Related illustration from a search in this journal for Wechsler —
|
|
Above: Dr. Harrison Pope, Harvard professor of psychiatry,
demonstrates the use of the Wechsler Adult Intelligence Scale (WAIS)
“block design” subtest.
— From a Log24 search for “Harrison Pope.”
A Midrash for Wikipedia
Midrash —
Related material —
________________________________________________________________________________
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
Fano-plane 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
3-space over the two-element 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 —
|
You, Xi-lin; Zhang, Peter. "Interality in Heidegger."
The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi. |
Some remarks from this journal on April 1, 2015 —
Manifest O
|
| 83-06-21 | An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | 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. |
The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."
Midrash from Philip Pullman . . .
"The 1929 Einstein-Carmichael Expedition"
I prefer the 1929 Emch-Carmichael expedition —
This is from . . .
“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
(Ch. 14 (pp. 693-746) of Handbook of Combinatorics,
Vol. I, MIT Press, 1995, edited by Ronald L. Graham,
Martin Grötschel, and László Lovász, Section 16 (p. 716))
From "Siri + Wechsler" in this journal —
|
For Little Man Tate —
Related material — Wechsler in this journal and
Mark and Lucille, Bill and Violet, Al and Regina,
|
Related material —
| Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
Another name for the Fano plane design — The Ghostly Hallows.
From a search in this journal for Ghostly —
“Perhaps only Shakespeare manages to create at the highest level
both images and people; and even Hamlet looks second-rate
compared with Lear .”
— Iris Murdoch, “Against Dryness,” 1961
Byline from a 2019 post — ‘GLOBE STAFF AND NEW SERVICES’ —
|
|
Above: Dr. Harrison Pope, Harvard professor of psychiatry,
demonstrates the use of the Wechsler Adult Intelligence Scale
“block design” subtest.
— From a Log24 search for “Harrison Pope.”
Related drama — Other posts tagged Plastic Elements.
Today’s earlier post “Binary Coordinates” discussed a Dec. 6
revision to the Wikipedia article on PG(3,2), the projective
geometry of 3 dimensions over the 2-element field GF(2).
The revision, which improved the article, was undone later today
by a clueless retired academic, one William “Bill” Cherowitzo,
a professor emeritus of mathematics at U. of Colorado at Denver.
(See his article “Adventures of a Mathematician in Wikipedia-land,”
MAA Focus , December 2014/January 2015.)
See my earlier remarks on this topic . . . specifically, on this passage —
“A 3-(16,4,1) block design has 140 blocks
of size 4 on 16 points, such that each triplet
of points is covered exactly once. Pick any
single point, take only the 35 blocks
containing that point, and delete that point.
The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points.”
As I noted on November 17, this is bullshit. Apparently Cherowitzo
never bothered to find out that an arbitrary “3-(16,4,1) block design”
(an example of a Steiner quadruple system ) does not yield a PG(3,2).
PG(3,2) is derived from the classical 3-(16,4,1) block design formed by the affine
space of 4 dimensions over GF(2). That design has 322,560 automorphisms.
In contrast, see a 3-(16,4,1) block design that is automorphism-free.
A related quotation:
“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
(Ch. 14 (pp. 693-746) of Handbook of Combinatorics,
Vol. I, MIT Press, 1995, edited by Ronald L. Graham,
Martin Grötschel, and László Lovász, Section 16 (p. 716))
See also the webpage Block Designs in Art and Mathematics
and Log24 posts tagged Plastic Elements.
From this journal on August 9, 2019 —
Perhaps not.
From an Instagram account, also on August 9, 2019 — (click to enlarge) —
The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.
A meditation, with apologies to Coleridge:
In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.
A related video —
Beware, beware, her flashing eyes, her floating hair:
Set design —
As opposed to block design —
For those who prefer greater clarity than is offered by Stevens . . .
The A section —
The B section —
"A paper from Helsinki in 2005 says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2)."
The exercise in the previous post was suggested by a passage
purporting to "use standard block design theory" that was written
by some anonymous author at Wikipedia on March 1, 2019:
Here "rm OR" apparently means "remove original research."
Before the March 1 revision . . .
The "original research" objected to and removed was the paragraph
beginning "To explain this further." That paragraph was put into the
article earlier on Feb. 28 by yet another anonymous author (not by me).
An account of my own (1976 and later) original research on this subject
is pictured below, in a note from Feb. 20, 1986 —
An image from a Log24 post of March 5, 2019 —
The following paragraph from the above image remains unchanged
as of this morning at Wikipedia:
"A 3-(16,4,1) block design has 140 blocks of size 4 on 16 points,
such that each triplet of points is covered exactly once. Pick any
single point, take only the 35 blocks containing that point, and
delete that point. The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points."
Exercise —
Prove or disprove the above assertion about a general "3-(16,4,1)
block design," a structure also known as a Steiner quadruple system
(as I pointed out in the March 5 post).
Relevant literature —
A paper from Helsinki in 2005* says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2),
from which PG(3,2) can be derived using the well-known process
from finite geometry described in the above Wikipedia paragraph.
* "The Steiner quadruple systems of order 16," by Kaski et al.,
Journal of Combinatorial Theory Series A Volume 113, Issue 8,
November 2006, pages 1764-1770.
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4-element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
WISC = Wechsler Intelligence Scale for Children
RISC = Reduced Instruction Set Computer or
Rust Inventory of Schizotypal Cognitions
See related material in earlier WISC RISC posts.
See also . . .
"Many parents ask us about the Block Design section
on the WISC and hope to purchase blocks and exercises
like those used on the WISC test. We explain that doing that
has the potential to invalidate their child's test results.
These Froebel Color Cubes will give you a tool to work with
your child on the skills tested for in the Block Design section
of the WISC in an ethical and appropriate way. These same
skills are applicable to any test of non-verbal reasoning like
the NNAT, Raven's or non-verbal sections of the CogAT or OLSAT. "
For a webpage that is perhaps un ethical and in appropriate,
see Block Designs in Art and Mathematics.
The New York Times online this evening —
"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."
Review —
|
Thursday, September 1, 2011
How It Works
|
See also 1984 Bricks in this journal.
Del Toro and the History of Mathematics ,
Or: Applied Bullshit Continues
For del Toro —
For the history of mathematics —
|
Thursday, September 1, 2011
How It Works
|
"William Christopher, best known for playing Father Mulcahy
on the hit sitcom M*A*S*H , died Saturday [Dec. 31, 2016] of
lung cancer, his agent confirmed to The Hollywood Reporter.
He was 84.
Christopher died at his home in Pasadena, with his wife by
his bedside, at 5:10 a.m. on New Year's Eve, according to a
statement from his agent."
— 5:59 PM PST 12/31/2016 by Meena Jang
Image reshown in this journal on the midnight (Eastern time)
preceding Christopher's death —
Related material —
From a Log24 search for "Deathly Hallows" —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
Those who prefer Latin with their theology
may search this journal for "In Nomine Patris."
The previous post, on the July 13 death of computer scientist Robert Fano,
suggests a review of "Deathly Hallows" posts in this journal. From that review —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
For further information, click the image below —
.
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 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field Ω = GF(8).
Related fiction: The Eight , by Katherine Neville.
Related non-fiction: 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— |
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. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | 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. |
The words: "symplectic polarity"—
The images:
The Natural Symplectic Polarity in PG(3,2)
Symmetry Invariance in a Diamond Ring
The Diamond-Theorem Correlation
Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).
My response —
Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid" …
… and remarks from a Log24 post of August 14, 2013 :
|
Norway dance (as interpreted by an American)
I prefer a different, Norwegian, interpretation of "the dance of four."
Related material: |
See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —
(Click image below to enlarge.)
Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :
From On Art and Magic (May 5, 2011) —
|
|
(Updated at about 7 PM ET on Dec. 3.)
This journal Tuesday, Oct. 28, 2014, at 5 PM ET:
"What is a tai chi master, and what is it that he unfolds?"
From an earlier post, Hamlet's father's ghost
on "the fretful porpentine":
Hamlet , Act 1, Scene 5 —
Ghost:
“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combinèd locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood."
Galway Kinnell:
"I roll
this way and that in the great bed, under
the quilt
that mimics this country of broken farms and woods"
— "The Porcupine"
For quilt-block designs that do not mimic farms or woods,
see the cover of Diamond Theory . See also the quotations
from Wallace Stevens linked to in the last line of yesterday's
post in memory of Kinnell.
"… a bee for the remembering of happiness" — Wallace Stevens
Or: The Confessions of Nat Tate
“A convincing lie is, in its own way, a tiny, perfect narrative.”
— William Boyd, “A Short History of the Short Story” (2006)
“A novel written in the first-person singular has certain powerful
narrative advantages, especially when it takes the form of a ‘confession.'”
— William Boyd, “Memoir of a Plagiarist” (1994)
From a Log24 post yesterday —
For Little Man Tate —
Related material — Wechsler in this journal and an earlier Siri Hustvedt
art novel, from 2003 —
Mark and Lucille, Bill and Violet, Al and Regina, etc., etc., etc. —
“Design is how it works.” — Steve Jobs
“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer (Ch. 14 (pp. 693-746),
Section 16 (p. 716) of Handbook of Combinatorics, Vol. I ,
MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel,
and László Lovász)
For some background on that Steiner system, see the footnote to
yesterday’s Christmas post.
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
|
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
(A prequel to Galois Field of Dreams)
The opening of Descartes' Dream ,
by Philip J. Davis and Reuben Hersh—
"The modern world,
our world of triumphant rationality,
began on November 10, 1619,
with a revelation and a nightmare."
For a revelation, see Battlefield Geometry.
For a nightmare, see Joyce's Nightmare.
Some later work of Descartes—
From "What Descartes knew of mathematics in 1628,"
by David Rabouin, CNRS-Univ. Paris Diderot,
Historia Mathematica , Volume 37, Issue 3,
Contexts, emergence and issues of Cartesian geometry,
August 2010, pages 428–459 —
Fig. 5. How to represent the difference between white, blue, and red
according to Rule XII [from Descartes, 1701, p. 34].
The 4×4 array of Descartes appears also in the Battlefield Geometry posts.
For its relevance to Galois's field of dreams, see (for instance) block designs.
This journal on June 18, 2008—
The Wechsler Cubes story continues with a paper from December 2009…
"Learning effects were assessed for the block design (BD) task,
on the basis of variation in 2 stimulus parameters:
perceptual cohesiveness (PC) and set size uncertainty (U)." —
(Click image for some background.)
The real intelligence test is, of course, the one Wechsler flunked—
investigating the properties of designs made with sixteen
of his cubes instead of nine.
"Design is how it works." — Steve Jobs
From a commercial test-prep firm in New York City—
From the date of the above uploading—
|
|
From a New Year's Day, 2012, weblog post in New Zealand—
From Arthur C. Clarke, an early version of his 2001 monolith—
"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."
The numerical (not crystal) pyramid above is related to a sort of
mathematical block design known as a Steiner system.
For its relationship to the graphic block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M24," which contains the following
version of the above numerical pyramid—
For graphic block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.
For the barbed tail of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.
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, 1984-2003."
“Design is how it works.” — Steven Jobs (See Symmetry and Design.)
“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
The name Carmichael is not to be found in Booher’s thesis. A book he does cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937. It should be dated 1931, as the following quotation shows—
From Log24 on Feb. 20, 2010—
“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”
– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240
Epigraph from Ch. 4 of Design Theory , Vol. I:
“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo XXXIX)
See also “Do you like apples?“
"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)
Today's American Mathematical Society home page—
Some related material—
The above Rowley paragraph in context (click to enlarge)—
"We employ Curtis's MOG …
both as our main descriptive device and
also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122
And the MOG incorporates the
Geometry of the 4×4 Square.
For this geometry's relation to "design"
in the graphic-arts sense, see
Block Designs in Art and Mathematics.
An article from cnet.com tonight —
For Jobs, design is about more than aesthetics
By: Jay Greene
… The look of the iPhone, defined by its seamless pane of glass, its chrome border, its perfect symmetry, sparked an avalanche of copycat devices that tried to mimic its aesthetic.
Virtually all of them failed. And the reason is that Jobs understood that design wasn't merely about what a product looks like. In a 2003 interview with the New York Times' Rob Walker detailing the genesis of the iPod, Jobs laid out his vision for product design.
''Most people make the mistake of thinking design is what it looks like,'' Jobs told Walker. "People think it's this veneer— that the designers are handed this box and told, 'Make it look good!' That's not what we think design is. It's not just what it looks like and feels like. Design is how it works.''
Related material: Open, Sesame Street (Aug. 19) continues… Brought to you by the number 24—
"By far the most important structure in design theory is the Steiner system
— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics , Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))
Two Blocks Short of a Design:
A sequel to this morning’s post on Douglas Hofstadter
Photo of Hofstadter by Mike McGrath taken May 13, 2006 |
Related material — See Lyche’s “Theme and Variations” in this journal
and Hofstadter’s “Variations on a Theme as the Essence of Imagination”
— Scientific American October 1982
A quotation from a 1985 book by Hofstadter—
“… we need to entice people with the beauties of clarity, simplicity, precision,
elegance, balance, symmetry, and so on.
Those artistic qualities… are the things that I have tried to explore and even
to celebrate in Metamagical Themas . (It is not for nothing that the word
‘magic’ appears inside the title!)”
The artistic qualities Hofstadter lists are best sought in mathematics, not in magic.
An example from Wikipedia —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
This morning's post, "Shining," gave James Hillman's 1976 remarks
on the distinction between soul and spirit .
The following images may help illustrate these concepts.
The distinction as illustrated by Jeff Bridges —
|
Soul
|
Spirit
|
The mirror has two faces (at least).
Postscript from a story, "The Zahir," in the Borges manner,
by Mark Jason Dominus (programmer of the quilt designs above)—
"I left that madhouse gratefully."
Dominus is also the author of…
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 finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element 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
** G20160 in Mitchell 1910, LF(3,22) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,22),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)
“… Kirkman has established an incontestable claim to be regarded as the founding father of the theory of designs.”
— “T.P. Kirkman, Mathematician,” by N.L. Biggs, Bulletin of the London Mathematical Society, Volume 13, Number 2 (March 1981), 97-120.
This paper is now available online for $12.
For more about this subject, see Design Theory, by Beth, Jungnickel, and Lenz, Cambridge U. Press, Volume I (2nd ed., 1999, 1120 pages) and Volume II (2nd ed., 2000, 513 pages).
For an apparently unrelated subject with the same name, see Graphic Design Theory: Readings from the Field, by Helen Armstrong (Princeton Architectural Press, 2009).
For what the two subjects have in common, see Block Designs in Art and Mathematics.
"Have your people
call my people."
— George Carlin
Diamond life, lover boy;
we move in space
with minimum waste
and maximum joy.
— Sade, quoted here on
Lincoln's Birthday, 2003
This is perhaps suitable
for the soundtrack of
the film "Blockheads"
(currently in development)–
Related material from Wikipedia:
"Uta Frith, in her book Autism: Explaining the Enigma,[5] addresses the superior performance of autistic individuals on the block design [link not in Wikipedia] test. This was also addressed in [an] earlier paper.[6] A particularly interesting article demonstrates the differences in construction time in the performance of the block design task by Asperger syndrome individuals and non-Asperger's individuals. An essential point here is that in an unsegmented version of the task, Asperger's individuals performed dramatically faster than non-Asperger's individuals: [7]."
5. Frith, Uta (2003). Autism: explaining the enigma (2nd ed. ). Cambridge, MA: Blackwell Pub. ISBN 0-631-22901-9.
6. Shah A, Frith U (Nov 1993). "Why do autistic individuals show superior performance on the block design task?". J Child Psychol Psychiatry 34 (8): 1351–64. PMID 8294523.
7. Caron MJ, Mottron L, Berthiaume C, Dawson M (Jul 2006). "Cognitive mechanisms, specificity and neural underpinnings of visuospatial peaks in autism". Brain 129 (Pt 7): 1789–802. doi:. PMID 16597652. "Fig 3".
Related material from a film (see Calvinist Epiphany, June 17):
For the relevance of this maxim to autism, see Markoff Process (March 4, 2009).
"By far the most important structure in design theory is the Steiner system
— "Block Designs," 1995, by Andries E. Brouwer
"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."
— The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)
"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."
The 1931 paper of Carmichael is now available online from the publisher for $10.
The Kohs Block Design
Intelligence Test
Samuel Calmin Kohs, the designer (but not the originator) of the above intelligence test, would likely disapprove of the "Aryan Youth types" mentioned in passing by a film reviewer in today's New York Times. (See below.) The Aryan Youth would also likely disapprove of Dr. Kohs.
1. Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…
Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.
2. Wechsler Cubes of a different sort (Log24, May 25, 2008)
3. Manohla Dargis in today's New York Times:
"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….
In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."
"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."
— Interview with Ken Jacobs, UC Berkeley, October 1999
5. For Dargis's "Aryan Youth types"–
From a Manohla Dargis
New York Times film review
of April 4, 2007
(Spy Wednesday) —
See also, from August 1, 2008
(anniversary of Hitler's
opening the 1936 Olympics) —
For Sarah Silverman —
and the 9/9/03 entry
Doonesbury,
August 21-22, 2008:
| CHANGE FEW CAN BELIEVE IN |
Page 67 —
“… Bill and Violet were married. The wedding was held in the Bowery loft on June 16th, the same day Joyce’s Jewish Ulysses had wandered around Dublin. A few minutes before the exchange of vows, I noted that Violet’s last name, Blom, was only an o away from Bloom, and that meaningless link led me to reflect on Bill’s name, Wechsler, which carries the German root for change, changing, and making change. Blooming and changing, I thought.”
For Hustvedt’s discussion of Wechsler’s art– sculptured cubes, which she calls “tightly orchestrated semantic bombs” (p. 169)– see Log24, May 25, 2008.
(after David Wechsler,
1896-1981, chief
psychologist at Bellevue)
These cubes are used to
make 3×3 patterns for
psychological testing.
Related 3×3 patterns appear
in “nine-patch” quilt blocks
and in the following–
| Don Park at docuverse.com, Jan. 19, 2007: “How to draw an Identicon
A 9-block is a small quilt using only 3 types of patches, out of 16 available, in 9 positions. Using the identicon code, 3 patches are selected: one for center position, one for 4 sides, and one for 4 corners. Positions and Rotations For center position, only a symmetric patch is selected (patch 1, 5, 9, and 16). For corner and side positions, patch is rotated by 90 degree moving clock-wise starting from top-left position and top position respectively.” |
| From a weblog by Scott Sherrill-Mix:
“… Don Park came up with the original idea for representing users with geometric shapes….” Claire | 20-Dec-07 at 9:35 pm | Permalink “This reminds me of a flash demo by Jarred Tarbell |
| Jared Tarbell at levitated.net, May 15, 2002:
“The nine block is a common design pattern among quilters. Its construction methods and primitive building shapes are simple, yet produce millions of interesting variations.  Figure A. Four 9 block patterns,
arbitrarily assembled, show the grid composition of the block. Each block is composed of 9 squares, arranged in a 3 x 3 grid. Each square is composed of one of 16 primitive shapes. Shapes are arranged such that the block is radially symmetric. Color is modified and assigned arbitrarily to each new block.
The basic building blocks of the nine block are limited to 16 unique geometric shapes. Each shape is allowed to rotate in 90 degree increments. Only 4 shapes are allowed in the center position to maintain radial symmetry.
 Figure B. The 16 possible shapes allowed |
Such designs become of mathematical interest when their size is increased slightly, from square arrays of nine blocks to square arrays of sixteen. See Block Designs in Art and Mathematics.
(This entry was suggested by examples of 4×4 Identicons in use at Secret Blogging Seminar.)
Hermann Hesse's 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–
"For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar's Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors."
From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–
A description of the work of Bill Wechsler, a fictional artist:
"Bill worked long hours on a series of autonomous pieces about numbers. Like O's Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has
— End of page 168 —
opened the door to his cell. The hidden rhyme is "free." Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn't visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe's color wheel and in Alfred Jensen's use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn't bother with the poet's meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.
The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page."
|
From 2002:
Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest. |
|
ZZ
Figures from the
Poem by Eugen Jost:
Mit Zeichen und Zahlen
Numbers and Names,
With numbers and names English translation A related poem:
Alphabets
From time to time
But if a savage
— Hermann Hesse (1943), |
"… 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 two-element field).
(To see how the display works,
try the Kaleidoscope Puzzle first.)
This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.
"The historical road
from the Platonic solids
to the finite simple groups
is well known."
— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture
"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of M24):
"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix
— Op. cit., p. 719
Finite Geometry of
the Square and Cube
and
Jewel in the Crown
"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
— H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth
Those who prefer stories to truth
may consult the Log24 entries
of March 1, 2, 3, 4, and 5.
They may also consult
the poet Rubén Darío:
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
In memory of
Rudolf Arnheim,
who died on
Saturday, June 9
From the Wikipedia article on Gestalt psychology prior to its modification on May 31, 2007:
“Emergence, reification, multistability, and invariance are not separable modules to be modeled individually, but they are different aspects of a single unified dynamic mechanism.
For a mathematical example of such a mechanism using the cubes of psychologists’ block design tests, see Block Designs in Art and Mathematics and The Kaleidoscope Puzzle.”
The second paragraph of the above passage refers to my own work.
Some Gestalt-related work of Arnheim:
Time of this entry:
1:06:18 AM ET.
Blitz by anonymous
New Delhi user
From Wikipedia on 31 May, 2007:
Shown below is a list of 25 alterations to Wikipedia math articles made today by user 122.163.102.246.
All of the alterations involve removal of links placed by user Cullinane (myself).
The 122.163… IP address is from an internet service provider in New Delhi, India.
The New Delhi anonymous user was apparently inspired by an earlier blitz by Wikipedia administrator Charles Matthews. (See User talk: Cullinane.)
Related material:
Ashay Dharwadker and Usenet Postings
and Talk: Four color theorem/Archive 2.
See also some recent comments from 122.163…
at Talk: Four color theorem.
May 31, 2007, alterations by
user 122.163.102.246:
The deletions should please Charles Matthews and fans of Ashay Dharwadker’s work as a four-color theorem enthusiast and as editor of the Open Directory sections on combinatorics and on graph theory.
There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.
— Cullinane 23:28, 31 May 2007 (UTC)
Serious
"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."
— Charles Matthews at Wikipedia, Oct. 2, 2006
"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."
— G. H. Hardy, A Mathematician's Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
Today’s birthday:
Tom Hanks, star of
“The Da Vinci Code”
Part III:
The Leonardo Connection
Part IV:
Nicholson’s Grail Quest
— Ben Nicholson in a 2005 interview
Nicholson’s quest has apparently lasted for some time. Promotional material for a 1996 Nicholson exhibit in Montreal says it “invites visitors of all ages to experience a contemporary architect’s search for order, meaning and logic in a world of art, science and mystery.” The title of that exhibit was “Uncovering Geometry.”
For web pages to which this same title might apply, see Quilt Geometry, Galois Geometry, and Finite Geometry of the Square and Cube.
* “Square Kufi” calligraphy is used in Islamic architectural ornament. I do not know what, if anything, is signified by Nicholson’s 6×12 example of “Kufi blocks” shown above.
101 101
— Ben Macintyre,
The London Times, June 4:
When Rimbaud Meets Rambo
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 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices. This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices. Such translations in the affine 3-space 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 Finite-Geometry Models.
For another application of the points-as-partitions technique, see Latin-Square 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.
God is in…
The Details
From an entry for Aug. 19, 2003 on
conciseness, simplicity, and objectivity:
Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest. Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life.
For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at |
On Harvard and psychiatry: see
The Crimson Passion:
A Drama at Mardi Gras
(February 24, 2004)
This is a reductio ad absurdum of the Harvard philosophy so eloquently described by Alston Chase in his study of Harvard and the making of the Unabomber, Ted Kaczynski. Kaczynski's time at Harvard overlapped slightly with mine, so I may have seen him in Cambridge at some point. Chase writes that at Harvard, the Unabomber "absorbed the message of positivism, which demanded value-neutral reasoning and preached that (as Kaczynski would later express it in his journal) 'there is no logical justification for morality.'" I was less impressed by Harvard positivism, although I did benefit from a course in symbolic logic from Quine. At that time– the early 60's– little remained at Harvard of what Robert Stone has called "our secret culture," that of the founding Puritans– exemplified by Cotton and Increase Mather.
From Robert Stone, A Flag for Sunrise:
"Our secret culture is as frivolous as a willow on a tombstone. It's a wonderful thing– or it was. It was strong and dreadful, it was majestic and ruthless. It was a stranger to pity. And it's not for sale, ladies and gentlemen."
Some traces of that culture:
|
A web page
|
A contemporary
Click on pictures for details. |
A more appealing view of faith was offered by PBS on Wednesday night, the beginning of this year's High Holy Days:
|
Armand Nicholi: But how can you believe something that you don't think is true, I mean, certainly, an intelligent person can't embrace something that they don't think is true — that there's something about us that would object to that.
Jeremy Fraiberg: Well, the answer is, they probably do believe it's true. Armand Nicholi: But how do they get there? See, that's why both Freud and Lewis was very interested in that one basic question. Is there an intelligence beyond the universe? And how do we answer that question? And how do we arrive at the answer of that question? Michael Shermer: Well, in a way this is an empirical question, right? Either there is or there isn't. Armand Nicholi: Exactly. Michael Shermer: And either we can figure it out or we can't, and therefore, you just take the leap of faith or you don't. Armand Nicholi: Yeah, now how can we figure it out? Winifred Gallagher: I think something that was perhaps not as common in their day as is common now — this idea that we're acting as if belief and unbelief were two really radically black and white different things, and I think for most people, there's a very — it's a very fuzzy line, so that — Margaret Klenck: It's always a struggle. Winifred Gallagher: Rather than — I think there's some days I believe, and some days I don't believe so much, or maybe some days I don't believe at all. Doug Holladay: Some hours. Winifred Gallagher: It's a, it's a process. And I think for me the big developmental step in my spiritual life was that — in some way that I can't understand or explain that God is right here right now all the time, everywhere. Armand Nicholi: How do you experience that? Winifred Gallagher: I experience it through a glass darkly, I experience it in little bursts. I think my understanding of it is that it's, it's always true, and sometimes I can see it and sometimes I can't. Or sometimes I remember that it's true, and then everything is in Technicolor. And then most of the time it's not, and I have to go on faith until the next time I can perhaps see it again. I think of a divine reality, an ultimate reality, uh, would be my definition of God. |
|
|
|
Gallagher seemed to be the only participant in the PBS discussion that came close to the Montessori ideals of conciseness, simplicity, and objectivity. Dr. Montessori intended these as ideals for teachers, but they seem also to be excellent religious values. Just as the willow-tombstone seems suited to Geoffrey Hill's style, the Pythagorean sangaku pictured above seems appropriate to the admirable Gallagher.
Ideas and Art
— Motto of
Plato's Academy
|
From Minimalist Fantasies,
All I want anyone to get out of my paintings, and all I ever get out of them, is the fact that you can see the whole idea without any confusion. … What you see is what you see.
Minimal Art remains too much a feat of ideation, and not enough anything else. Its idea remains an idea, something deduced instead of felt and discovered.
The artists even questioned whether art needed to be a tangible object. Minimalism … Conceptualism — suddenly art could be nothing more than an idea, a thought on a piece of paper….
There was a period, a decade or two ago, when you could hardly open an art journal without encountering the quotation from Frank Stella I used as an epigraph. The bit about “what you see is what you see” was reproduced ad nauseam. It was thought by some to be very deep. In fact, Stella’s remarks—from a joint interview with him and Donald Judd—serve chiefly to underscore the artistic emptiness of the whole project of minimalism. No one can argue with the proposition that “what you see is what you see,” but there’s a lot to argue with in what he calls “the fact that you can see the whole idea without any confusion.” We do not, of course, see ideas. Stella’s assertion to the contrary might be an instance of verbal carelessness, but it is not merely verbal carelessness. At the center of minimalism, as Clement Greenberg noted, is the triumph of ideation over feeling and perception, over aesthetics. |
|
From How Not Much Is a Whole World, Decades on, it's curious how much Minimalism, the last great high modern movement, still troubles people who just can't see why … a plain white canvas with a line painted across it
should be considered art. That line might as well be in the sand: on this side is art, it implies. Go ahead. Cross it. …. The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated, either. Philosophical ambition and formal modesty still constitute Minimalism's bottom line. If what results can sometimes be more fodder for the brain than exciting to look at, it can also have a serene and exalted eloquence…. That line in the sand doesn't separate good art from bad, or art from nonart, but a wide world from an even wider one. |
I maintain that of course
we can see ideas.
Example: the idea of
invariant structure.
"What modern painters
are trying to do,
if they only knew it,
is paint invariants."
— James J. Gibson, Leonardo,
Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978
For a discussion
of how this works, see
Block Designs,
4×4 Geometry, and
Diamond Theory.
Incidentally, structures like the one shown above are invariant under an important subgroup of the affine group AGL(4,2)… That is to say, they are not lost in translation. (See previous entry.)
Modernism as a Religion
In light of the controversy over Mel Gibson's bloody passion play that opens today, some more restrained theological remarks seem in order. Fortunately, Yale University Press has provided a
From a review by Adam White Scoville of Iain Pears's novel titled An Instance of the Fingerpost:
"Perhaps we are meant to see the story as a cubist retelling of the crucifixion, as Pilate, Barabbas, Caiaphas, and Mary Magdalene might have told it. If so, it is sublimely done so that the realization gradually and unexpectedly dawns upon the reader. The title, taken from Sir Francis Bacon, suggests that at certain times, 'understanding stands suspended' and in that moment of clarity (somewhat like Wordsworth's 'spots of time,' I think), the answer will become apparent as if a fingerpost were pointing at the way."
Recommended related
By others:
Inside Modernism: Relativity Theory, Cubism, Narrative, Thomas Vargish and Delo E. Mook, Yale University Press, 1999
Signifying Nothing: The Fourth Dimension in Modernist Art and Literature
Corpus Hypercubus,
by Dali. Not cubist,
perhaps "hypercubist."
By myself:
The Crucifixion of John O'Hara
The Da Vinci Code and Symbology at Harvard
Material that is related, though not
Diamonds and Whirls
New applets have rotating 3D versions of the diamond and whirl cubes in Block Designs.
Gestalt Update
Updated Block Designs page with material on Gestalt aesthetics and the work of James J. Gibson.
Intelligence Test
From my August 31, 2002, entry quoting Dr. Maria Montessori on conciseness, simplicity, and objectivity:
|
|
Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.
Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life.
|
Pope |
Nicholi |
|
Old |
New |
For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at
For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at
The Institute for Advanced Study.
For the meaning of life, see
The Gospel According to Jill St. John,
whose birthday is today.
The Triangular God
From the New York Times of June 10, 2003:
As Spinoza noted, “If a triangle could speak, it would say… that God is eminently triangular.”
— “Giving God a Break,” by Nicholas D. Kristof
Related material:
The figure above is by
Robert Anton Wilson.
From “The Cocktail Party,” Act One, Scene One, by T. S. Eliot:
UNIDENTIFIED GUEST [Sings]:
Tooryooly toory-iley
What’s the matter with One Eyed Riley?[Exit.]
JULIA: Edward, who is that dreadful man?
From T. S. Eliot, The Complete Poems and Plays, 1909-1950 (Harcourt, Brace and Company, 1952), page 144:
“The end is where we start from.”
From the end of that same book:
“And me be-in’ the One-Eyed Ri-ley”
For more on this song, see
Reilly’s Daughter (with midi tune),
See also my previous journal entry of
and the perceptive analysis of the Shakti-Shiva symbol that I quoted on May 25, 2003.
Here is a note from Sept. 15, 1984, for those who would like to
block that metaphor.
See also Block Designs from the Cabinet of Dr. Montessori and Sacerdotal Jargon.
Mental Health Month:
Springtime for Wagner
“And now what you’ve all been waiting for…
— Colin Hay as Zac in the film “Cosi“
“When I sought those who would sympathize with my plans, I had only you, the friends of my particular art, my most personal work and creation, to turn to.”
— Wagner’s address at the ceremony for the laying of the foundation stone of the Festival Theater in Bayreuth, May 22 (Wagner’s birthday), 1872
“The new computer package DISCRETA which was created in Bayreuth is in the process of permanent development.”
— “A Computer Approach to the Enumeration of Block Designs Which Are Invariant With Respect to a Prescribed Permutation Group”
The above is a preprint from Dresden.
See, too, the work of Bierbrauer, who received his doctorate at Mainz in 1977 and taught at Heidelberg from 1977 to 1994. Bierbrauer’s lecture notes give a particularly good background for the concepts involved in my Diamond Theory, in the tradition of Witt and Artin. See
Introduction to Group Theory
and Applications,
by Jürgen Bierbrauer, 138 pp., PostScript
THE MONTESSORI METHOD: CHAPTER VI
“Let all thy words be counted.”
Dante, Inf., canto X.
CONCISENESS, SIMPLICITY, OBJECTIVITY.
…Dante gives excellent advice to teachers when he says, “Let thy words be counted.” The more carefully we cut away useless words, the more perfect will become the lesson….
Another characteristic quality of the lesson… is its simplicity. It must be stripped of all that is not absolute truth…. The carefully chosen words must be the most simple it is possible to find, and must refer to the truth.
The third quality of the lesson is its objectivity. The lesson must be presented in such a way that the personality of the teacher shall disappear. There shall remain in evidence only the object to which she wishes to call the attention of the child….
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Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale “block design” subtest.
Mathematicians mean something different by the phrase “block design.”
A University of London site on mathematical design theory includes a link to my diamond theory site, which discusses the mathematics of the sorts of visual designs that Professor Pope is demonstrating. For an introduction to the subject that is, I hope, concise, simple, and objective, see my diamond 16 puzzle.
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Initial Xanga entry. Updated Nov. 18, 2006.
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