Saturday, August 1, 2026
Friday, July 31, 2026
The “Schoolgirl Space”* PG(3.2) . . .
From Fano to Whitehead to Cullinane
260731-Fano-1892-PG(3,2)…AI_Overview.jpg
...AI_Overview.jpg)
260731-Square_Model_of_PG(3,2)-by-Cullinane…AI_Overview.jpg
-by-Cullinane...AI_Overview.jpg)
* See Kirkman's schoolgirl problem in Wikipedia, esp.
the sections "Galois Geometry" and "Spreads and Packing."
The geometry of what might be called "Schoolgirl Space"
is, as the above image "Geometry of the Six Element Set"
indicates, the same as the geometry of the "Hexastigm"
described by Richmond (King's College, Cambridge,
March 30, 1899) —
"On the Figure of Six Points in Space of Four Dimensions,"
by H. W. Richmond, Quarterly Journal of Pure and Applied
Mathematics, Vol. 31 (1900), pp. 125-160.
This is available on the Web at Scribd.com . . .
https://www.scribd.com/document/424182244/
Richmond-QuarterlyJournal-1900-SixPoints .
Richmond's excellent historical survey, together with the Cullinane
square model of PG(3,2), provide the connection of finite geometry
to earlier, non-finite geometry that is lacking in some later treatments
of the subject . . . in particular, the connection to finite geometry of
studies of the classical sixty Pascal lines of the Hexagrammum.
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From Fano to Whitehead to Cullinane
From Fano to Whitehead to Cullinane
Sunday, September 22, 2019
Whitehead and the Relativity Problem
"This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them."
— Hermann Weyl, The Classical Groups,
Princeton University Press, 1946, p. 16
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Tuesday, August 11, 2026
Process Theology . . . New Process
260810-Alexander_Graham_Bell-and-wife-1903-tetrahedral-kite-kiss.jpg
Sunday, May 10, 2026
Entities:
A Schoolgirl Problem for Zorro Ranch
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A Schoolgirl Problem for Zorro Ranch
A Schoolgirl Problem for Zorro Ranch
Sunday, December 28, 2025
“Square Model Harvard” . . . AI Overview, 2025
A rival model of the Fano space, also in the
"Mathematics/History of Science" category —
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Sunday, November 9, 2025
Bloomberg, Berners-Lee, and
Chaos vs. “Rigid Tables”
A rather different perspective . . . Static Pyramid vs. Dytnamic Array —
Two other views of Whitehead's work . . .
Related images: Parmy Olson herself and "the test of time" on Dec. 11, 2024,
as well as a geometric tomb raider, also on Dec. 11, 2024.
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Chaos vs. “Rigid Tables”
Chaos vs. “Rigid Tables”
Tuesday, August 5, 2025
Heisenberg’s “Geometry and Kinematics” —
Arrangements and Movements
The title is from remarks by Heisenberg in the previous post.
Illustrated below are some classic arrangements . . .
The simplex at left is rather static, while the 4×4 array
at right is surprisingly dynamic, giving rise to a group
of 322,560 movements.
Static Simplex vs. Dynamic Array —
Related Disney artifacts —
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Arrangements and Movements
Arrangements and Movements
Thursday, July 17, 2025
Entity Song
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Wednesday, July 16, 2025
From Christian’s Maelstrom
Christian Lorentzen in The New York Times yesterday —
"We may be … passing into an era of disembodied media entities —
an unholy maelstrom of websites, YouTube channels and,
worst of all, podcasts."
Midrash —
"… this book centers on the division of Quality
into the Static and the Dynamic." — Wikipedia
Examples of such division —
Static Simplex vs. Dynamic Array —
For the dynamics of the array, see the diamond theorem group
of 322,560 permutations and its generalization to the dynamics
of a 4x4x4 cube.
For the Whitehead labels on the static simplex, vide . . .
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Thursday, March 9, 2023
Meditation on a URL:
Putting the “de” in
https://www.hu-berlin.de/
From other posts tagged Tetrahedron vs. Square —
A Scholium for Chomsky —
The ABC of words —
A nutshell —
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Putting the “de” in
https://www.hu-berlin.de/
Putting the “de” in
https://www.hu-berlin.de/
Wednesday, February 15, 2023
Mathematics and Narrative . . .
Mathematics:
From Log24 "Pyramid Game" posts —
The letter labels, but not the tetrahedron, are from Whitehead’s
The Axioms of Projective Geometry (Cambridge U. Press, 1906), page 13.
Narrative:
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Wednesday, May 25, 2022
Mexico City Blues
The New York Times yesterday ("2022-05-24T21:54:19.000Z")
on a Saturday, May 21, death —
"Colin Cantwell, an animator, conceptual artist and computer expert
who played significant production roles in seminal science fiction films
like '2001: A Space Odyssey,' 'Star Wars' and 'WarGames,' died
on May 21 at his home in Colorado Springs, Colo. He was 90."
Cantwell at Teotihuacan pyramid, September 26, 2019 —
A different image, also from September 26, 2019,
in other Log24 posts tagged Pyramid Game —
The letter labels, but not the tetrahedron, are from Whitehead’s
The Axioms of Projective Geometry (Cambridge U. Press, 1906),
page 13.
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Monday, February 7, 2022
Morphart Meets Morph Art
Warren (PA) Public Library's Instagram
on January 21, 2022 —
Morphart —
Morph Art — from Raiders of the Lost Coordinates
"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.
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Sunday, January 16, 2022
Space
"Leave a space." — Stoppard, "Jumpers."
"Read something that means something." — The New Yorker
Related material — Bialystock Configurations :
Malgorzata Prazmowska and Krzysztof Prazmowski,
University of Bialystok, Bialystok, Poland
malgpraz@math.uwb.edu.pl and krzypraz@math.uwb.edu.pl
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Wednesday, April 14, 2021
Raiders of the Lost Coordinates . . .
A pyramid scheme† in memory of the late Bernie Madoff —
 |
The above passage from Whitehead’s 1906 book suggests
that the tetrahedral model may be older than Polster thinks.*
This is shown by . . .
† See also “Profzi Scheme.”
* For some related work of the above “D. Mesner,” see
Mesner, D. (1967). “Sets of Disjoint Lines in PG(3, q),”
Canadian Journal of Mathematics, 19, 273-280.
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Thursday, March 11, 2021
Underworld Type
" LaTeX is widely used in academia[3][4]
for the communication and
publication of scientific documents
in many fields . . . ." — Wikipedia
Related academic remarks —
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Wednesday, February 17, 2021
Raiders of the Lost Coordinates . . .
From other posts tagged Tetrahedron vs. Square —
"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.
Illustration (central detail a from the above tetrahedral figure) —
A Harvard Variation
from Timothy Leary —
The topics of Harvard and Leary suggest some other cultural
history, from The Coasters — "Poison Ivy" and "Yakety Yak."
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Sunday, January 3, 2021
Grey Advertising
Clay Risen memorializes an advertising executive —
“Ms. Young rose to prominence as an executive with Grey Advertising,
where she began in 1959, standing out as one of the few women and
one of the few Asian-Americans at the firm, which was then a power
in its field.”
Related art from Log24 on the date of Ms. Young’s death —
“A colour is eternal.
It haunts time like a spirit.”
— Alfred North Whitehead
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Saturday, December 26, 2020
Memorial
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Sunday, July 5, 2020
It’s Still the Same Old Story …
“He recounted the story of Adam and Eve, who were banished
from paradise because of their curiosity. Their inability to resist
the temptation of the forbidden fruit. Which itself was a metaphorical
stand-in for knowledge and power. He urged us to find the restraint
needed to resist the temptation of the cube—the biblical apple
in modern garb. He urged us to remain in Eden until we were able
to work out the knowledge the apple offered, all by ourselves.”
— Richards, Douglas E.. The Enigma Cube (Alien Artifact Book 1)
(pp. 160-161). Paragon Press, 2020. Kindle Edition.
The biblical apple also appears in the game, and film, Assassin's Creed .
Related material —
See the cartoon version of Alfred North Whitehead in the previous post,
and some Whitehead-related projective geometry —
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Tuesday, January 21, 2020
Eye-in-the-Pyramid Points
"it remains only to choose a pleasing arrangement of {1, 2, … 7}
to label the eye-in-the-pyramid points.
there are, as it’ll turn out, 168 of ’em that’ll work."
— Comment at a weblog on November 27, 2010.
See also Log24 on that date.
The 11/27/2010 comment was on a post dated November 23, 2010.
See also Log24 on that date.
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Wednesday, October 23, 2019
Art-Historical Narrative*
"Leonardo was something like what we now call a Conceptual artist,
maybe the original one. Ideas — experiments, theories — were
creative ends in themselves."
— Holland Cotter in the online New York TImes this evening
From other Log24 posts tagged Tetrahedron vs. Square —
* Phrase from the previous post, "Overarching Narrative."
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Tuesday, October 15, 2019
Inside the Fire Temple
(The title refers to Log24 posts now tagged Fire Temple.)
In memory of a New Yorker cartoonist who
reportedly died at 97 on October 3, 2019 …
"Read something that means something."
— New Yorker advertising slogan
From posts tagged Tetrahedron vs. Square —
This journal on October 3 —
"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.
Illustration (central detail a from the above tetrahedral figure) —
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Tuesday, October 8, 2019
Kummer at Noon
The Hudson array mentioned above is as follows —
See also Whitehead and the
Relativity Problem (Sept. 22).
For coordinatization of a 4×4
array, see a note from 1986
in the Feb. 26 post Citation.
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Monday, October 7, 2019
Oblivion
(A sequel to Simplex Sigillum Veri and
Rabbit Hole Meets Memory Hole)
" Wittgenstein does not, however, relegate all that is not inside the bounds
of sense to oblivion. He makes a distinction between saying and showing
which is made to do additional crucial work. 'What can be shown cannot
be said,' that is, what cannot be formulated in sayable (sensical)
propositions can only be shown. This applies, for example, to the logical
form of the world, the pictorial form, etc., which show themselves in the
form of (contingent) propositions, in the symbolism, and in logical
propositions. Even the unsayable (metaphysical, ethical, aesthetic)
propositions of philosophy belong in this group — which Wittgenstein
finally describes as 'things that cannot be put into words. They make
themselves manifest. They are what is mystical' " (Tractatus 6.522).
— Stanford Encyclopedia of Philosophy , "Ludwig Wittgenstein"
|
From Tractatus Logico-Philosophicus by Ludwig Wittgenstein.
(First published in Annalen der Naturphilosophie ,1921. 5.4541 The solutions of the problems of logic must be simple, since they set the standard of simplicity. Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined — a priori — to form a self-contained system. A realm subject to the law: Simplex sigillum veri. |
Somehow, the old Harvard seal, with its motto "Christo et Ecclesiae ,"
was deleted from a bookplate in an archived Harvard copy of Whitehead's
The Axioms of Projective Geometry (Cambridge U. Press, 1906).
In accordance with Wittgenstein's remarks above, here is a new
bookplate seal for Whitehead, based on a simplex —
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Saturday, October 5, 2019
Friday, September 27, 2019
Algebra for Schoolgirls
The 15 points of the finite projective 3-space PG(3,2)
arranged in tetrahedral form:
The letter labels, but not the tetrahedral form,
are from The Axioms of Projective Geometry , by
Alfred North Whitehead (Cambridge U. Press, 1906).
The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."
Screen Rant on July 31, 2019:
A Google Search sidebar this morning:
Apocalypse Soon! —
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Wednesday, September 25, 2019
Language Game
Previous posts now tagged Pyramid Game suggest …
A possible New Yorker caption: " e . . . (ab) . . . (cd) . "
Caption Origins —
Playing with shapes related to some 1906 work of Whitehead:
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Tuesday, September 24, 2019
Emissary
|
Thursday, September 12, 2019
Tetrahedral Structures
|
Playing with shapes related to some 1906 work of Whitehead:
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