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
(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.
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:
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.
(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 .
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.
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 —
For further details, see finitegeometry.org/sc/35/hudson.html.
“… 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"
"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."
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.
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."
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.
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
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 —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7, W (E7):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .
Denote the d-dimensional hypercube by
"… after coloring the sixty-four vertices of
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."
— From "Kummer's 166 ," section 12 of Coxeter's 1950
"Self-dual Configurations and Regular Graphs"
Just as the 4×4 square represents the 4-dimensional
hypercube
so the 4x4x4 cube represents the 6-dimensional
hypercube
For religious interpretations, see
Nanavira Thera (Indian) and
I Ching geometry (Chinese).
See also two professors in The New York Times
discussing images of the sacred in an op-ed piece
dated Sept. 26 (Yom Kippur).
260409-Oct_8_2019-Kummer_at_Noon.jpg
For those who prefer Trudeau's "Story Theory of Truth" —
260409-All_Souls_Hootenanny-poster-Oct_8_2019.jpg
Related cultural artifacts . . .
This journal on All Souls' Day 2025 —
260409-Greetings-from-Derry-sign.jpg
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 —
From "The Practice of Mathematics, Part 1" by Robert P. Langlands —
|
My feeling for the Greeks as mathematicians is every bit as inadequate as that for the youthful Gauss. I do not know whence came their curiosity and depth. Perhaps no-one does. We live in a highly structured environment dedicated to research. We earn our living by it and we pin our hopes of recognition on it, but the questions we ask and the problems we solve are determined more by tradition, more by our colleagues than by our own natural and spontaneous curiosity. We are seldom playful; our efforts are never simply for our own amusement. A brief romp with Greek mathematics in which we examine the construction of the pentagon at length may be an occasion to capture briefly the ludible spirit of the Greeks An hour is also not enough for an adequate understanding of analytic geometric and complex numbers nor for a presentation of the algebra required for Gauss’s construction [of the 17-sided regular polygon]. The complex numbers are an enormously effective tool that swallows the geometry, but it will be good to ask ourselves how. Moreover the four-fold or sixteen-fold algebraic symmetry is far more subtle than the five-fold or seventeen-fold geometric symmetry. Since it will reappear again and in spades when, and if, we discuss Galois and Kummer, it is best to get used to it now. |
For a less biblical "locked box" overview, see
all other posts now tagged Kummerhenge.
… in posts tagged Kummer-Sept-2013.
Musical accompaniment suggested by
tonight's earlier Paradise Dreams —
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.
See also https://m759.net/wordpress/?s=Dirac+skew+anticommuting.
For fans of a different sort of space . . .
See also the Wikipedia article on Bloom.
See as well "The Thing and I."
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].
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 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 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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