At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, but
rather at the ends of a cube's diagonal axis of symmetry.
See some related illustrations below.
Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.
The exceptional role of 0 and ∞in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
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.
In a small bedroom to Foredragssalen populate
Josefine Lyche exhibition with a group sculptures
that are part of the work group 4D Ambassador
(2014-2015). Together they form an installation
where she uses light to amplify the feeling of
stepping into a new dimension, for which the title
suggests, this "ambassadors" for a dimension we
normally do not have access to. "Ambassadors"
physical forms presents nonphysical phenomena.
Lyches works have in recent years been placed
in something one might call an "esoteric direction"
in contemporary art, and defines itself this
sculpture group humorous as "glam-minimalist."
She has in many of his works returned to basic
geometric shapes, with hints to the occult,
"new space-age", mathematics and where
everything in between.
" 'The jury is still out on how long – and whether – people are actually
going to understand this.' It took the world 150 years to realize
the true power of the printing press . . . ." — Cade Metz
"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig
An earlier presentation of the above
seven partitions of the eightfold cube:
Related mathematics:
The use of binary coordinate systems
as a conceptual tool
Natural physical transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic transformations of vector spaces over GF(2).
This was apparently not previously known.
The colors surrounding Watson's body in the above
"bandeau" photo suggest a review. A search in this journal
for Green+Orange+Black yields . . .
In the above image, the "hard core of objectivity" is represented
by the green-and-white eightfold cube. The orange and black are,
of course, the Princeton colors.
Art is magic delivered from
the lie of being truth.
— Theodor Adorno, Minima moralia,
London, New Left Books, 1974, p. 222
(First published in German in 1951.)
The director, Carol Reed, makes…
impeccable use of the beauty of black….
— V. B. Daniel on The Third Man
I see your ironical smile.
— Hans Reichenbach
Adorno, The Third Man, and Reichenbach
are illustrated below (l. to r.) above the names of
cities with which they are associated.
Identical copies of the above image are being offered for sale
on three websites as representing a Masonic "cubic stone."
None of the three sites say where, exactly, the image originated.
Image searches for "Masonic stone," "Masonic cube," etc.,
fail to yield any other pictures that look like the above image —
that of a 2x2x2 array of eight identical subcubes.
For purely mathematical — not Masonic — properties of such
an array, see "eightfold cube" in this journal.
The websites offering to sell the questionable image —
Metaphysics does have a catalytic effect, which has been described in a very beautiful text by the mathematician André Weil:
Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another.4
André Weil cuts to the quick here: he conjures these 'murky reflections', these 'furtive caresses', the 'theory of Galois that Lagrange touches … with his finger through a screen that he does not manage to pierce.' He is a connoisseur of these metaphysical 'fogs' whose dissipation at one point heralds their reforming at another. It would be better to talk here of a horizon that tilts thereby revealing a new space of gestures which has not as yet been elucidated and cut out as structure.
4 A. Weil, 'De la métaphysique aux mathématiques', (Oeuvres, vol. II, p. 408.)
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012, note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
These references will not appeal to those who enjoy modernism as a religion.
(For such a view, see Rosalind Krauss on grids and another writer's remarks
on the religion's 100th anniversary this year.)
Some related nihilist philosophy from Cormac McCarthy —
"Forms turning in a nameless void."
Comments Off on “Was ist Raum?” — Bauhaus Founder Walter Gropius
Exercise: Explain why the lead article in the November issue of Notices of the American Mathematical Society misquotes Weyl
and gives the misleading impression that the example above,
the eightfold cube , might be part of the mathematical pursuit
known as geometric group theory .
"… 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."
"It’s important, as art historian Reinhard Spieler has noted,
that after a brief, unproductive stay in Paris, circa 1907,
Kandinsky chose to paint in Munich. That’s where he formed
the Expressionist art group Der Blaue Reiter (The Blue Rider) —
and where he avoided having to deal with cubism."
"Denn um zu wiederholen, was ich anfangs sagte:
in dem Geheimnis der Einheit von Ich und Welt,
Sein und Geschehen, in der Durchschauung des
scheinbar Objectiven und Akzidentellen als
Veranstaltung der Seele glaube ich den innersten Kern
der analytischen Lehre zu erkennen." (GW IX 488)
An Einheit-Geheimnis that is perhaps* more closely related
to pure mathematics** —
"What is the nature of the original unity
that throws itself apart in this separation,
and in what sense are the separated ones
here as the essence of the abyss?
Here it cannot be a question of any kind of 'dialectic,'
but only of the essence of the ground
(that is, of truth) itself." [Tr. by Google]
" Welcher Art ist die ursprüngliche Einheit,
daß sie sich in diese Scheidung auseinanderwirft,
und in welchem Sinn sind die Geschiedenen
hier als Wesung der Ab-gründigkeit gerade einig?
Hier kann es sich nicht um irgend eine »Dialektik«
handeln, sondern nur um die Wesung des Grundes
(der Wahrheit also) selbst."
Exercise: Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points ∞, 0, 1, 2, 3 … 22 of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on
cube-part permutations by the Mathieu group M24.
Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.
The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.
"… 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 2x2x2eightfold cube ."
— Steven H. Cullinane on April 19, 2016 — The Folding.
Hurt’s dies natalis (date of death, in the saints’ sense) was,
it now seems, 25January 2017, not 27.
A connection, for fantasy fans, between the Philosopher’s Stone
(represented by the eightfold cube) and the Deathly Hallows
(represented by the usual Fano-plane figure) —
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Essentially the same figure as above appears also in the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?” Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91. JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
"One day — 'I don’t know exactly why,' he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.
Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical: a solid, static object that is also fluid…." — Alexandra Alter
Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
Comments Off on “The Ultimate Epistemological Fact”
“I held him in the highest respect and was delighted
to find him living in a room above mine on the same
staircase. I frequently met him walking up or down
the stairs, but I was too shy to start a conversation.”
Frank Close on Ron Shaw:
“Shaw arrived there in 1949 and moved into room K9,
overlooking Jesus Lane. There is nothing particularly
special about this room other than the coincidence that
its previous occupant was Freeman Dyson.”
If Minimalist artist Donald Judd is known as a writer at all, it’s likely for one important text— his 1965 essay “Specific Objects,” in which he observed the rise of a new kind of art that collapsed divisions between painting, sculpture, and other mediums. But Judd was a prolific critic, penning shrewd reviews for various publications throughout his career—including ARTnews . With a Judd retrospective going on view this Sunday at the Museum of Modern Art in New York, ARTnews asked New York Times co-chief art critic Roberta Smith— who, early in her career, worked for Judd as his assistant— to comment on a few of Judd’s ARTnews reviews. How would she describe his critical style? “In a word,” she said, “great.” . . . .
And then there is Temple Eight, or Ex Fano Apollinis —
He reached Delos. There one night he secretly 46
carried off, from the much-revered sanctuary of
Apollo, several ancient and beautiful statues, and
had them put on board his own transport. Next
day, when the inhabitants of Delos saw their sanc-
tuary stripped of its treasures, they were much
distressed . . . .
Delum venit. Ibi ex fano Apollinis religiosissimo
noctu clam sustulit signa pulcherrima atque anti-
quissima, eaque in onerariam navem suam conicienda
curavit. Postridie cum fanum spoliatum viderent ii
qui Delum incolebant, graviter ferebant . . . .
"Although art is fundamentally everywhere and always the same,
nevertheless two main human inclinations, diametrically opposed
to each other, appear in its many and varied expressions. ….
The first aims at representing reality objectively, the second subjectively."
A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :
The Curtis octads are related to symmetries of the square.
Aitchison's octads are instead related to symmetries of the cube.
Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .
" Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view:
' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' "
13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010).
(Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010)
"The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' 1), writing (of which 'spacing' is said to be 'the fundamental property' 2) and deconstruction (with one of Derrida's last major texts, Le Toucher: Jean-Luc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' 3)."
1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D .
2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW .
3 Jacques Derrida, Le Toucher, Jean-Luc Nancy (Paris: Galilée, 2000), p. 207.
. . . .
"… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,16 we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…."
16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207-227.
. . . .
… "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'in-between' more 'originary' than the play of differences it in-forms. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a pre-ontological 'opening' – or, shall we say, 'in-between.'
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices(espacements ) between
the blocks. Such group actions provide an elementary picture of the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some …
Einstein, "Geometry and Experience," lecture before the
Prussian Academy of Sciences, January 27, 1921–
… This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics.
Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. The very word geometry, which, of course, means earth-measuring, proves this. For earth-measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies.
Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. ….
Later in the same lecture, Einstein discusses "the theory of a finite
universe." Of course he is not using "finite" in the sense of the field
of mathematics known as "finite geometry " — geometry with only finitely
many points.
Nevertheless, his remarks seem relevant to the Fano plane , an
axiomatically defined entity from finite geometry, and the eightfold cube,
a physical object embodying the properties of the Fano plane.
I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed.
First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe.
Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart …
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research …
Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience.
The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is being, after all,
beyond the arbitrary flux of existence."
Neither does the above materialization of 281 as the page
number of her philosophical remark.
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them.
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.
Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure above
than they are an eight-cube mechanical puzzle like the one below.
Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."
“It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde understood, they
were not only theories but … hot news,
straight from the studios, from the scene.”
* See the term interality in this journal.
For many synonyms, see
“The Human Seriousness of Interality,”
by Peter Zhang, Grand Valley State University, China Media Research 11(2), 2015, 93-103.
"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."
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.
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 title refers to the Chinese book the I Ching ,
the Classicof 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.
"Clearly, there is a spirit of openhandedness in post-conceptual art
uses of the term 'Conceptualism.' We can now endow it with a
capital letter because it has grown in scale from its initial designation
of an avant-garde grouping, or various groups in various places, and
has evolved in two further phases. It became something like a movement,
on par with and evolving at the same time as Minimalism. Thus the sense
it has in a book such as Tony Godfrey’s Conceptual Art. … Beyond that,
it has in recent years spread to become a tendency, a resonance within
art practice that is nearly ubiquitous." — Terry Smith, 2011
"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."
"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."
“The man who lives in contact with what he believes to be a living Church
is a man always expecting to meet Plato and Shakespeare to-morrow
at breakfast.”
"When Analogies Fail," by Alexander Stern,
a doctoral candidate in philosophy at Notre Dame, in The Chronicle of Higher Education online September 11, 2016.
(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 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 2x2x2eightfold 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 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:
"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."
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, 1984-2003."
Click image above for a view of pages 50-51 of a new KUNSTforum
article showing two photos relevant to my own work — those labeled
"after S. H. Cullinane."
(The phrase "den pensjonerte Oxford-professoren 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."
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 real-world physical object?
2. Which groups have actually been represented as the group of symmetries of some real-world 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 7-point 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
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