Not So Blank —
McLuhan's "Retrieves" part —
From Hudson's 1905 classic
Kummer's Quartic Surface —
For those who prefer bullshit, a first-rate example of the genre —
Wednesday, February 25, 2026
Saturday, December 27, 2025
Raiders of the Lost Overview
For a less biblical "locked box" overview, see
all other posts now tagged Kummerhenge.
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Friday, December 26, 2025
Folk Etymology
For those who might guess that the name Hirzebruch in German
means "heartbreak" . . .
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After Hirzebruch
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Monday, September 1, 2025
Friday, August 22, 2025
Friday, May 2, 2025
Japan and Germany in the arXiv
Will the above arXiv date, May 10, 2024 . . . live in infamy ?
. . . Vide May 10, 2024, in this journal . . .
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Fact Hunt
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Wednesday, March 12, 2025
Dirac, Kummer, Hudson, and the 16 Points
From the January 2025
Bulletin of the American Mathematical Society :
Some background for the above article's conclusion —
For some related material . . .
Search for "Hudson Kummer Quartic" in Log24.
A song for Singer . . .
"I've got this problem when I'm reading a book
Know there's an ending, so I can't help but look"
— Early James, "I Got This Problem" lyrics
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Sunday, February 23, 2025
Quantum Chip?
Perhaps … For news of greater permanent interest,
see The Angel Particle (Log24, Dec. 7, 2018).
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Sunday, October 6, 2024
For New York Times Fans*
Who Prefer Witchcraft to Reason
* See Fire Temple as well as the previous post and . . .
Letters to Goya, by James Magee, October 5, 2019.
(That 2019 Magee performance was at The Crowley Theater
in Marfa . . . NOT named for Aleister Crowley.)
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Who Prefer Witchcraft to Reason
Who Prefer Witchcraft to Reason
Tuesday, July 2, 2024
Arrière-garde
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Monday, October 30, 2023
Red October Revisited
A New Yorker piece from October 7th, 2023 —
"Terry Bisson's History of the Future" . . .
The "May 19th" name "was derived from the birthdays
of Ho Chi Minh and Malcolm X." — Wikipedia
And then there is the May 19 Gestalt . . .
For a prequel of sorts, see a May 19, 2023, arXiv paper —
Related Log24 reading: Other posts tagged Kummerhenge.
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Wednesday, October 4, 2023
Quantum Dots: “The Thing and I” Continues.
See as well "The Thing and I."
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Tuesday, May 30, 2023
The Tour
In memory of a co-founder of Hollywood's "Magic Castle"
who reportedly died at 92 on Sunday . . .
From posts that were tagged "Blake Tour" on Sunday —
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Tuesday, April 25, 2023
For the Crimson Abyss
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Monday, October 10, 2022
Seeking the Path
See also a Log24 search for "The Path."
Related material from a similar search
for "Nanavira Thera" —
"I am glad you have discovered that the situation is comical:
ever since studying Kummer I have been, with some difficulty,
refraining from making that remark."
— Nanavira Thera, Seeking the Path [Early Letters, 17 July 1958].
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Monday, August 1, 2022
Review
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 —
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Sunday, June 26, 2022
Friday, March 18, 2022
Found† in Space*
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Tuesday, March 15, 2022
The Rosenhain Symmetry
See other posts now so tagged.
Hudson's Rosenhain tetrads, as 20 of the 35 projective lines in PG(3,2),
illustrate Desargues's theorem as a symmetry within 10 pairs of squares
under rotation about their main diagonals:
See also "The Square Model of Fano's 1892 Finite 3-Space."
The remaining 15 lines of PG(3,2), Hudson's Göpel tetrads, have their
own symmetries . . . as the Cremona-Richmond configuration.
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Sunday, March 6, 2022
Overarching Symmetries
By the Daniel J. Peterson whose Swarthmore honors thesis was quoted
here last night —
"What, then, is the relationship between theory-relative symmetries
(physical symmetries) and theory-independent symmetries
(overarching symmetries)? My statement of this problem is
a bit abstract, so let’s look at an example: classical Newtonian gravity
and classical electromagnetism . . . ."
— Prospects for a New Account of Time Reversal
by Daniel J. Peterson, Ph.D. dissertation, U. Mich., 2013, p. 16.
Another 2013 approach to the word "overarching" and sytmmetries —
Other terms of interest: Tenet , Nolanism , and Magic for Liars .
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Sunday, February 20, 2022
4×4 Nomenclature
The geometry of the 4×4 square may be associated with the name
Galois, as in "the Galois tesseract," or similarly with the name Kummer.
Here is a Google image search using the latter name —
(Click to enlarge.)
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Saturday, January 29, 2022
On the Diamond-Theorem Group* of Order 322,560
Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.
This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.
* See a Log24 post of June 1, 2013.
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Friday, August 20, 2021
Space Note
"Consider the six-dimensional vector space ( 𝔽2 )6
over the two-element field 𝔽2 ."
— Page 23 of "The Universal Kummer Threefold,"
arXiv:1208.1229v3, 12 June 2013, by Qingchun Ren,
Steven V. Sam, Gus Schrader, and Bernd Sturmfels.
An illustration of that space from 1981 —
The above recollection of the Kummer Threefold remark was suggested by
recent posts now tagged Smallfield . . .
|
"Third Man – an elderly American railway bum, "Art to which I fix my celebrated signature." — "Third Man" in Victor Snaith's play "Changing Stations"
|
If we read the above "art" as a scythe blade to which the "signature" —
Snaith ("the crooked handle or shaft of a scythe") — is attached,
an image of the late art critic Robert Hughes comes to mind:
That image of Hughes appeared here in a post of June 17, 2015 —
"Slow Art, Continued" — that also referenced the Kummer Threefold
paper above.
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Thursday, June 3, 2021
Notes towards the Redefinition of Culture
"My little horse must think it queer" — Robert Frost.
This is from a poem mentioned here on December 22, 2004,
in a post titled "The Longest Night."
Related material from December 21, 2004 —
And then there is the Timeless Square . . .
See "Framed" (May 30) and "In Memory of Ernst Eduard Kummer" (May 14).
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Friday, May 14, 2021
In Memory of Ernst Eduard Kummer
(29 January 1810 – 14 May 1893)
See as well some earlier references to diamond signs here .
The proper context for some diamond figures that I am interested in
is the 4×4 array that appears, notably, in Hudson's 1905 classic
Kummer's Quartic Surface . Hence this post's "Kummerhenge" tag,
suggested also by some monumental stonework at Tufte's site.
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Tuesday, December 8, 2020
Another Damned Dyslexic
The Importance of Being Ernst —
In the above Wikipedia revision today, the anonymous user “Redactedentity”
found that the article Kummer surface omitted Kummer’s first name
and so changed “authorlink=Ernst Kummer|last=Kummer” to
“authorlink=Ernst Kummer|last=Ernst Kummer.” This fixed the
omission but makes no sense as a statement of parameters.
“Redactedentity” was apparently unable to read the following page,
which explains that “last=” is for the author’s last name —
Of course, this revision may be merely an instance of trolling or of
the sort of humor sometimes found among people with the following
interests:
See also Pazouzou.
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Thursday, December 3, 2020
Brick Joke
The "bricks" in posts tagged Octad Group suggest some remarks
from last year's HBO "Watchmen" series —
Related material — The two bricks constituting a 4×4 array, and . . .
"(this is the famous Kummer abstract configuration )"
— Igor Dolgachev, ArXiv, 16 October 2019.
As is this —
.
The phrase "octad group" does not, as one might reasonably
suppose, refer to symmetries of an octad (a "brick"), but
instead to symmetries of the above 4×4 array.
A related Broomsday event for the Church of Synchronology —
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Monday, November 23, 2020
In the Garden of Adding
“In the garden of Adding,
Live Even and Odd….”
— The Midrash Jazz Quartet in
City of God , by E. L. Doctorow
Related material — Schoolgirls and Six-Set Geometry.
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Thursday, November 19, 2020
Set Design and the Schoolgirl Problem
Underlying Structure of the Design —
Schoolgirl Problem —
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Sunday, November 15, 2020
In Memoriam
https://www.straitstimes.com/singapore/
ntuc-learninghub-ceo-kwek-kok-kwong-dies-aged-53
“Mountain, not fountain.” — Nabokov
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Tuesday, October 6, 2020
Only Connect : Bulk Apperception* Continues.
The above Vanity Fair article was republished on the Web by VF
on September 3, 2013. See also this journal on that date.
Related religious remarks —
* “Bulk apperception” is a phrase from Westworld. See Log24 notes.
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Saturday, August 15, 2020
Wednesday, April 29, 2020
Curtis at Pilsen, Thursday, July 5, 2018
For an account by R. T. Curtis of how he discovered the Miracle Octad Generator,
see slides by Curtis, “Graphs and Groups,” from his talk on July 5, 2018, at the
Pilsen conference on algebraic graph theory, “Symmetry vs. Regularity: The first
50 years since Weisfeiler-Leman stabilization” (WL2018).
See also “Notes to Robert Curtis’s presentation at WL2018,” by R. T. Curtis.
Meanwhile, here on July 5, 2018 —
| Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oft-repeated and ever-different metaphor of identity.
— Paz, Octavio. The Monkey Grammarian |
The 2018 Log24 post containing the above Paz quote goes on to quote
remarks by Lévi-Strauss. Paz’s phrase “series of transparent strata”
suggests a review of other remarks by Lévi-Strauss in the 2016 post
“Key to All Mythologies.“
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Tuesday, March 24, 2020
The Amsterdam Connection
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Thursday, February 20, 2020
Tale
“Continue to exercise caution with stories that can only be
corroborated by dead guys. Fabricated stories are almost
never made up out of whole cloth, but are made by stitching
together generally known facts with bits of uncheckable fantasy.”
— Intelligence officer Frank Anderson, who reportedly
died on January 27, 2020.
This journal on that date —
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Monday, February 10, 2020
Notes for Doctor Sleep
Or: Plato's Cave.
See also this journal on November 9, 2003 …
A post on Wittgenstein's "counting pattern" —
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Friday, February 7, 2020
Correspondences
The 15 2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)
The 35 3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)
The 56 3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)
Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.
See also Correspondences in this journal.
* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .
-
Gonzalez-Dorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P3.
American Mathematical Society, Providence, RI, 1994. -
Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 3031-3057.
Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:
See as well the word correspondences in
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).
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Friday, December 20, 2019
Identity Theory*
The van Dam cited by Polster should not be confused
with the fictional Vandamm of "North by Northwest."
See Pursued by a Biplane (Log24, May 23, 2017).
* For the title, see posts tagged March 8, 2018.
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Saturday, November 16, 2019
Logic in the Spielfeld
"A great many other properties of E-operators
have been found, which I have not space
to examine in detail."
— Sir Arthur Eddington, New Pathways in Science ,
Cambridge University Press, 1935, page 271.
The following 4×4 space, from a post of Aug. 30, 2015,
may help:
The next time she visits an observatory, Emma Stone
may like to do a little dance to …
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Sunday, November 3, 2019
Kummerhenge: 200 Years
http://www.math.lsa.umich.edu/~idolga/KummerOliver.pdf
is a preprint of an Oct. 10, 2019, talk by Igor Dolgachev —
Kummer Surfaces: 200 Years of Study.
The preprint is also available on the arXiv:
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Friday, October 11, 2019
The Flynn Legacy
James R. Flynn (born in 1934), "is famous for his discovery of
the Flynn effect, the continued year-after-year increase of IQ
scores in all parts of the world." —Wikipedia
His son Eugene Victor Flynn is a mathematician, co-author
of the following chapter on the Kummer surface—
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Wednesday, October 9, 2019
The Joy of Six
Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
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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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Saturday, October 5, 2019
Friday, October 4, 2019
Kummerhenge
(Continued.)
The previous post suggests a review of
the following mathematical landmark —
The cited article by Kummer is at . . .
https://archive.org/details/monatsberichtede1864kn/page/246 .
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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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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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Saturday, September 14, 2019
Landscape Art
From "Six Significant Landscapes," by Wallace Stevens (1916) —
VI
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses —
As, for example, the ellipse of the half-moon —
Rationalists would wear sombreros.
But see "cones, waving lines, ellipses" in Kummer's Quartic Surface
(by R. W. H. T. Hudson, Cambridge University Press, 1905) and their
intimate connection with the geometry of the 4×4 square.
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Friday, August 16, 2019
Nocciolo
A revision of the above diagram showing
the Galois-addition-table structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
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Saturday, May 4, 2019
The Chinese Jars of Shing-Tung Yau
The title refers to Calabi-Yau spaces.
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A less "cosmic" but still noteworthy code — The Golay code.
This resides in a 12-dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related Calabi-Yau "Chinese jar" first described in detail in 1905 —
A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —
ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf
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Friday, March 29, 2019
Front-Row Seed
"This outer automorphism can be regarded as
the seed from which grow about half of the
sporadic simple groups…." — Noam Elkies
Closely related material —
The top two cells of the Curtis "heavy brick" are also
the key to the diamond-theorem correlation.
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Thursday, March 28, 2019
Culture
The previous post, "Dream of Plenitude," suggests . . .
"So here's to you, Nordstrom-Robinson . . . ."
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Tuesday, February 26, 2019
Citation
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies's "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K6 and to the 15 2-subsets of a 6-set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2-subsets of a 6-set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6-set
in the work of other authors in 1994 and 2002 . . .
-
Gonzalez-Dorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P3.
American Mathematical Society, Providence, RI, 1994. -
Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 3031-3057.
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Monday, February 18, 2019
Sacerdotal K6, Continued
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Saturday, February 16, 2019
Melancholy for Dürer
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Wednesday, February 13, 2019
April 18, 2003 (Good Friday), Continued
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
— Igor Shafarevitch, 1973 remark published as above in 1982.
"Perhaps."
— Steven H. Cullinane, February 13, 2019
|
From Log24 on Good Friday, April 18, 2003 — . . . What, indeed, is truth? I doubt that the best answer can be learned from either the Communist sympathizers of MIT or the “Red Mass” leftists of Georgetown. For a better starting point than either of these institutions, see my note of April 6, 2001, Wag the Dogma. See, too, In Principio Erat Verbum , which notes that “numbers go to heaven who know no more of God on earth than, as it were, of sun in forest gloom.” Since today is the anniversary of the death of MIT mathematics professor Gian-Carlo Rota, an example of “sun in forest gloom” seems the best answer to Pilate’s question on this holy day. See
“Examples are the stained glass windows Motto of Plato’s Academy † The Exorcist, 1973 |
Detail from an image linked to in the above footnote —
"And the darkness comprehended it not."
Id est :
A Good Friday, 2003, article by
a student of Shafarevitch —
"… there are 25 planes in W . . . . Of course,
replacing {a,b,c} by the complementary set
does not change the plane. . . ."
Of course.
See. however, Six-Set Geometry in this journal.
Comments Off on April 18, 2003 (Good Friday), Continued
Tuesday, February 12, 2019
A Long Time
This journal on the above date, October 17, 2008 —
“Every musician wants to do something of lasting quality,
something which will hold up for a long time, and
I guess we did it with ‘Stairway.'”
— Jimmy Page on “Stairway to Heaven“
Scholium —
"Kummer " in German means "sorrow."
Related material —
Other posts now tagged Dolmen.
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Thursday, February 7, 2019
Geometry of the 4×4 Square: The Kummer Configuration
From the series of posts tagged Kummerhenge —
A Wikipedia article relating the above 4×4 square to the work of Kummer —
A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24 with the aid of the MOG.
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Saturday, January 26, 2019
Installasjon
The above cryptic search result indicates that there may
soon be a new Norwegian art installation based on this page
of Eddington (via Log24) —
See also other posts tagged Kummerhenge.
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Friday, December 14, 2018
Small Space Odyssey
References in recent posts to physical space and
to mathematical space suggest a comparison.
Physical space is well known, at least in the world
of mass entertainment.
Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.
A figure from each space —
The source of the Conway-Sloane brick —
Quote from a mathematics writer —
“Looking carefully at Golay’s code is like staring into the sun.”
The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.
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Thursday, December 13, 2018
Space Art
For Oslo artist Josefine Lyche, excerpts
from a Google image search today —
Material related to Lyche's experience as an adolescent with a ZX Spectrum computer —
Click "Hello World" for a larger image.
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Wednesday, December 12, 2018
Kummerhenge Continues.
Those pleased by what Ross Douthat today called
"The Return of Paganism" are free to devise rituals
involving what might be called "the sacred geometry
of the Kummer 166 configuration."
As noted previously in this journal,
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also earlier posts also tagged "Kummerhenge" and
another property of the remarkable Kummer 166 —
For some related literary remarks, see "Transposed" in this journal.
Some background from 2001 —
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An Inscape for Douthat
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
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Friday, December 7, 2018
The Angel Particle
(Continued from this morning)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also other Log24 posts tagged Kummerhenge.
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Sunday, December 2, 2018
Symmetry at Hiroshima
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
|
http://www.math.sci.hiroshima-u.ac.jp/ branched/files/2018/abstract/Aitchison.txt Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve. |
See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
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Thursday, November 22, 2018
Rosenhain and Göpel Meet Kummer in Projective 3-Space
For further details, see finitegeometry.org/sc/35/hudson.html.
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Geometric Incarnation
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
Note also the four 4×4 arrays surrounding the central diamond
in the chi of the chi-rho page of the Book of Kells —
From a Log24 post
of March 17, 2012
"Interlocking, interlacing, interweaving"
— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science
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Sunday, November 18, 2018
Space Music
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Thursday, November 8, 2018
Geometry Lesson
From "The Trials of Device" (April 24, 2017) —
See also Wittgenstein in a search for "Ein Kampf " in this journal.
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Saturday, September 22, 2018
Minimalist Configuration
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Monday, July 16, 2018
Greatly Exaggerated Report
"The novel has a parallel narrative that eventually
converges with the main story."
— Wikipedia on a book by Foer's novelist brother
Public Squares
An image from the online New York Times
on the date, July 6,
of the above Atlantic article —
An image from "Blackboard Jungle," 1955 —
"Through the unknown, remembered gate . . . ."
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Thursday, July 12, 2018
Kummerhenge Illustrated
“… the utterly real thing in writing is the only thing that counts…."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"Omega is as real as we need it to be."
— Burt Lancaster in "The Osterman Weekend"
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Friday, July 6, 2018
Something
"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."
— T. S. Eliot, "Burnt Norton," 1936
"Read something that means something."
— Advertising slogan for The New Yorker
The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —
"… we have sixteen elements with which to form a group-structure" —
See as well posts tagged Dirac and Geometry.
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Friday, June 29, 2018
For St. Stanley
The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis
(birth into heaven) of St. Stanley Kubrick.
Related material from March 7, 2014,
in this journal —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
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Wednesday, June 27, 2018
The Letterman Intro
"Summerfield, Kummerhenge. Kummerhenge, Summerfield."
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Taken In
A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :
|
From Mere Christianity , by C. S. Lewis (1952) —
"Book IV – Beyond Personality: I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully. You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares. Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels. Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine. In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already. You may ask, "If we cannot imagine a three-personal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like. . . . . |
But beware of being drawn into the personal life of the Happy Family .
https://www.jstor.org/stable/24966339 —
"The colorful story of this undertaking begins with a bang."
And ends with …
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
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Sunday, June 24, 2018
For 6/24
A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M24 —
Related material —
See too the diamond-theorem correlation.
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Saturday, June 23, 2018
Meanwhile …
Backstory for fiction fans, from Log24 on June 11 —
Related non -fiction —
See as well the structure discussed in today's previous post.
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Plan 9 from Inner Space
From Nanavira Thera, "Early Letters," in Seeking the Path —
"nine possibilities arising quite naturally" —
Compare and contrast with Hudson's parametrization of the
4×4 square by means of 0 and the 15 2-subsets of a 6-set —
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Friday, June 22, 2018
For the Late Charles Krauthammer
"… lo lidchok et haketz …."
— Acceptance speech, Guardian of Zion award, 2002
Also on February 20, 2012 —
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Thursday, June 21, 2018
Models of Being
A Buddhist view —
“Just fancy a scale model of Being
made out of string and cardboard.”
— Nanavira Thera, 1 October 1957,
on a model of Kummer’s Quartic Surface
mentioned by Eddington
A Christian view —
A formal view —
From a Log24 search for High Concept:
See also Galois Tesseract.
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Dirac and Geometry (continued)
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
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Cavell’s Matrix
From an obituary for Stanley Cavell, Harvard philosopher
who reportedly died at 91 on Tuesday, June 19:
The London Review of Books weblog yesterday —
"Michael Wood reviewed [Cavell’s]
Philosophy the Day after Tomorrow in 2005:
'The ordinary slips away from us. If we ignore it, we lose it.
If we look at it closely, it becomes extraordinary, the way
words or names become strange if we keep staring at them.
The very notion turns into a baffling riddle.' "
See also, in this journal, Tuesday morning's Ici vient M. Jordan and
this morning's previous post.
Update of 3:24 AM from my RSS feed —
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Tuesday, June 19, 2018
Ici vient M. Jordan
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Saturday, June 16, 2018
Kummer’s (16, 6) (on 6/16)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See too "The Ruler of Reality" in this journal.
Related material —
A more esoteric artifact: The Kummer 166 Configuration . . .
An array of Göpel tetrads appears in the background below.
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
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For June 16
"But perhaps the desire for story
is what gets us into trouble to begin with."
— Sarah Marshall on June 5, 2018
"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule…
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities…’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence … is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."
— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018
Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce —
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Monday, June 11, 2018
Arty Fact
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "Uh-Oh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computer-eye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
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Friday, February 16, 2018
Two Kinds of Symmetry
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.
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Friday, December 22, 2017
Thursday, November 30, 2017
The Matrix for Quantum Mystics
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976
I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
.
From a Log24 post of June 26-27, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
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Wednesday, November 29, 2017
Definitions
See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.
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Tuesday, October 31, 2017
One Fell Shmoop
https://www.shmoop.com/no-country-for-old-men/coin-symbol.html —
"You know the date on this coin?"
Related material —
This journal on March 7, 2014 —
From Klein’s 1893 Lectures on Mathematics —
“The varieties introduced by Wirtinger may be called
Kummer varieties….” — E. Spanier, 1956
From the "varieties introduced by Wirtinger" link above —
.
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Saturday, September 9, 2017
Mathematics and Metaphysics
"For use of the Kummer surface in Buddhist metaphysics . . ."
is a phrase from a search in this journal for Nanavira.
See as well Buddhism in the previous post.
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Saturday, September 2, 2017
A Touchstone
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Inter-universal Teichmuller Theory," by S. Mochizuki.
* Specifically, Jacobi and Kummer (along with theta functions).
I do not know of any direct connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.
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Wednesday, July 26, 2017
Icon Parking
For the title, see Icon Parking in a search for 54th in this journal.
For related iconic remarks, click on either image below.
.
This post was suggested by the Dec. 30, 2016, date of the
death in Nuremberg of mathematician Wolf Barth. The first
image above is from a mathematics-related work by
John von Neumann discussed here on that date.
See also Wolf Barth in this journal for posts that largely
concern not the above Barth, but an artist of the same name.
For posts on the mathematician only, see Barth + Kummer.
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Monday, June 26, 2017
Upgrading to Six
This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.
Related material (click to enlarge) —
Detail —
From the work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Updates from the morning of June 27 —
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
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Tuesday, May 23, 2017
Friday, April 14, 2017
Hudson and Finite Geometry
The above four-element sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements. This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
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Wednesday, March 29, 2017
Art Space, Continued
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent 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 —
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Sunday, December 11, 2016
Complexity to Simplicity via Hudson and Rosenhain*
*The Hudson of the title is the author of Kummer's Quartic Surface (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.
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Thursday, November 3, 2016
Triple Cross
(Continued … See the title in this journal, as well as Cube Bricks.)
Cube Bricks 1984 —
Related material —
Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge and Nanavira Thera.
See as well Bill Murray's 1984 film "The Razor's Edge" …
Movie poster from 1984 —
"A thin line separates
love from hate,
success from failure,
life from death."
Three other dualities, from Nanavira Thera in 1959 —
"I find that there are, in every situation,
three independent dualities…."
(Click to enlarge.)
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Friday, September 16, 2016
A Counting-Pattern
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Tuesday, September 13, 2016
Parametrizing the 4×4 Array
The previous post discussed the parametrization of
the 4×4 array as a vector 4-space over the 2-element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface —
Hudson in 1905:
These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":
A recap by Cullinane in 2013:
Click images for further details.
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Monday, September 12, 2016
The Kummer Lattice
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 "
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.
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Tuesday, July 12, 2016
Klein and Kummer Configurations in 1889
Further details from Edmund Hess in 1889* related to
last night's remarks on the Klein 6015 configuration
and the Kummer 166 configuration —
* Edmund Hess, "Beiträge zur Theorie der räumlichen Configurationen.
Ueber die Klein'sche Configuration Cf. (60₁₅, 30₆) und einige
bemerkenswerthe aus dieser ableitbare räumliche Configurationen."
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Group Elements and Skew Lines
The following passage by Igor Dolgachev (Good Friday, 2003)
seems somewhat relevant (via its connection to Kummer's 166 )
to previous remarks here on Dirac matrices and geometry —
Note related remarks from E. M. Bruins in 1959 —
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Friday, June 3, 2016
Bruins and van Dam
A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:
- Kummer and Dirac (May 25)
- Framework (May 25)
- Expanding the Spielraum (May 26)
- Dorje (May 26)
The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:
- Anticommuting Dirac Matrices as Skew Lines (Nov. 20, 2015)
- Dirac and Line Geometry (Nov. 23, 2015)
- Einstein and Geometry (Nov. 27, 2015)
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Wednesday, May 25, 2016
Kummer and Dirac
From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng and H. van Dam,
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239
(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)
" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "
" 1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is
related to that of Kummer’s 166 configuration . . . ."
[4]
O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef
E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135
F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished
A remark of my own on the structure of Kummer’s 166 configuration . . . .
See as well yesterday morning's post.
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Tuesday, May 24, 2016
Rosenhain and Göpel Revisited
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
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Tuesday, February 9, 2016
Cubism
The hexagons above appear also in Gary W. Gibbons,
"The Kummer Configuration and the Geometry of Majorana Spinors,"
1993, in a cube model of the Kummer 166 configuration —
Related material — The Religion of Cubism (May 9, 2003).
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Thursday, December 3, 2015
Overarching Symmetry
From p. 34 of the preprint "Snapshots of Conformal Field Theory,"
by Katrin Wendland, arXiv, 11 April 2014 —
50. Gannon, T.: Much ado about Mathieu (arXiv:1211.5531 [math.RT])
85. Taormina, A., Wendland, K.: The overarching finite symmetry group
of Kummer surfaces in the Mathieu group M24. JHEP 08, 125 (2013)
86. Taormina, A., Wendland, K.: Symmetry-surfing the moduli space
of Kummer K3s (arXiv:1303.2931 [hep-th])
87. Taormina, A., Wendland, K.: A twist in the M24 moonshine story
(arXiv:1303.3221 [hep-th])
The Wendland paper was published on Jan. 7, 2015, in
Mathematical Aspects of Quantum Field Theories ,
edited by Damien Calaque and Thomas Strobl
(Springer Mathematical Physics Studies), pages 89-129.
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Monday, November 23, 2015
Dirac and Line Geometry
Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —
His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.
For a biography of Bruins translated by Google, click here.
For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 39-52 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992 . (Springer, 2012,
originally published by Kluwer in 1993.)
For more-recent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post.
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Friday, September 25, 2015
A Great Moonshine
|
Saturday, March 14, 2015
Introduction to Yau
|
-embedding-in-PG(3,2)-Planat-Saniga-1000w.jpg){kind=link}
A note on the somewhat distant relation —
See also Kummer in this journal.
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Saturday, July 4, 2015
Context
Some context for yesterday's post on a symplectic polarity —
This 1986 note may or may not have inspired some remarks
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .
See also the diamond-theorem correlation.
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Wednesday, June 17, 2015
Slow Art, Continued
The title of the previous post, "Slow Art," is a phrase
of the late art critic Robert Hughes.
Example from mathematics:
-
Göpel tetrads as subsets of a 4×4 square in the classic
1905 book Kummer's Quartic Surface by R. W. H. T. Hudson.
These subsets were constructed as helpful schematic diagrams,
without any reference to the concept of finite geometry they
were later to embody.
-
Göpel tetrads (not named as such), again as subsets of
a 4×4 square, that form the 15 isotropic projective lines of the
finite projective 3-space PG(3,2) in a note on finite geometry
from 1986 —
-
Göpel tetrads as these figures of finite geometry in a 1990
foreword to the reissued 1905 book of Hudson:
Click the Barth passage to see it with its surrounding text.
Related material:
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Wednesday, June 10, 2015
Epistemic* Tetrads
"Those that can be obtained…." —
Related music video: Waterloo.
* "In defense of the epistemic view of quantum states:
a toy theory," by Robert W. Spekkens, Perimeter Institute
for Theoretical Physics, Waterloo, Canada
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Tuesday, February 3, 2015
Expanding the Spielraum
A short poem by several authors:
"The role of
the 16 singular points
on the Kummer surface
is now played by
the 64 singular points
on the Kummer threefold."
— From Remark 2.4 on page 9 of
"The Universal Kummer Threefold,"
by Qingchun Ren, Steven V Sam,
Gus Schrader, and Bernd Sturmfels,
http://arxiv.org/abs/1208.1229v3,
August 6, 2012 — June 12, 2013.
See also "Expanded Field" in this journal.
Illustration from "Sunday School," July 20, 2014.
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