Gell-Mann Meets Bosch . . .
At Hiroshima . . .
* The Bosch cuboctahedron is from an exhibition at Napoli in 2021.
See also, from that exhibition's starting date,
the Log24 post Desperately Seeking Symmetry.
Gell-Mann Meets Bosch . . .
At Hiroshima . . .
* The Bosch cuboctahedron is from an exhibition at Napoli in 2021.
See also, from that exhibition's starting date,
the Log24 post Desperately Seeking Symmetry.
At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, as
in the Gell-Mann picture above, but rather at the ends of one of
a cube's four diagonal axes of symmetry.
See some related illustrations below.
Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.
The exceptional role of 0 and ∞ in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —
Transposition of 0 and ∞ in the knight coordinatization
induces the symplectic polarity of PG(3,2) discussed by
(for instance) Anne Duncan in 1968.
Note the three quadruplets of parallel edges in the 1984 figure above.
The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —
Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.
Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —
And then, more simply, there is the Galois tesseract —
For parts of my own world in June 2010, see this journal for that month.
The above Galois tesseract appears there as follows:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
Anne Duncan in 1968 on a 1960 paper by Robert Steinberg —
_______________________________________________________________________________
Related remarks in this journal — Steinberg + Chevalley.
Related illustrations in this journal — 4×4.
Related biographical remarks — Steinberg Deathdate.
On the wife of the fictional billionaire Byron Gogol —
Continuing the theme of independence, a less fictional Byron . . .
"This is the worst trip I've ever been on"
That song was played at the end of the TV series
"The Resort," which concluded today.
Part I —
Also in May 1986 —
86-05-08… A linear complex related to M24 . Anatomy of the polarity pictured in the 86-04-26 note. 86-05-26… The 2-subsets of a 6-set are the points of a PG(3,2).
Beutelspacher's model of the 15 points of PG(3,2) |
Part II — (36 years later)
In memory of a founder of MCC Theater:
A musical rendition of the ending of the classic
1947 E. B. White short story
"The Second Tree from the Corner."
"In my beginning is my end." — T. S. Eliot
Two readings from December 7, 2008 —
A related quotation, suggested by the now-deceased professor David Lavery
of Middle Tennessee State University —
Lavery, a professor of English, was born in 1949 in Oil CIty, Pa.
The Peacock series "The Resort" yesterday presented its concept
of "a room outside of time" (the Pasaje ) as a hole in the ground.
"I saw a werewolf with a Chinese menu in his hand
Walking through the streets of Soho in the rain"
See other posts now tagged Structure Character.
See as well the Tolkien release date in Wondertale posts.
Season 1 of "The Resort" will end as "The Lord of the Rings:
The Rings of Power" begins.
For seekers of the Pasaje — "The Room Outside of Time" —
"The Vision is of what the transliteration of their collaborative
Great Music into a material reality would be like. They are
shown that the Music has a point, has a result and effect
beyond its composition and singing: it amounts to no less
than a highly detailed template commensurate with the entire
history – beginning to end – of a material, 'physical' Universe
that could exist inside 'time'."
See Ballet Blanc and Black Art in this journal.
From the former:
"A blank underlies the trials of device."
— Wallace Stevens
From the latter:
From "A special configuration of 12 conics and generalized Kummer surfaces,"
by David Kohel, Xavier Roulleau, and Alessandra Sarti.
(arXiv:2004.11421 (math), submitted on 23 Apr 2020 (v1),
last revised 17 May 2021 (this version, v2)) —
"… we study the set C12 of conics that contain at least 6 points in P9. One has
Theorem 1. The set C12 has cardinality 12. Each conic in C12 contains exactly
6 points in P9 and through each point in P9 there are 8 conics. The sets (P9, C12)
form therefore a (98, 126)-configuration.
The configuration (P9, C12) has interesting symmetries, e.g. there are 8 conics
among the 12 passing through a fixed point q in P9 and the 8 points in P9 \ {q},
which form a 85 point-conic configuration. The freeness of the arrangement of
curves C12 is studied in [19], where we learned that this configuration has been
also independently discovered in [11]."
[11] Dolgachev I., Laface A., Persson U., Urzúa G.,
"Chilean configuration of conics, lines and points," preprint.
(arXiv:2008.09627 (math), submitted on 21 Aug 2020)
[19] Pokora P., Szemberg T.,
"Conic-line arrangements in the complex projective plane," preprint
(arXiv:2002.01760 (math), submitted on 5 Feb 2020 (v1),
last revised 10 Feb 2022 (this version, v3))
From the Stillwell remembrance, a Shenitzer quote —
"An English major may or may not be a novelist or a poet,
but would undoubtedly be expected to be able to evaluate
a novel or a poem. The term 'English major' implies some
historical, philosophical, and evaluative training and
competence. It is sad but true that the term 'mathematician'
does not imply corresponding training and competence."
Related material — The previous post, and posts tagged Super-8.
"… the new geometries … provide the best example of
the power of the human mind, for the mind had to defy
and overcome habit, intuition, and sense perceptions
to produce these geometries."
— Morris Kline, Mathematics in Western Culture ,
Oxford University Press, 1953, page 430.
Points as Cuts —
"This discussion, intended to define the nature and the largest common denominator of all games, has at the same time the advantage of placing their diversity in relief and enlarging very meaningfully the universe ordinarily explored when games are studied. In particular, these remarks tend to add two new domains to this universe: that of wagers and games of chance, and that of mimicry and interpretation. Yet there remain a number of games and entertainments that still have imperfectly defined characteristics— for example, kite-flying and top-spinning, puzzles such as crossword puzzles, the game of patience, horsemanship, seesaws, and certain carnival attractions."
— Caillois, Roger. 1913-1978, in
Translated by Meyer Barash from Les jeux et les hommes . |
See also Caillois in the previous post.
See as well the discussion of
"the flag of the nude above the holiday hotel,"
on pages 108-109 in
Leon Surette, “Wallace Stevens, Roger Caillois and
‘The Pure Good of Theory',” Paideuma , Vol. 32,
Nos. 1-2-3 (Spring, Fall and Winter 2003), pp. 95-122
"And, as with all retold tales that are in people's hearts,
there are only good and bad things and black and white
things and good and evil things and no in-between anywhere."
— John Steinbeck, author's epigraph to The Pearl
From the Season 4 finale of Westworld :
uploading Dolores's pearl at Hoover Dam —
For those who prefer greater theological simplicity . . .
Optimus Prime on a different Hoover Dam figure, that of
the AllSpark: "Before time began, there was the Cube."
Simplifying even more . . .
“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”
– Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975
As above, Black's theology forms a cube.
A Web search shows that the above 2014 photo is, specifically,
from June 26, 2014, when the guest speaker was Jules Feiffer.
See as well this journal on June 26, 2014.
Westworld Season 4 Episode 8 (Finale)
Christina: Where am I?
Read more at: |
From a college botany laboratory in the 1915
D. H. Lawrence novel The Rainbow —
"Suddenly she had passed away into
an intensely-gleaming light of knowledge."
A later passage in the same novel, under
a metaphorical Tree of Life —
"She passed away as on a dark wind, far, far away,
into the pristine darkness of paradise, into the original
immortality. She entered the dark fields of immortality."
Some will prefer . . .
For further context, see posts tagged Screw Theory.
See also Dark Fields in this journal.
See also "Abstract Signature" in this journal.
The art-school phrase "line of action" has a mathematical counterpart.
Some background . . .
In memoriam —
For those who prefer bullshit . . .
“There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities.”
— Article at Zoltan Dienes’s website
This quote is from a Log24 post of Feb. 6, 2014,
The Representation of Minus One.
"Leave a space." — Tom Stoppard, "Jumpers"
See also Lily Collins's recent ice-cream-cone post.
The number 105 displayed in that post may suggest,
to sufferers from apophenia, the date 1/05.
See that date in this journal. For the color of Collins's
ice cream — lavender — see posts now tagged Space X.
"Snowman author Raymond Briggs dies aged 88"
See as well a Log24 search: Stoppard + "Leave a space" .
Related literary notes:
"Jurassic World: Maisie Lockwood Adventures 2: The Yosemite Six
will be released on September 27, 2022."
Thanks for the warning.
Of greater interest to some: The Number Six.
A phrase from the above scene: "the metaphysics of identity."
I prefer a May 1986 looking-glass from pure mathermatics.
See more? Yes, Yes!
* See this morning's Rimshot Muse.
** See Lillian Roth in Madam Satan (1930, pre-Code).
The centrolinden.com address in the previous post
suggests a search for Navarre in this journal that yields…
“D’exterieur en l’interieur entre
Qui va par moi, et au milieu du centre
Me trouvera, qui suis le point unique,
La fin, le but de la mathematique;
Le cercle suis dont toute chose vient,
Le point ou tout retourne et se maintient.”
— Marguerite de Navarre
Related philosophical reflections . . .
Waxing poetic . . .
"In the Garden of Adding live Even and Odd" — E. L. Doctorow
To wit:
1/2 + 1/3 = 5/6, since the LCM of 2 and 3 is 6.
See as well . . .
Progressive Matrices
A sample Raven's Progressive Matrices test item —
Update of 10 AM ET Sunday, August 7, 2022 —
See as well Siobhan Roberts on geometry in The New York Times
on March 22, 2022, and a Log24 post on geometry on that date.
The above obituary reports a death that happened on July 23.
Also on that date . . . Myth Space and Date Note.
"“Quantity has a quality all its own.”
— James Stavridis, quoted on Aug. 5, 2022.
In order, approximately, of increasing popularity:
Sean Scully, artist, whose work is the subject of
the recent book and exhibition, "The Shape of Ideas."
Vincent Scully, architectural historian at Yale.
Vincent Edward ("Vin") Scully, "Voice of the Dodgers"
8/2
* See Kipnis in this journal. For instance . . .
The trait of Derrida is mentioned also in
the paper from yesterday's Gefüge post.
Related material — The Eightfold Cube.
See also . . .
"… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty."
— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010
"Schufreider shows that a network of linguistic relations
is set up between Gestalt, Ge-stell, and Gefüge, on the
one hand, and Streit, Riß, and Fuge, on the other . . . ."
— From p. 14 of French Interpretations of Heidegger ,
edited by David Pettigrew and François Raffoul.
State U. of New York Press, Albany, 2008. (Links added.)
One such "network of linguistic relations" might arise from
a non-mathematician's attempt to describe the diamond theorem.
(The phrase "network of linguistic relations" appears also in
Derrida's remarks on Husserl's Origin of Geometry .)
For more about "a system of slots," see interality in this journal.
The source of the above prefatory remarks by editors Pettigrew and Raffoul —
"If there is a specific network that is set up in 'The Origin of the Work of Art,'
a set of structural relations framed in linguistic terms, it is between
Gestalt, Ge-stell and Gefüge, on the one hand, and Streit, Riß and Fuge,
on the other; between (as we might try to translate it)
configuration, frame-work and structure (system), on the one hand, and
strife, split (slit) and slot, on the other. On our view, these two sets go
hand in hand; which means, to connect them to one another, we will
have to think of the configuration of the rift (Gestalt/Riß) as taking place
in a frame-work of strife (Ge-stell/Streit) that is composed through a system
of slots (Gefüge/Fuge) or structured openings."
— Quotation from page 197 of Schufreider, Gregory (2008):
"Sticking Heidegger with a Stela: Lacoue-Labarthe, art and politics."
Pp. 187-214 in David Pettigrew & François Raffoul (eds.),
French Interpretations of Heidegger: An Exceptional Reception.
State University of New York Press, 2008.
Update at 5:14 AM ET Wednesday, August 3, 2022 —
See also "six-set" in this journal.
"There is such a thing as a six-set."
— Saying adapted from a 1962 young-adult novel.
From Log24 posts tagged Art Space —
From a paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader,
and Bernd Sturmfels —
Two such considerations —
The reference to Vallega-Neu in posts that last night were tagged
The Ereignis Sanction leads to . . .
Heidegger’s ‘Contributions to Philosophy.’ An Introduction .
(Indiana University Press, 2003).
That book is about . . .
Martin Heidegger, Contributions to Philosophy (From Enowning) ,
trans. Parvis Emad and Kenneth Maly (Bloomington:
Indiana University Press, 1999). German edition:
Beiträge zur Philosophie (vom Ereignis) ,
ed. F.-W. von Herrmann, Gesamtausgabe, vol. 65
(Frankfurt a. M.: Klostermann, 1989).
* See today's news and a Log24 search for "Philippine."
The posts of February 1, 2, and 3, 2020, have now
been tagged "The Ereignis Sanction."
"… the tesseract, identified with a figure too inclusive,
contradictory, and all-pervasive to be seen as a character,
connects multiple dimensions in a manner counter to
ordinary thought…."
— Catherine Flynn, "From Dowel to Tesseract" (2016)
A mnemonic from a course titled
“Traditionally, there are two modalities, namely,
|
For less rigorous remarks, search Log24 for Modal Diamond Box.
Or: The Sontag Puzzlement
Wikipedia on "Heavenly Creatures" —
"Juliet introduces Pauline to the idea of 'the Fourth World',
a Heaven without Christians where music and art are
celebrated. Juliet believes she will go there when she dies.
Certain actors and musicians have the status of saints in
this afterlife, such as singer Mario Lanza, with whom
both girls are obsessed."
Related material — Sontag + Camp .
(A sequel to the previous post — "To the Lighthouse")
From that same date . . .
Log24 on August 5, 2002 — "To really know a subject you've got to learn a bit of its history." — John Baez, August 4, 2002
"We both know what memories can bring; — Joan Baez, April 1975 "Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 nonoverlapping regions." — History of Mathematics at St. Andrews "Who would not be rapt by the thought of such marvels?" — Saint Bonaventure on the Trinity |
"Who would not be rapt?" . . . Cristin Milioti? —
From a review of "The Resort" at another website —
"The phrase 'The attempt to recall your past
is a waste of time' is repeated throughout the series."
A waste of time? … Perhaps. In "The Resort," Milioti is drawn
into an investigation of fictional events from December 2007.
A check of my own memories from that December
may or may not be a waste of time, but it yields a
page from a book that I fondly recall . . .
The artist Jennifer Bartlett reportedly died at 81 on July 25.
An image excerpted from Log24 posts that were tagged
"Butterfly Song" on that date, with an added quotation from
a 1918 poem by Wallace Stevens —
OPINION DYNAMICS ON DISCOURSE SHEAVES By JAKOB HANSEN AND ROBERT GHRIST arXiv:2005.12798 (math) [Submitted on 26 May 2020] Funding: This work was funded by the Office of the Assistant Secretary of Defense Research & Engineering through a Vannevar Bush Faculty Fellowship, ONR N00014-16-1-2010. HANSEN — Department of Mathematics, University of Pennsylvania, Philadelphia, PA (jhansen@math.upenn.edu) GHRIST — Department of Mathematics and Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA (ghrist@math.upenn.edu) |
See as well this journal on the above date (26 May 2020) —
Related literary notes — On April 28 The New York Times
reported a death from the above date (Tuesday, April 26, 2022).
See a followup in the Times today on "New York literary royalty."
In memory of a political figure who reportedly died on Sunday —
Note the approximate target of the holy nib.
Modernism, Fiction and Mathematics
by Nina Engelhardt
(Edinburgh Critical Studies in Modernist Culture)
From a review by Johann A. Makowsky in
Notices of the American Mathematical Society,
November 2020, pp. 1589-1595 —
"Engelhardt’s goal in this study is to put the interplay
between fiction and mathematical conceptualizations
of the world into its historical context. She sees her work
as a beginning for further studies on the role of mathematics,
not only modern, in fiction in the wider field of literature and
science. It is fair to say that in her book Nina Engelhardt does
succeed in giving us an inspiring tour d’horizon of this interplay."
Another such tour —
On the title of Westworld Season 4 Episode 5, "Zhuangzi" —
A song for Teddy: "Across my dreams, with nets of wonder . . ."
See Zhuangzi also in the 2022 Black Rock CIty manifesto, "Waking Dreams" . . .
Some may prefer their own, less collective, manifestations.
Magic Mikes Continues:
"I get no kick from champagne…." — Cole Porter
See too another item with the BRC "Waking Dreams" date —
From the new URL mythspace.org, which forwards to . . .
http://m759.net/wordpress/?tag=mythspace —
From Middlemarch (1871-2), by George Eliot, Ch. III — "Dorothea by this time had looked deep into the ungauged reservoir of Mr. Casaubon's mind, seeing reflected there in vague labyrinthine extension every quality she herself brought; had opened much of her own experience to him, and had understood from him the scope of his great work, also of attractively labyrinthine extent. For he had been as instructive as Milton's 'affable archangel;' and with something of the archangelic manner he told her how he had undertaken to show (what indeed had been attempted before, but not with that thoroughness, justice of comparison, and effectiveness of arrangement at which Mr. Casaubon aimed) that all the mythical systems or erratic mythical fragments in the world were corruptions of a tradition originally revealed. Having once mastered the true position and taken a firm footing there, the vast field of mythical constructions became intelligible, nay, luminous with the reflected light of correspondences. But to gather in this great harvest of truth was no light or speedy work." |
See also the term correspondence in this journal.
Oxford University Press Blog
|
Note the above Oxford University Press date. Also on that date —
Tuesday September 29, 2009
|
"Is a puzzlement." — Oscar Hammerstein II
“Not games. Puzzles. Big difference. That’s a whole other matter.
All art — symphonies, architecture, novels — it’s all puzzles.
The fitting together of notes, the fitting together of words have
by their very nature a puzzle aspect. It’s the creation of form
out of chaos. And I believe in form.”
— Stephen Sondheim, in Stephen Schiff,
“Deconstructing Sondheim,”
The New Yorker, issue of March 8, 1993, p. 76
"When you come to a fork in the road, take it." — Yogi Berra
The Story Game —
Two players of interest . . .
Further combinatorial properties* of 24261120 may
be investigated with the aid of a 9×9 square grid, and
perhaps (eventually) also with its triangular counterpart —
.
* Cap sets, gerechte designs, etc.
"Sean Kelly, a wry master of literary and musical parodies
who helped infuse National Lampoon with the sharp-edged
and often crude humor it became known for, died on July 11
in Manhattan. He was 81."
— Richard Sandomir in the online New York Times today
This journal on July 11 —
Mark Zuckerberg at Harvard, May 25, 2017 —
"No one writes math formulas on glass. That’s not a thing."
Mosses from an Old Manse (Hawthorne Recycled)
Click the above image to enlarge it.
This lockscreen suggests happy memories of J3 . . .
From The New York Times this afternoon —
Transylvania III, a 1973 tapestry made of horsehair and goat hair.
Backstory —
“Somehow, a message had been lost on me. Groups act .
The elements of a group do not have to just sit there,
abstract and implacable; they can do things, they can
‘produce changes.’ In particular, groups arise
naturally as the symmetries of a set with structure.”
— Thomas W. Tucker, review of Lyndon’s Groups and Geometry
in The American Mathematical Monthly , Vol. 94, No. 4
(April 1987), pp. 392-394.
"…groups are invariably best studied through their action on some structure…."
— R. T. Curtis, “Symmetric Generation of the Higman-Sims Group” in
Journal of Algebra 171 (1995), pp. 567-586.
Related material — Other posts now tagged Groups Act.
"Oldenburg … died on Monday at his home and studio in
the Soho section of Manhattan." — Martha Schwendener, NYT
Some may prefer other concepts of shape. For instance …
… and, from Log24 on the above Yalebooks date —
Related material:
From "Higman- Sims Graph," a webpage by A. E. Brouwer — "Similar to the 15+35 construction of the Hoffman-Singleton graph is the 30+70 construction of the Higman-Sims graph. In the former the starting point was that the lines of PG(3,2) can be labeled with the triples in a 7-set such that lines meet when the corresponding triples have 1 element in common. This time we label the lines of PG(3,2) with the 4+4 splits of an 8-set, where intersecting lines correspond to splits with common refinement 2+2+2+2. Clearly, both descriptions of the lines of PG(3,2) are isomorphic. Take as vertices of the Higman-Sims graph the 15 points and 15 planes of PG(3,2) together with the 70 4-subsets of an 8-set. Join two 4-sets when they have 1 element in common. A 4-set determines a 4+4 split and hence a line in PG(3,2), and is adjacent to the points and planes incident with that line. A plane is adjacent to the nonincident points. This yields the Higman-Sims graph." |
See also PG(3,2) in this journal.
Her lips are pips
I call her hips
“Twirly” and “Whirly.”
(Pips are the dots on dice. The above "choose us" image in the form of a
St. Bridget's cross is from Twirly Industries, a sportswear maker in Pakistan.)
See as well a Polish poet's meditation
quoted here on St. Bridget's Day, 2012:
For the Lily of the title, see The New York Times online tonight
on the life of a socialite-philanthropist who reportedly died on
July 9, 2022.
Related art —
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