A phrase by Aitchison at Hiroshima . . .
"The proof of the above is a relabelling of the Klein quartic . . . ."
Related art — A relabelling of the Klein quadric by Curtis bricks:
Tuesday, January 14, 2025
Thursday, August 22, 2024
Locke & Key: Alpha & Omega
A more sophisticated alpha and omega . . .
— R. T. Curtis, "A New Combinatorial Approach to M 24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society (1976),
A related key . . .
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.
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Wednesday, December 28, 2022
The Santa Fe Institute as Magisterium Wannabe
"The novelist Cormac McCarthy has been a fixture around
the Santa Fe Institute since its embryonic stages in the
early 1980s. Cormac received a MacArthur Award in 1981
and met one of the members of the board of the MacArthur
Foundation, Murray Gell-Mann, who had won the Nobel Prize
in physics in 1969. Cormac and Murray discovered that they
shared a keen interest in just about everything under the sun
and became fast friends. When Murray helped to found the
Santa Fe Institute in 1984, he brought Cormac along, knowing
that everyone would benefit from this cross-disciplinary
collaboration." — https://www.santafe.edu/news-center/news/
cormac-and-sfi-abiding-friendship
Joy Williams, review of two recent Cormac McCarthy novels —
"McCarthy has pocketed his own liturgical, ecstatic style
as one would a coin, a ring, a key, in the service of a more
demanding and heartless inquiry through mathematics and
physics into the immateriality, the indeterminacy, of reality."
A Demanding and Heartless Coin, Ring, and Key:
COIN
RING
"We can define sums and products so that the G-images of D generate
an ideal (1024 patterns characterized by all horizontal or vertical "cuts"
being uninterrupted) of a ring of 4096 symmetric patterns. There is an
infinite family of such 'diamond' rings, isomorphic to rings of matrices
over GF(4)."
KEY
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.
For those who prefer a "liturgical, ecstatic style" —
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Saturday, December 28, 2019
Caballo Blanco
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
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“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight
“The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”
— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.
“It must be remarked that these 8 heptads are the key to an elegant proof….”
— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.
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Wednesday, April 4, 2018
The Key
"The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, 'heptads' …."
— George M. Conwell, "The 3-space PG (3, 2) and Its Group,"
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.
For those who, like the author of The Eight (a novel in which today's
date figures prominently), prefer fiction —
See as well . . .
Literary theorists may, if they wish, connect
cabalistically the Insidious address "414"
with the date 4/14 of the above post, and
the word Appletree with the biblical Garden.
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Sunday, March 1, 2009
Sunday March 1, 2009
Solomon's Cube
continued
"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."
— G. H. Hardy, A Mathematician's Apology
"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.
"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."
— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
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Saturday, February 28, 2009
Saturday February 28, 2009
Mathematics
and Narrative
continued
and Narrative
continued
Narrative:
xxx
Mathematics:
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference, (July 2000), Springer, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
Mathematics:
"Regular graphs are considered, whose automorphism groups are permutation representations P of the orthogonal groups in various dimensions over GF(2). Vertices and adjacencies are defined by quadratic forms, and after graphical displays of the trivial isomorphisms between the symmetric groups S2, S3, S5, S6 and corresponding orthogonal groups, a 28-vertex graph is constructed that displays the isomorphism between S8 and O6 + (2)."
— J. Sutherland Frame in "Orthogonal Groups over GF(2) and Related Graphs," Springer Lecture Notes in Mathematics vol. 642, Theory and Applications of Graphs (Proceedings, Michigan, May 11–15, 1976), edited by Y. Alavi and D. R. Lick, pp. 174-185
"One has O+(6) ≅ S8, the symmetric group of order 8!…."
— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint. This paper gives some context in superstring theory for the following work of Frame:
[F1] J.S. Frame, The classes and representations of the group of 27 lines and 28 bitangents, Annali
di Mathematica Pura ed Applicata, 32 (1951) 83–119.
[F2] J.S. Frame, Some characters of orthogonal groups over the field of two elements, In: Proc. of the
Second Inter. Conf. on the Theory of Groups, Lecture Notes in Math., Vol. 372, pp. 298–314,
Springer, 1974.
[F3] J. S. Frame, Degree polynomials for the orthogonal groups over GF(2), C. R. Math. Rep. Acad.
Sci. Canada 2 (1980) 253–258.
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