^{Typographical:} »
Eightfold Cube:
"… 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 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 —
Three links with a Borges flavor—
Related material
The 236 in yesterday evening's NY lottery may be
viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.
A new illustration for that topic—
This shows a reconcilation of the triples described by Sloane
in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.
Note that the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of Sloane's 8_{3} configuration—
The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.
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.
Continued from yesterday evening
Today's mathematical birthday —
Claude Chevalley, 11 Feb. 1909 – 28 June 1984.
Chevalley's daughter, Catherine Chevalley, wrote about For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged. The way my father worked, it seems that this was what counted most, this production of an object which then became inert— dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons. 
Recent scholarly news suggests a search for Chapel Hill
in this journal. That search leads to Transformative Hermeneutics.
Those who, like Professor Eucalyptus of Wallace Stevens's
New Haven, seek God "in the object itself" may contemplate
yesterday's afternoon post on Eightfold Design in light of the
Transformative post and of yesterday's New Haven remarks and
Chapel Hill events.
… 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 new Log24 tag "Eightfold Metaphysics" used in the previous post
suggests a review of posts that were tagged "The Reality Blocks" on May 24.
Then there is, of course, the May 24 death of Murray GellMann, who
hijacked from Buddhism the phrase "eightfold way."
See GellMann in this journal and May 24, 2003.
Related material on automorphism groups —
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order168 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 eightcube mathematical structure above
than they are an eightcube 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."
"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 —
Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:
A more interesting account of these 35 structures:
"It is commonly known that there is a bijection between
the 35 unordered triples of a 7set [i.e., the 35 partitions
of an 8set into two 4sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.
For some context, see Eightfold Geometry by Steven H. Cullinane.
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— 
Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —
For Tschira in this journal, see Stiftung .
For some Würfel illustrations, see this morning's post
Manifest O. A related webpage —
The title was suggested by
http://benmarcus.com/smallwork/manifesto/.
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —

An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001  Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. 
831016  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915  Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. 
"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.
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
Verbum theme:
A search today for Verbum in this journal yielded
a Georgetown University Chomskyite, Professor
David W. Lightfoot.
"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."
Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.
See Tecumseh Fitch in this journal.
For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.
That birthday was also the upload date for the following:
See esp. the comments section.
The second Logos figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—
Click for further details.
…. 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.
R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott's remarks.
For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 19842003."
"The space in which a film takes place"—
See Eightfold Geometry, linked to here on the date of Boyle's death.
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 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 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's Pennsylvania lottery numbers suggest the following meditations…
Midday: Lot 497, Bloomsbury Auctions May 15, 2008– Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of SpaceTime."*
Evening: 5/07, 2008, in this journal– "Forms of the Rock."
Related material:
A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009.
The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space."
A related quotation by Lefebvre (cf. 2/22 2009):
"… an epochmaking event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered… the space… of classical perspective and geometry…."
— Page 25 of The Production of Space (Blackwell Publishing, 1991)
This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910.
In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3space PG(3, 2) and Its Group." This paper, while perhaps neither epochmaking nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.†
* Ending on Stephen King's birthday, 2008
† Mardi Gras
Wittgenstein's Lasting Significance, edited by Max Kölbel and Bernhard Weiss, published by Routledge, 2004–
Page 168:
"Wittgenstein told Norman Malcolm that 'a serious and good philosophical work could be written that would consist entirely of jokes (without being facetious)' (Malcolm 1999: 64)."
Malcolm, N. (1999) "Wittgenstein: A Memoir," in F.A. Flowers (ed.) Portraits of Wittgenstein, vol. 3, Bristol: Thoemmes Press, pp. 60112
The lasting significance here is perhaps in the page numbers.
Design at Harvard:
Natural or Unnatural?
From the Harvard Graduate School of Design–
Call for Entries: The Space of Representation
DEADLINE FEBRUARY 27, 2009 8PM EST
"According to Henri Lefebvre, the social production of space has three components: spatial practice, the representation of space, and the space of representation. The latter two are integral to both design and the review process."
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 
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:

"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.
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.)
The title is from Bachelard.
I prefer Stevens:
The rock is the habitation of the whole, Its strength and measure, that which is near, point A In a perspective that begins again At B: the origin of the mango's rind. It is the rock where tranquil must adduce Its tranquil self, the main of things, the mind, The starting point of the human and the end, That in which space itself is contained, the gate To the enclosure, day, the things illumined By day, night and that which night illumines, Night and its midnightminting fragrances, Night's hymn of the rock, as in a vivid sleep.
— Wallace Stevens,
"The Rock," 1954
Joan Ockman in Harvard Design Magazine (Fall 1998):
"'We are far removed from any reference to simple geometrical forms,' Bachelard wrote…."
No, we are not. See Log24, Christmas 2005:
More on Bachelard from Harvard Design Magazine:
"The project of discerning a loi des quatre éléments would preoccupy him until his death…."
For such a loi, see Theme and Variations 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.)
or, The Eightfold Cube
Every permutation of the plane's points that preserves collinearity is a symmetry of the plane. The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)
The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle. It does not, however, indicate where the other 162 symmetries come from.
Shown below is a new model of this same projective plane, using partitions of cubes to represent points:
The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.
(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations. But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results. This illustrates the difference between affine and projective spaces over the binary field GF(2). In a related 2x2x2 cubic model of the affine 3space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cubeslices. This is clearly a subgroup of the group generated by permuting 1x1x2 cubeslices. Such translations in the affine 3space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)
To view the cubes model in a wider context, see Galois Geometry, Block Designs, and FiniteGeometry Models.
For another application of the pointsaspartitions technique, see LatinSquare Geometry: Orthogonal Latin Squares as Skew Lines.
For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way. For more on the mathematics of cubic models, see Solomon's Cube.


Example:





Initial Xanga entry. Updated Nov. 18, 2006.
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