Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:
Hudson in 1905 —
Golomb in 1967 —
It is also likely that some figures illustrating Walsh functions as
two-color square arrays were published prior to 1976.
Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345-353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M 24 ,"
arXiv.org > hep-th > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
|
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array |
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
By Rahem D. Hamid, Harvard Crimson Staff Writer
Wednesday, November 08, 2023 at 12:44 am ET
Harvard Dean of Science Christopher W. Stubbs is stepping down
at the end of the academic year, Faculty of Arts and Sciences Dean
Hopi E. Hoekstra announced at a faculty meeting Tuesday.
. . . .
A professor in Physics and Astronomy, Stubbs will continue to advise
Hoekstra on issues regarding artificial intelligence, according to Hoekstra.
Stubbs has made the incorporation of AI at Harvard a priority in recent months
and will be teaching a course on generative AI in the spring.
Musical accompaniment suggested by the previous Log24 post —
♫ "Deans could get no keener reception in a deanery."
Google's new update page for its Bard AI experiment yesterday:
"We've updated Bard with better capabilities for math and logic."
Better, but still faulty.
Exercise: Correct the errors in the following —
(The worst errors are "1997" and "inspired by.")
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare’s Tempest , 1956
|
“Propriation1 gathers the rift-design2 of the saying
— p. 415 of Heidegger‘s Basic Writings ,
“Das Ereignis versammelt den Aufriß der Sage — Heidegger, Weg zur Sprache 1. “Mirror-Play of the Fourfold” |
From a post of Friday, March 30, 2012 —
In memory of actor Warren Stevens—
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare’s Tempest , 1956
Posts from the date of Stevens’s death are tagged Requiem Day.
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
|
Log24 on January 31, 2015 —
|
The 1913 puzzle from above, claiming priority —
A more sophisticated puzzle related to the previous post —
For "the Trojan family" —
Related material on the late Solomon W. Golomb —
"While at JPL, Sol had also been teaching some classes
at the nearby universities: Caltech, USC and UCLA. In
the fall of 1962, following some changes at JPL—and
perhaps because he wanted to spend more time with
his young children— he decided to become a full-time
professor. He got offers from all three schools. He
wanted to go somewhere where he could 'make
a difference'. He was told that at Caltech 'no one has
any influence if they don’t at least have a Nobel Prize',
while at UCLA 'the UC bureaucracy is such that no one
ever has any ability to affect anything'. The result was
that—despite its much-inferior reputation at the time—
Sol chose USC. He went there in the spring of 1963 as
a Professor of Electrical Engineering—and ended up
staying for 53 years." — Stephen Wolfram, 5/25/16
See also Priority (Nov. 25) and "What's in a Name" (Dec. 1).
For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.
The above passage suggests a meditation on this morning's
New York Times * —
"When shall we three meet again?" — William Shakespeare
“We three have scattered, leaving only me behind
to clean up the scene,” Ms. Yang wrote.
“I am alone, missing us three.” — Amy Qin
Continued from Once Upon a Matrix (November 27, 2015).
Click image below to enlarge.
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare's Tempest , 1956
Midrash —
"Remember me to Herald Square."
Yesterday's post suggests a review of the following —
|
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups" Pages 8-9: Substructures of S(5, 8, 24) An octad is a block of S(5, 8, 24). Theorem 5.1
Let B0 be a fixed octad. The 30 octads disjoint from B0
the design of the points and affine hyperplanes in AG(4, 2), Proof…. … (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2) An explicit construction of the vector space is also easy….) |
Related material: Posts tagged Priority.
In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…
Hafner's abstract:
We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
|
Narrative:
Aooo.
Happy birthday to Stephen King.
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare's Tempest , 1956
From "The Onto-theological Origin of Play:
Heraclitus and Plato," by Yücel Dursun, in
Lingua ac Communitas Vol 17 (October 2007)—
"Heraclitus’s Aion and His Transformations
The saying is as follows:
αἰὼν παῖς ἐστι παίζων, πεττεύων·
παιδὸς ἡ βασιληίη
(Aion is a child playing draughts;
the kingship is the child’s)
(Krell 1972: 64).*
* KRELL, David Farrell.
“Towards an Ontology of Play:
Eugen Fink’s Notion of Spiel,”
Research in Phenomemology ,
2, 1972: 63-93.
This is the translation of the fragment in Greek by Krell.
There are many versions of the translation of the fragment….."
See also Child's Play and Froebel's Magic Box.
Update of May 5— For some background
from the date May 4 seven years ago, see
The Fano Plane Revisualized.
For some background on the word "aion,"
see that word in this journal.
In memory of actor Warren Stevens—
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare's Tempest , 1956
Some other dialogue—
"Where is the cat?" he asked at last.
"Where is the box?"
"Here."
"Where's here?"
"Here is now."
"We used to think so," I said,
"but really we should use larger boxes."
— "Schrödinger's Cat,"
by Ursula K. Le Guin (1974)
There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A8 with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M24.
A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.
Edge says that
It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….
Excerpts from the Edge paper—
Excerpts from the Moore paper—
Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439
* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72
** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss, Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.
From a search in this journal for "Krell"—
Dialogue from an American adaptation of Shakespeare's Tempest—
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
– Taken from a video, Forbidden Planet Monster Attack
From yesterday's A Manifold Showing—
"Propriation gathers the rift-design of the saying and unfolds it
in such a way that it becomes the well-joined structure of a manifold showing."
(p. 415 of Heidegger's Basic Writings, edited by David Farrell Krell,
HarperCollins paperback, 1993)
German versions found on the Web—
„Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des vielfältigen Zeigens.“ 323
323 Heidegger, Weg zur Sprache, S. 259.
"Das Regende im Zeigen der Sage ist das Eignen. Es erbringt das An- und Abwesen in sein jeweilig Eigenes, aus dem dieses sich an ihm selbst zeigt und nach seiner Art verweilt. Das erbringende Eignen, das die Sage als die Zeige in ihrem Zeigen regt, heiße das Ereignen. Es er-gibt das Freie der Lichtung, in die Anwesendes anwähren, aus der Abwesendes entgehen und im Entzug sein Währen behalten kann. Was das Ereignen durch die Sage ergibt, ist nie Wirkung einer Ursache, nicht die Folge eines Grundes. Das erbringende Eignen, das Ereignen, ist gewährender als jedes Wirken, Machen und Gründen. Das Ereignende ist das Ereignis selbst – und nichts außerdem. Das Ereignis, im Zeigen der Sage erblickt, läßt sich weder als ein Vorkommnis noch als ein Geschehen vorstellen, sondern nur im Zeigen der Sage als das Gewährende erfahren. Es gibt nichts anderes, worauf das Ereignis noch zurückführt, woraus es gar erklärt werden könnte. Das Ereignen ist kein Ergebnis (Resultat) aus anderem, aber die Er-gebnis, deren reichendes Geben erst dergleichen wie ein `Es gibt' gewährt, dessen auch noch `das Sein' bedarf, um als Anwesen in sein Eigenes zu gelangen. Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des Vielfältigen Zeigens. Das Ereignis ist das Unscheinbarste des Unscheinbaren, das Einfachste des Einfachen, das Nächste des Nahen und das Fernste des Fernen, darin wir Sterbliche uns zeitlebens aufhalten." 8
8 M. Heidegger: Unterwegs zur Sprache. S. 258 f.
From Google Translate:
"The event brings together the outline of the legend and unfolds it to the structure of the manifold showing."
On Misplaced Concreteness—
An excerpt from China and Vietnam: The Politics of Asymmetry, by Brantly Womack (Cambridge U. Press, 2006)—
The book is intended to be a contribution to the general theory of international relations as well as to the understanding of China and Vietnam, but I give greater priority to “the case” rather than to the theory. This is a deliberate methodological decision. As John Gerring has argued, case studies are especially appropriate when exploring new causal mechanisms.2 I would argue more broadly that the “case” is the reality to which the theory is secondary. In international relations theory, “realism” is often contrasted to “idealism,” but surely a more basic and appropriate meaning of “realism” is to give priority to reality rather than to theory. The philosopher Alfred North Whitehead defined the Fallacy of Misplaced Concreteness as “neglecting the degree of abstraction involved when an actual entity is considered merely so far as it exemplifies certain categories of thought.”3 In effect, the concept is taken as the concrete reality, and actual reality is reduced to a mere appendage of data. Misplaced Concreteness may well be the cardinal sin of modern social science. It is certainly pandemic in international relations theory, where a serious consideration of the complexities of real political situations is often dismissed as mere “area studies.” Like the Greek god Anteus who was sustained by touching his Mother Earth, theory is challenged and rejuvenated by planting its feet in thick reality.
2 John Gerring, "What Is a Case Study and What Is It Good For?"
American Political Science Review 98:2 (May 2004), pp. 341-54
3 Alfred North Whitehead, Process and Reality
(New York: Harper, 1929), p. 11
Remarks—
"Whitehead defined the Fallacy of Misplaced Concreteness…."
The phrase "misplaced concreteness" occurs in the title of a part of an exhibition, "Theme and Variations," by artist Josefine Lyche (Oslo, 2009). I do not know what Lyche had in mind when she used the phrase. A search for possible meanings yielded the above passage.
"In international relations theory, “realism” is often contrasted to “idealism….”
For a more poetic look at "realism" and "idealism" and international relations theory, see Midsummer Eve's Dream.
Related material:
Dialogue from Forbidden Planet —
Dialogue from another story —
“Sizewise?”
“Brainwise, but what they did was multiply me by myself into a quadratic.”
— Psychoshop, by Bester and Zelazny, 1998 paperback, p. 7
“… which would produce a special being– by means of that ‘cloned quadratic crap.’ [P. 75] The proper term sounds something like ‘Kaleideion‘….”
“So Adam is a Kaleideion?”
She shook her head.
“Not a Kaleideion. The Kaleideion….”
— Psychoshop, 1998 paperback, p. 85
|
“Kaleidoscope turning…
Shifting pattern within
|
“When life itself seems lunatic,
who knows where madness lies?”
— For the source, see
Joyce’s Nightmare Continues.
Dialogue from the classic film Forbidden Planet—
"… Which makes it a gilt-edged priority that one of us gets into that Krell lab and takes that brain boost."
— Taken from a video (5:18-5:24 of 6:09) at David Lavery's weblog in the entry of Tuesday, April 7.
(Cf. this journal on that date.)
Thanks to Professor Lavery for his detailed notes on his viewing experiences.
My own viewing recently included, on the night of Good Friday, April 10, the spiritually significant film Indiana Jones and the Kingdom of the Crystal Skull.
The mystic circle of 13 aliens at the end of that film, together with Leslie Nielsen's Forbidden Planet remark quoted above, suggests the following:
|
"The aim of Conway’s game M13 is to get the hole at the top point and all counters in order 1,2,…,12 when moving clockwise along the circle." —Lieven Le Bruyn
The illustration is from the weblog entry by Lieven Le Bruyn quoted below. The colored circles represent 12 of the 13 projective points described below, the 13 radial strokes represent the 13 projective lines, and the straight lines in the picture, including those that form the circle, describe which projective points are incident with which projective lines. The dot at top represents the "hole."
From "The Mathieu Group M12 and Conway’s M13-Game" (pdf), senior honors thesis in mathematics by Jeremy L. Martin under the supervision of Professor Noam D. Elkies, Harvard University, April 1, 1996–
"Let P3 denote the projective plane of order 3. The standard construction of P3 is to remove the zero point from a three-dimensional vector space over the field F3 and then identify each point x with -x, obtaining a space with
Conway [3] proposed the following game…. Place twelve numbered counters on the points… of P3 and leave the thirteenth point… blank. (The empty point will be referred to throughout as the "hole.") Let the location of the hole be p; then a primitive move of the game consists of selecting one of the lines containing the hole, say There is an obvious characterization of a move as a permutation in S13, operating on the points of P3. By limiting our consideration to only those moves which return the hole to its starting point…. we obtain the Conway game group. This group, which we shall denote by GC, is a subgroup of the symmetric group S12 of permutations of the twelve points…, and the group operation of GC is concatenation of paths. Conway [3] stated, but did not prove explicitly, that GC is isomorphic to the Mathieu group M12. We shall subsequently verify this isomorphism. The set of all moves (including those not fixing the hole) is given the name M13 by Conway. It is important that M13 is not a group…." [3] John H. Conway, "Graphs and Groups and M13," Notes from New York Graph Theory Day XIV (1987), pp. 18–29. Another exposition (adapted to Martin's notation) by Lieven le Bruyn (see illustration above):
"Conway’s puzzle M13 involves the 13 points and 13 lines of P3. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the 'hole' (the unique point having no counter) and interchanging the counters on the two other points of the line determined by the first two points. In the picture [above] the lines are represented by dashes around the circle in between two counters and the points lying on this line are those that connect to the dash either via a direct line or directly via the circle. In the first part we saw that the group of all reachable positions in Conway's M13 puzzle having the hole at the top position contains the sporadic simple Mathieu group M12 as a subgroup." |
For the religious significance of the circle of 13 (and the "hole"), consider Arthur and the 12 knights of the round table, et cetera.
"At least partly, the Bourne movies
are a 21st-century Frankenstein story."
On Prof. Gian-Carlo Rota of MIT,
found dead on April 19, 1999–
"He made it a priority to
start any sort of meeting with
a long drawn-out hello…."
2007:
Minimalism
"It's become our form of modern classicism."
— Nancy Spector in
the New York Times of April 23, 2004
Part I: Aesthetics
In honor of the current Guggenheim exhibition, "Singular Forms" — A quotation from the Guggenheim's own website:
"Minimalism refers to painting or sculpture
Discuss these seven points
in relation to the following:
Form,
by S. H. Cullinane
Mark Rothko's reference
to geometry as a "swamp"
and his talk of "the idea" in art
Michael Kimmelman's
remarks on ideas in art
ART WARS:
Judgment Day
(2003, 10/07)
Part II: Theology
Today's previous entry, "Skylark," concluded with an invocation of the Lord. Of course, the Lord one expects may not be the Lord that appears.
"… the idea that, in art at least, less is more.
It is an idea surely as old, as enduringly attractive and as ubiquitous as its opposite. In the beginning was the Word: only later came the Bible, not to mention the three-decker Victorian novel. The oracle at Delphi did not say, 'Exhaustive analysis and comprehension of one's own psyche may be prerequisite to an understanding of one's behavior and of the world at large'; it said, 'Know thyself.' Such inherently minimalist genres as oracles (from the Delphic shrine of Apollo to the modern fortune cookie), proverbs, maxims, aphorisms, epigrams, pensees, mottoes, slogans and quips are popular in every human century and culture–especially in oral cultures and subcultures, where mnemonic staying power has high priority–and many specimens of them are self-reflexive or self-demonstrative: minimalism about minimalism. 'Brevity is the soul of wit.' "
Another form of the oracle at Delphi, in minimalist prose that might make Hemingway proud:
"He would think about Bert. Bert was an interesting man. Bert had said something about the way a gambler wants to lose. That did not make sense. Anyway, he did not want to think about it. It was dark now, but the air was still hot. He realized that he was sweating, forced himself to slow down the walking. Some children were playing a game with a ball, in the street, hitting it against the side of a building. He wanted to see Sarah.
When he came in, she was reading a book, a tumbler of dark whiskey beside her on the end table. She did not seem to see him and he sat down before he spoke, looking at her and, at first, hardly seeing her. The room was hot; she had opened the windows, but the air was still. The street noises from outside seemed almost to be in the room with them, as if the shifting of gears were being done in the closet, the children playing in the bathroom. The only light in the room was from the lamp over the couch where she was reading.
He looked at her face. She was very drunk. Her eyes were swollen, pink at the corners. 'What's the book,' he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
She blinked up at him, smiled sleepily, and said nothing.
'What's the book?' His voice had an edge now.
'Oh,' she said. 'It's Kierkegaard. Soren Kierkegaard.' She pushed her legs out straight on the couch, stretching her feet. Her skirt fell back a few inches from her knees. He looked away.
'What's that?' he said.
'Well, I don't exactly know, myself." Her voice was soft and thick.
He turned his face away from her again, not knowing what he was angry with. 'What does that mean, you don't know, yourself?'
She blinked at him. 'It means, Eddie, that I don't exactly know what the book is about. Somebody told me to read it once, and that's what I'm doing. Reading it.'
He looked at her, tried to grin at her — the old, meaningless, automatic grin, the grin that made everbody like him — but he could not. 'That's great,' he said, and it came out with more irritation than he had intended.
She closed the book, tucked it beside her on the couch. She folded her arms around her, hugging herself, smiling at him. 'I guess this isn't your night, Eddie. Why don't we have a drink?'
'No.' He did not like that, did not want her being nice to him, forgiving. Nor did he want a drink.
Her smile, her drunk, amused smile, did not change. 'Then let's talk about something else,' she said. 'What about that case you have? What's in it?' Her voice was not prying, only friendly, 'Pencils?'
'That's it,' he said. 'Pencils.'
She raised her eyebrows slightly. Her voice seemed thick. 'What's in it, Eddie?'
'Figure it out yourself.' He tossed the case on the couch."
— Walter Tevis, The Hustler, 1959,
Chapter 11
See, too, the invocation of Apollo in
A Mass for Lucero, as well as
GENERAL AUDIENCE OF JOHN PAUL II
Wednesday 15 January 2003:
"The invocation of the Lord is relentless…."
and
JOURNAL ENTRY OF S. H. CULLINANE
Wednesday 15 January 2003:
Karl Cullinane —
"I will fear no evil, for I am the
meanest son of a bitch in the valley."
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