further:
“Yes. Vide ‘The Eightfold Cube.'”
Through the
Looking Glass:
A Sort of Eternity
From the new president's inaugural address:
"… in the words of Scripture, the time has come to set aside childish things."
The words of Scripture:
9  For we know in part, and we prophesy in part. 
10  But when that which is perfect is come, then that which is in part shall be done away. 
11  When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things. 
12 
For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.

"through a glass"—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
Notsochildish:
Three planes through
the center of a cube
that split it into
eight subcubes:
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3space
over the field of real numbers,
but rather in the finite Galois
3space over the 2element field.
Galois age fifteen,
drawn by a classmate.
These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
Related material:
The central aim of Western religion–
"Each of us has something to offer the Creator... the bridging of masculine and feminine, life and death. It's redemption.... nothing else matters."  Martha Cooley in The Archivist (1998) The central aim of Western philosophy– Dualities of Pythagoras as reconstructed by Aristotle: Limited Unlimited Odd Even Male Female Light Dark Straight Curved ... and so on .... "Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy." — Jamie James in The Music of the Spheres (1993)
"In the garden of Adding — The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000) A quotation today at art critic Carol Kino's website, slightly expanded:
"Art inherited from the old religion — Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space: "We have now reached
"Space: what you — James Joyce, Ulysses 
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:
“… we do not need to accept Jung’s theory as true in order to find it illuminating.”
The same is true of Jung’s remarks on synchronicity.
For example —
Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s fourdiamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 19842003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
“What, then, is a frame, and what is frame work?”
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.
Part I: The White Cube
Part II: Inside
Part III: Outside
For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).
For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.
For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation
(U. of Chicago Press, 1999):
From Chapter 3,
"Sutures* of Structures," p. 58: "What, then, is a frame, and what is frame work? This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'" * P. 61:

Wikipedia on Rubik's 2×2×2 "Pocket Cube"–
"Any permutation of the 8 corner cubies is possible (8! positions)."
Some pages related to this claim–
Analyzing Rubik's Cube with GAP
Online JavaScript Pocket Cube.
The claim is of course trivially true for the unconnected subcubes of Froebel's Third Gift:
See also:
MoMA Goes to Kindergarten,
Tea Privileges,
and
"Ad Reinhardt and Tony Smith:
A Dialogue,"
an exhibition opening today
at Pace Wildenstein.
For a different sort
of dialogue, click on the
artists' names above.
For a different
approach to S_{8},
see Symmetries.
"With humor, my dear Zilkov.
Always with a little humor."
 The Manchurian Candidate
On a book by Margaret Wertheim:
“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)
There are also other sorts of space.
This photo may serve as an
introduction to a different
sort of space.
See The Eightfold Cube.
For the religious meaning
of this small space, see
For a related larger space,
see the entry and links of
St. Augustine’s Day, 2006.
Observations suggested by an article on author Lewis Hyde– “What is Art For?“– in today’s New York Times Magazine:
Margaret Atwood (pdf) on Lewis Hyde’s
Trickster Makes This World: Mischief, Myth, and Art —
“Trickster,” says Hyde, “feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation.” (71) As Hyde says, “… almost everything that can be said about psychopaths can also be said about tricksters,” (158), although the reverse is not the case. “Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists.” (159)
What is “the next world”? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the conartist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
The Trickster
and the Paranormal
and
Martin Gardner on
a disappearing cube —
“What happened to that… cube?”
Apollinax laughed until his eyes teared. “I’ll give you a hint, my dear. Perhaps it slid off into a higher dimension.” “Are you pulling my leg?” “I wish I were,” he sighed. “The fourth dimension, as you know, is an extension along a fourth coordinate perpendicular to the three coordinates of threedimensional space. Now consider a cube. It has four main diagonals, each running from one corner through the cube’s center to the opposite corner. Because of the cube’s symmetry, each diagonal is clearly at right angles to the other three. So why shouldn’t a cube, if it feels like it, slide along a fourth coordinate?” — “Mr. Apollinax Visits New York,” by Martin Gardner, Scientific American, May 1961, reprinted in The Night is Large 
this illustration in
Beware of Gardner’s
“clearly” and other lies.
On May 4, 2005, I wrote a note about how to visualize the 7point Fano plane within a cube.
Last month, John Baez showed slides that touched on the same topic. This note is to clear up possible confusion between our two approaches.
From Baez’s Rankin Lectures at the University of Glasgow:
The statement is, however, true of the eightfold cube, whose eight subcubes correspond to points of the linear 3space over the twoelement field, if “planes through the origin” is interpreted as planes within that linear 3space, as in Galois geometry, rather than within the Euclidean cube that Baez’s slides seem to picture.
This Galoisgeometry interpretation is, as an article of his from 2001 shows, actually what Baez was driving at. His remarks, however, both in 2001 and 2008, on the planecube relationship are both somewhat trivial– since “planes through the origin” is a standard definition of lines in projective geometry– and also unrelated– apart from the possibility of confusion– to my own efforts in this area. For further details, see The Eightfold Cube.
(Continued from Sept. 22–
“A Rose for Ecclesiastes.”)
From Kibler’s
“Variations on a Theme of
Heisenberg, Pauli, and Weyl,”
July 17, 2008:
“It is to be emphasized
that the 15 operators…
are underlaid by the geometry
of the generalized quadrangle
of order 2…. In this geometry,
the five sets… correspond to
a spread of this quadrangle,
i.e., to a set of 5 pairwise
skew lines….”
— Maurice R. Kibler,
July 17, 2008
For ways to visualize
this quadrangle,
see Inscapes.
Related material
A remark of Heisenberg “… die Schönheit… [ist] die
richtige Übereinstimmung der Teile miteinander und mit dem Ganzen.” “Beauty is the proper conformity 
"Credences of Summer," VII,
by Wallace Stevens, from
"Three times the concentred 
One possibility —
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:
"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Another possibility —
A more modest object —
the 4×4 square.
Update of Aug. 2021 —
Kostant's poetic comparison might be applied also to this object.
More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.
For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.
* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–
If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that
"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."
The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
“This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It’s probably the best way for a nonmathematician to appreciate the symmetry of Klein’s quartic. It’s a 3holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron’s 4 corners, and 8 for each of its 6 edges.”
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
The Relativity Theory
of Kindergarten Blocks
(Continued from
January 16, 2008)
“Hmm, next paper… maybe
‘An Unusually Complicated
Theory of Something.'”
Something:
From Friedrich Froebel,
who invented kindergarten:
An Unusually
Complicated Theory:
From Christmas 2005:
For the eightfold cube
as it relates to Klein’s
simple group, see
“A Reflection Group
of Order 168.”
For an even more
complicated theory of
Klein’s simple group, see
From Log24 on June 27, 2008,
the day that comicbook artist
Michael Turner died at 37 —
Van Gogh (by Ed Arno) in
The Paradise of Childhood
(by Edward Wiebé):
For Turner’s photoopportunity,
click on Lara.
In his memory:
a cartoon by Arno combined
with material shown here,
under the heading
“From the Cartoon Graveyard,”
on May 27, the date of
Arno’s death —
Related material:
Yesterday’s entry. The key part of
that entry is of course the phrase
“the antics of a drunkard.”
Ray Milland in
“The Lost Weekend”
(see June 25, 10:31 AM)–
“I’m van Gogh
painting pure sunlight.”
It is not advisable,
in all cases,
to proceed thus far.
Undertakings bring misfortune.
Nothing that would further.
“Brian O’Doherty, an Irishborn artist,
before the [Tuesday, May 20] wake
of his alter ego* ‘Patrick Ireland’
on the grounds of the
Irish Museum of Modern Art.”
— New York Times, May 22, 2008
THE IMAGE
Thus the superior man
understands the transitory
in the light of
the eternity of the end.
Another version of
the image:
See 2/22/08
and 4/19/08.
Michael Kimmelman in today’s New York Times—
“An essay from the ’70s by Mr. O’Doherty, ‘Inside the White Cube,’ became famous in art circles for describing how modern art interacted with the gallery spaces in which it was shown.”
Brian O’Doherty, “Inside the White Cube,” 1976 Artforum essays on the gallery space and 20thcentury art:
“The history of modernism is intimately framed by that space. Or rather the history of modern art can be correlated with changes in that space and in the way we see it. 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.”
“Nothing that would further.”
— Hexagram 54
…. Now thou art an 0 
“…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done. Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known. It is sovereign or it is nothing, and if it is nothing then man was born dead.”
— The Greater Trumps,
by Charles Williams, Ch. 14
"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."
— "Was Modernism Born
in Toddler Toolboxes?"
by Trip Gabriel, New York Times,
April 10, 1997
Figure 1 —
Concept from 1819:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.
(Footnote 3)
Figure 3 —
The Third Gift, 1906:
Figure 4 —
Solomon's Cube,
1981 and 1983:
Figure 5 —
Design Cube, 2006:
The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the twoelement field).
(To see how the display works,
try the Kaleidoscope Puzzle first.)
Part I:
Part II:
This figure is related to
the mathematics of
reflection groups.
Part III:
— George Steiner in Grammars of Creation
Inverse Canon —
From Werner Icking Music Archive:
Bach, Fourteen Canons
on the First Eight Notes
of the Goldberg Ground,
No. 11 —
At a different site —
an mp3 of the 14 canons.
That Crown of Thorns,
by Timothy A. Smith
Click on the image for a larger version
and an expansion of some remarks
quoted here on Christmas 2005.
For further details, see
Finite Geometry of
the Square and Cube
and The Eightfold Cube.
In Defense of
Plato’s Realism
(vs. sophists’ nominalism–
see recent entries.)
Plato cited geometry,
notably in the Meno,
in defense of his realism.
Consideration of the
Meno’s diamond figure
leads to the following:
Click on image for details.
As noted in an entry,
Plato, Pegasus, and
the Evening Star,
linked to
at the end of today’s
previous entry,
the “universals”
of Platonic realism
are exemplified by
the hexagrams of
the I Ching,
which in turn are
based on the seven
trigrams above and
on the eighth trigram,
of all yin lines,
not shown above:
K’un
The Receptive
_____________________________________________
Update of Nov. 30, 2013:
From a littleknown website in Kuala Lumpur:
(Click to enlarge.)
The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.
M. Scott Peck,
People of the Lie
“Far in the woods they sang their unreal songs, Secure. It was difficult to sing in face Of the object. The singers had to avert themselves Or else avert the object.” — Wallace Stevens, 
Today is June 25,
anniversary of the
birth in 1908 of
Willard Van Orman Quine.
Quine died on
Christmas Day, 2000.
Today, Quine’s birthday, is,
as has been noted by
Quine’s son, the point of the
calendar opposite Christmas–
i.e., “AntiChristmas.”
If the AntiChrist is,
as M. Scott Peck claims,
a spirit of unreality, it seems
fitting today to invoke
Quine, a student of reality,
and to borrow the title of
Quine’s Word and Object…
Word:
An excerpt from
“Credences of Summer”
by Wallace Stevens:
“Three times the concentred self takes hold, three times The thrice concentred self, having possessed The object, grips it — “Credences of Summer,” VII, 
Object:
From Friedrich Froebel,
who invented kindergarten:
From Christmas 2005:
Click on the images
for further details.
For a larger and
more sophisticaled
relative of this object,
see yesterday’s entry
At Midsummer Noon.
The object is real,
not as a particular
physical object, but
in the way that a
mathematical object
is real — as a
pure Platonic form.
“It’s all in Plato….”
— C. S. Lewis
Gift of the Third Kind
Background:
Art Wars and
Russell Crowe as
Santa's Helper.
From Christmas 2005:
Related material from
Pittsburgh:
… and from Grand Rapids:
Related material
for Holy Saturday:
Bernard Holland in The New York Times on Monday, May 20, 1996:
“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday….”
Log24 on Monday,
Dec. 18, 2006: “I did a column in — Martin Gardner (pdf) “… the entire profession — Joan S. Birman (pdf)
Lottery on Friday,
Dec. 22, 2006:

“Art history was very personal
through the eyes of Ad Reinhardt.”
— Robert Morris,
Smithsonian Archives
of American Art
“A set having three members is a
single thing wholly constituted by
its members but distinct from them.
After this, the theological doctrine
of the Trinity as ‘three in one’
should be child’s play.”
— Max Black,
Caveats and Critiques:
Philosophical Essays in
Language, Logic, and Art
Geometry
from Point
to Hyperspace
by Steven H. Cullinane
Euclid is “the most famous
geometer ever known
and for good reason:
for millennia it has been
his window
that people first look through
when they view geometry.”
— Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace,
by Leonard Mlodinow
“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”
— Paul Halmos in
I Want To Be a Mathematician
Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.
Each of these spaces
has infinitely many points.
Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:


From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4point affine plane,
which we may call
Galois’s Window.
The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3space
and Galois affine hyperplane
in greater detail.
For more on these cases, see
The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
LatinSquare Geometry, and
Geometry of the 4×4 Square.
(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)
These 8 and 16point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.
The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
Elements of Finite Geometry.
“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 tomorrow at breakfast.”
Natasha Wescoat, 2004 

Not to mention Euclid and Picasso  
(Click on pictures for details. Euclid is represented by Alexander Bogomolny, Picasso by Robert Foote.)

See also works by the late Arthur Loeb of Harvard’s Department of Visual and Environmental Studies.
“I don’t want to be a product of my environment. I want my environment to be a product of me.” — Frank Costello in The Departed
For more on the Harvard environment,
see today’s online Crimson:
The Harvard Crimson, Online Edition 
Sunday, Oct. 8, 2006 
POMP AND Friday, Oct. 6: The Ringling Bros. Barnum & Bailey Circus has come to town, and yesterday the animals were disembarked near MIT and paraded to their temporary home at the Banknorth Garden. 
OPINION At Last, a By THE CRIMSON STAFF The Trouble By SAHIL K. MAHTANI 
A Living Church
continued from March 27
— G. K. Chesterton
Shakespearean Fool 
as well as
and the remarks
of Oxford professor
Marcus du Sautoy,
who claims that
"the right side of the brain
is responsible for mathematics."
Let us hope that Professor du Sautoy
is more reliable on zeta functions,
his real field of expertise,
than on neurology.
The picture below may help
to clear up his confusion
between left and right.
His confusion about
pseudoscience may not
be so easily remedied.
flickr.com/photos/jaycross/3975200/
(Any resemblance to the film
"Hannibal" is purely coincidental.)
"What is it, Major Lawrence,
that attracts you personally
to the desert?"
"It's clean."
Visible Mathematics,
continued —
"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."
— Anne Morrow Lindbergh,
Earth Shine, p. xii
— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Related material:
(in Arabic, ka'b)
and
101 101
— Ben Macintyre,
The London Times, June 4:
When Rimbaud Meets Rambo
“Room 101 was the place where
your worst fears were realised
in George Orwell’s classic
Nineteen EightyFour.
Classics Illustrated —
Click on picture for details.
(For some mathematics that is actually
from 1984, see Block Designs
and the 2005 followup
The Eightfold Cube.)
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