Thursday, February 7, 2019
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 finitegeometry properties of the 4×4 square as
a finite affine 4space — properties that are of use in studying the Mathieu
group M_{24 }with the aid of the MOG.
Comments Off on Geometry of the 4×4 Square: The Kummer Configuration
Tuesday, July 12, 2016
Further details from Edmund Hess in 1889* related to
last night's remarks on the Klein 60_{15} configuration
and the Kummer 16_{6} 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."
Verhandlungen der Kaiserlichen LeopoldinischCarolinischen
Deutschen Akademie der Naturforscher, Vol.55, No. 2, pp. 98167
Comments Off on Klein and Kummer Configurations in 1889
Sunday, November 3, 2019
Comments Off on Kummerhenge: 200 Years
Tuesday, October 8, 2019
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.
Comments Off on Kummer at Noon
Friday, October 4, 2019
(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 .
Comments Off on Kummerhenge
Wednesday, December 12, 2018
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 16_{6 }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 16_{6} —
For some related literary remarks, see "Transposed" in this journal.
Some background from 2001 —
Comments Off on Kummerhenge Continues.
Thursday, November 22, 2018
Comments Off on Rosenhain and Göpel Meet Kummer in Projective 3Space
Saturday, September 22, 2018
Comments Off on Minimalist Configuration
Thursday, July 12, 2018
“… 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"
Comments Off on Kummerhenge Illustrated
Thursday, June 21, 2018
See also the Omega Matrix in this journal.
Comments Off on Kummerhenge
Saturday, June 16, 2018
"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 16_{6} 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."
Comments Off on Kummer’s (16, 6) (on 6/16)
Monday, September 12, 2016
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 4space 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 M_{24 }"
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 twosubsets of a sixset. See the post of Sept. 13.
Comments Off on The Kummer Lattice
Wednesday, May 25, 2016
From "Projective Geometry and PTSymmetric 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 spin½ 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 PluckerKlein correspondence between lines of
a threedimensional projective space and points of a quadric
in a fivedimensional 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 16_{6} 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 16_{6} configuration . . . .
See that structure in this journal, for instance —
See as well yesterday morning's post.
Comments Off on Kummer and Dirac
Friday, March 7, 2014
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 64point vector space and
to the Weyl group of type E_{7}, W (E_{7}):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑfunctions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441445
of Volume 10 of his Collected Mathematical Papers .
Comments Off on Kummer Varieties
Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6volume
Principles of Geometry .
Pointline incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
Comments Off on The Large Desargues Configuration
Tuesday, February 19, 2013
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103104.
The PisanskiServatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to redescribing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the PisanskiServatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
Comments Off on Configurations
Thursday, September 27, 2012
Denote the ddimensional hypercube by γ_{d} .
"… after coloring the sixtyfour vertices of γ_{6}
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 16_{6 }," section 12 of Coxeter's 1950
"Selfdual Configurations and Regular Graphs"
Just as the 4×4 square represents the 4dimensional
hypercube γ_{4 }over the twoelement Galois field GF(2),
so the 4x4x4 cube represents the 6dimensional
hypercube γ_{6} over GF(2).
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 oped piece
dated Sept. 26 (Yom Kippur).
Comments Off on Kummer and the Cube
Tuesday, February 14, 2012
The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title,
that of a novel by the author of The Exorcist .
The Ninth Configuration —
The ninth* in a list of configurations—
"There is a (2^{d1})_{d} configuration
known as the Cox configuration."
— MathWorld article on "Configuration"
For further details on the Cox 32_{6} configuration's Levi graph,
a model of the 64 vertices of the sixdimensional hypercube γ_{6 },
see Coxeter, "SelfDual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413455, 1950.
This contains a discussion of Kummer's 16_{6} as it
relates to γ_{6 }, another form of the 4×4×4 Galois cube.
See also Solomon's Cube.
* Or tenth, if the fleeting reference to 11_{3} configurations is counted as the seventh—
and then the ninth would be a 15_{3} and some related material would be Inscapes.
Comments Off on The Ninth Configuration
Friday, March 18, 2011
The OnLine Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.
From that article:
 DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
 EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.
The following corrects the word "unique" in the example.
* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
The correction was made at about 11:50 AM on March 20, 2011.
_____________________________________________________________
Update of March 21
The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—
In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term setconfigurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to setconfigurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.
Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number of configurations in the resulting theory, as the above (8_3) examples show.
Update of March 22 (itself updated on March 25)
For further background on configurations, see Dolgachev—
Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.
Update of March 27
See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order (1937), pp. 4243. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2subsets of a 6set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 89, 1617, 4445, 7677, 7879, and 80.
Comments Off on Defining Configurations*
Friday, December 7, 2018
(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.
Comments Off on The Angel Particle
Tuesday, February 9, 2016
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 16_{6} configuration —
Related material — The Religion of Cubism (May 9, 2003).
Comments Off on Cubism
Saturday, September 21, 2013
The Kummer 16_{6} configuration is the configuration of sixteen
6sets within a 4×4 square array of points in which each 6set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
Comments Off on Geometric Incarnation
Tuesday, December 8, 2020
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 Kummerlast=Kummer” to
“authorlink=Ernst Kummerlast=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.
Comments Off on Another Damned Dyslexic
Thursday, December 3, 2020
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 —
Comments Off on Brick Joke
Monday, November 23, 2020
“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 SixSet Geometry.
Comments Off on In the Garden of Adding
Thursday, November 19, 2020
Underlying Structure of the Design —
Schoolgirl Problem —
Comments Off on Set Design and the Schoolgirl Problem
Sunday, November 15, 2020
https://www.straitstimes.com/singapore/
ntuclearninghubceokwekkokkwongdiesaged53
“Mountain, not fountain.” — Nabokov
Comments Off on In Memoriam
Tuesday, October 6, 2020
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.
Comments Off on Only Connect : Bulk Apperception* Continues.
Saturday, August 15, 2020
Wednesday, April 29, 2020
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 WeisfeilerLeman 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 oftrepeated and everdifferent metaphor of identity.
— Paz, Octavio. The Monkey Grammarian
(Kindle Locations 11851191).
Arcade Publishing. Kindle Edition. 
The 2018 Log24 post containing the above Paz quote goes on to quote
remarks by LéviStrauss. Paz’s phrase “series of transparent strata”
suggests a review of other remarks by LéviStrauss in the 2016 post
“Key to All Mythologies.“
Comments Off on Curtis at Pilsen, Thursday, July 5, 2018
Tuesday, March 24, 2020
Some mathematics from Ghent —
Earlier, in Amsterdam . . .
See as well Dirac and Geometry.
Comments Off on The Amsterdam Connection
Thursday, February 20, 2020
“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 —
Comments Off on Tale
Monday, February 10, 2020
Comments Off on Notes for Doctor Sleep
Friday, February 7, 2020
The 15 2subsets of a 6set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)
The 35 3subsets of a 7set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)
The 56 3subsets of an 8set 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 6set also appeared
in the work of other authors in 1994 and 2002 . . .
 GonzalezDorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P^{3}.
American Mathematical Society, Providence, RI, 1994.
 Dolgachev, Igor, and Keum, JongHae,
“Birational Automorphisms of Quartic Hessian Surfaces.”
Trans. Amer. Math. Soc. 354 (2002), 30313057.
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).
Comments Off on Correspondences
Friday, December 20, 2019
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.
Comments Off on Identity Theory*
Saturday, November 16, 2019
"A great many other properties of Eoperators
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 …
Comments Off on Logic in the Spielfeld
Friday, October 11, 2019
James R. Flynn (born in 1934), "is famous for his discovery of
the Flynn effect, the continued yearafteryear increase of IQ
scores in all parts of the world." —Wikipedia
His son Eugene Victor Flynn is a mathematician, coauthor
of the following chapter on the Kummer surface—
Comments Off on The Flynn Legacy
Wednesday, October 9, 2019
Note that in the pictures below of the 15 twosubsets of a sixset,
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."
Comments Off on The Joy of Six
Saturday, October 5, 2019
Comments Off on Midnight Landmarks
Tuesday, September 24, 2019
Playing with shapes related to some 1906 work of Whitehead:
Comments Off on Emissary
Sunday, September 22, 2019
"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
Comments Off on Whitehead and the Relativity Problem
Saturday, September 14, 2019
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 rightangled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses —
As, for example, the ellipse of the halfmoon —
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.
Comments Off on Landscape Art
Friday, August 16, 2019
(Continued)
A revision of the above diagram showing
the Galoisadditiontable structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
Comments Off on Nocciolo
Friday, March 29, 2019
"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 diamondtheorem correlation.
Comments Off on FrontRow Seed
Thursday, March 28, 2019
The previous post, "Dream of Plenitude," suggests . . .
"So here's to you, NordstromRobinson . . . ."
Comments Off on Culture
Tuesday, February 26, 2019
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 K_{6} and to the 15 2subsets of a 6set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2subsets of a 6set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6set
in the work of other authors in 1994 and 2002 . . .

GonzalezDorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P^{3}.
American Mathematical Society, Providence, RI, 1994.

Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 30313057.
Comments Off on Citation
Monday, February 18, 2019
From yesterday's post on sacerdotal jargon —
A related note from May 1986 —
Comments Off on Sacerdotal K_{6}, Continued
Saturday, February 16, 2019
The title refers to a 1514 engraving.
See also Angel Particle in this journal.
Comments Off on Melancholy for Dürer
Wednesday, February 13, 2019
"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 GianCarlo Rota, an example of “sun in forest gloom” seems the best answer to Pilate’s question on this holy day. See
The Shining of May 29.
“Examples are the stained glass windows
of knowledge.” — Vladimir Nabokov
AGEOMETRETOS MEDEIS EISITO
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, SixSet Geometry in this journal.
Comments Off on April 18, 2003 (Good Friday), Continued
Tuesday, February 12, 2019
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.
Comments Off on A Long Time
Saturday, January 26, 2019
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
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 12dimensional
finite space of the Golay code, is less well known.
A figure from each space —
The source of the ConwaySloane brick —
Quote from a mathematics writer —
“Looking carefully at Golay’s code is like staring into the sun.”
— Richard Evan Schwartz
The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.
Comments Off on Small Space Odyssey
Wednesday, December 12, 2018
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."
Comments Off on An Inscape for Douthat
Thursday, November 22, 2018
"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 chirho 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
Comments Off on Geometric Incarnation
Sunday, November 18, 2018
Update of Nov. 19 —
"Design is how it works." — Steve Jobs
See also www.cullinane.design.
Comments Off on Space Music
Thursday, November 8, 2018
From "The Trials of Device" (April 24, 2017) —
See also Wittgenstein in a search for "Ein Kampf " in this journal.
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Monday, July 16, 2018
"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 . . . ."
— T. S. Eliot, Four Quartets
Comments Off on Greatly Exaggerated Report
Friday, July 6, 2018
"… 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 groupstructure" —
See as well posts tagged Dirac and Geometry.
Comments Off on Something
Wednesday, June 27, 2018
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:
or First Steps in the Doctrine of the Trinity”
. . . .
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 twodimensional world, you still get straight lines, but many lines make one figure. In a threedimensional 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 superpersonal—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 threepersonal 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 threepersonal 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 …
Martin Gardner on Galois—
“Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a ‘personality disorder.’ His anger was
paranoid and unremitting.”
Comments Off on Taken In
Sunday, June 24, 2018
A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M_{24} —
Related material —
See too the diamondtheorem correlation.
Comments Off on For 6/24
Saturday, June 23, 2018
Backstory for fiction fans, from Log24 on June 11 —
Related non fiction —
See as well the structure discussed in today's previous post.
Comments Off on Meanwhile …
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 2subsets of a 6set —
Comments Off on Plan 9 from Inner Space
Friday, June 22, 2018
"… lo lidchok et haketz …."
— Acceptance speech, Guardian of Zion award, 2002
Also on February 20, 2012 —
Comments Off on For the Late Charles Krauthammer
Thursday, June 21, 2018
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.
Comments Off on Models of Being
"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 "supermathematician" 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.
Comments Off on Dirac and Geometry (continued)
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 —
Comments Off on Cavell’s Matrix
Tuesday, June 19, 2018
Comments Off on Ici vient M. Jordan
Saturday, June 16, 2018
"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 —
Comments Off on For June 16
Monday, June 11, 2018
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, "UhOh" —
— 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 computereye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
Comments Off on Arty Fact
Friday, September 29, 2017
(Some Remarks for Science Addicts)
Principles —
Personalities —
* See "Tradition Twelve."
Comments Off on Principles Before Personalities*
Saturday, September 2, 2017
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Interuniversal 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.
Comments Off on A Touchstone
Sunday, December 11, 2016
*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.
Comments Off on Complexity to Simplicity via Hudson and Rosenhain*
Friday, September 16, 2016
Wittgenstein, 1939
Dolgachev and Keum, 2002
For some related material, see posts tagged Priority.
Comments Off on A CountingPattern
Tuesday, September 13, 2016
The previous post discussed the parametrization of
the 4×4 array as a vector 4space over the 2element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2subsets of a 6set, 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 2element 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.
Comments Off on Parametrizing the 4×4 Array
Tuesday, July 12, 2016
The following passage by Igor Dolgachev (Good Friday, 2003)
seems somewhat relevant (via its connection to Kummer's 16_{6} )
to previous remarks here on Dirac matrices and geometry —
Note related remarks from E. M. Bruins in 1959 —
Comments Off on Group Elements and Skew Lines
Tuesday, May 24, 2016
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 manyfaceted 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 4space over
the twoelement 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
Comments Off on Rosenhain and Göpel Revisited
Monday, November 23, 2015
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 3952 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 morerecent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post.
Comments Off on Dirac and Line Geometry
Wednesday, June 10, 2015
"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
Comments Off on Epistemic* Tetrads
Monday, February 10, 2014
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Comments Off on Mystery Box III: Inside, Outside
Sunday, September 22, 2013
From yesterday —
"… a list of group theoretic invariants
and their geometric incarnation…"
— David Lehavi on the Kummer 16_{6} configuration in 2007
Related material —
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
"This is not theology; this is mathematics."
— Steven H. Cullinane on four quartets
To wit:
Click to enlarge.
Comments Off on Incarnation, Part 2
Saturday, September 21, 2013
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine." An example— the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M_{24}. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4element
subsets of a 16element 4×4 array. It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the twoelement Galois field.
(See Diamond Theory in 1937.)
Thursday, September 5, 2013
(Continued from yesterday)
The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3space." The affine 4space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
nonisotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."

Narrative:
Aooo.
Happy birthday to Stephen King.
Comments Off on Mathematics and Narrative (continued)
Thursday, September 5, 2013
(Continued from yesterday)
The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3space." The affine 4space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
nonisotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."
Comments Off on Moonshine II
Wednesday, September 4, 2013
Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine." An example— the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M_{24}. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4element
subsets of a 16element 4×4 array. It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the twoelement Galois field.
(See Diamond Theory in 1937.)
A Google search documents the moonshine
relating Rosenhain's and Göpel's 19thcentury work
in complex analysis to M_{24 } via the book of Hudson and
the geometry of the 4×4 square.
Comments Off on Moonshine
Saturday, August 17, 2013
The following excerpt from a January 20, 2013, preprint shows that
a Galoisgeometry version of the large Desargues 15_{4}20_{3} configuration,
although based on the nineteenthcentury work of Galois* and of Fano,**
may at times have twentyfirstcentury applications.
Some context —
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 15_{4}20_{3} configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106132.
Comments Off on UptoDate Geometry
Monday, April 1, 2013
Background: Rosenhain and Göpel Tetrads in PG(3,2)
Introduction:
The Large Desargues Configuration
Added by Steven H. Cullinane on Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6volume
Principles of Geometry .
Pointline incidence in this larger configuration is,
as noted in the post of April 1 that follows
this introduction, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.

A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desarguestheorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desarguestheorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original VeblenYoung figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above DesarguesRosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finitegeometry version of Baker's Vol. I
frontispiece, see Sunday evening's finitegeometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3Space.
In summary… the following classicalgeometry figures
are closely related to the Galois geometry PG(3,2):
Volume I of Baker's Principles
has a cover closely related to
the Rosenhain tetrads in PG(3,2)

Volume IV of Baker's Principles
has a cover closely related to
the Göpel tetrads in PG(3,2)

Foundations
(click to enlarge)

Higher Geometry
(click to enlarge)

Comments Off on Desargues via Rosenhain
Sunday, March 10, 2013
(Continued)
The 16point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 16_{6} configuration.
Some closely related material:
Comments Off on Galois Space
Sunday, September 23, 2012
"We live together, we act on, and react to, one another; but always and in all circumstances we are by ourselves. The martyrs go hand in hand into the arena; they are crucified alone. Embraced, the lovers desperately try to fuse their insulated ecstasies into a single selftranscendence; in vain. By its very nature every embodied spirit is doomed to suffer and enjoy in solitude. Sensations, feelings, insights, fancies – all these are private and, except through symbols and at second hand, incommunicable. We can pool information about experiences, but never the experiences themselves. From family to nation, every human group is a society of island universes. Most island universes are sufficiently like one another to permit of inferential understanding or even of mutual empathy or "feeling into." Thus, remembering our own bereavements and humiliations, we can condole with others in analogous circumstances, can put ourselves (always, of course, in a slightly Pickwickian sense) in their places. But in certain cases communication between universes is incomplete or even nonexistent. The mind is its own place, and the places inhabited by the insane and the exceptionally gifted are so different from the places where ordinary men and women live, that there is little or no common ground of memory to serve as a basis for understanding or fellow feeling. Words are uttered, but fail to enlighten. The things and events to which the symbols refer belong to mutually exclusive realms of experience."
— The Doors of Perception
"Greet guests with a touch of glass."
— The Perception of Doors
Comments Off on Point Counterpoint
In Like Flynn
From the Wall Street Journal site Friday evening—
ESSAY September 21, 2012, 9:10 p.m. ET
Are We Really Getting Smarter?
Americans' IQ scores have risen steadily over the past century.
James R. Flynn examines why.

No, thank you. I prefer the ninth configuration as is—
Why? See Josefine Lyche's art installation "Grids, you say?"
Her reference there to "High White Noon" is perhaps
related to the use of that phrase in this journal.
The phrase is from a 2010 novel by Don DeLillo.
See "Point Omega," as well as Lyche's "Omega Point,"
in this journal.
The Wall Street Journal author above, James R. Flynn (born in 1934),
"is famous for his discovery of the Flynn effect, the continued
yearafteryear increase of IQ scores in all parts of the world."
—Wikipedia
His son Eugene Victor Flynn is a mathematician, coauthor
of the following chapter on the Kummer surface—
For use of the Kummer surface in Buddhist metaphysics, see last night's
post "Occupy Space (continued)" and the letters of Nanavira Thera from the
late 1950s at nanavira.blogspot.com.
These letters, together with Lyche's use of the phrase "high white noon,"
suggest a further quotation—
You know that it would be untrue
You know that I would be a liar
If I was to say to you
Girl, we couldn't get much higher
See also the Kummer surface at the web page Configurations and Squares.
Comments Off on Plan 9 (continued)–
Saturday, September 22, 2012
(Continued)
"The word 'space' has, as you suggest, a large number of different meanings."
— Nanavira Thera in [Early Letters. 136] 10.xii.1958
From that same letter (links added to relevant Wikipedia articles)—
Space (ākāsa) is undoubtedly used in the Suttas
to mean 'what/where the four mahābhūtas are not',
or example, the cavities in the body are called ākāsa
M.62—Vol. I, p. 423). This, clearly, is the everyday
'space' we all experience—roughly, 'What I can move
bout in', the empty part of the world. 'What you can't
ouch.' It is the 'space' of what Miss Lounsberry has so
appily described as 'the visible world of our five
senses'. I think you agree with this. And, of course, if
this is the only meaning of the word that we are
going to use, my 'superposition of several spaces' is
disqualified. So let us say 'superposition of several
extendednesses'. But when all these
extendednesses have been superposed, we get
'space'—i.e. our normal spacecontaining visible
world 'of the five senses'. But now there is another
point. Ākāsa is the negative of the four mahābhūtas,
certainly, but of the four mahābhūtas understood
in the same everyday sense—namely, solids (the
solid parts of the body, hair, nails, teeth, etc.),
liquids (urine, blood, etc.), heat and processes
(digestion) and motion or wind (N.B. not 'air').
These four, together with space, are the normal
furniture of our visible world 'of the five senses',
and it is undoubtedly thus that they are intended
in many Suttas. But there is, for example, a Sutta
(I am not sure where) in which the Ven. Sariputta
Thera is said to be able to see a pile of logs
successively as paṭhavi, āpo, tejo, and vāyo; and
it is evident that we are not on the same level.
On the everyday level a log of wood is solid and
therefore pathavi (like a bone), and certainly not
āpo, tejo, or vāyo. I said in my last letter that I
think that, in this second sense—i.e. as present in,
or constitutive of, any object (i.e. = rupa)—they
are structural and strictly parallel to nama and can
be defined exactly in terms of the Kummer
triangle. But on this fundamental level ākāsa has
no place at all, at least in the sense of our normal
everyday space. If, however, we take it as equivalent
to extendedness then it would be a given arbitrary
content—defining one sense out of many—of which
the four mahābhūtas (in the fundamental sense) are
the structure. In this sense (but only in this sense—
and it is probably an illegitimate sense of ākāsa)
the four mahābhūtas are the structure of space
(or spatial things). Quite legitimately, however, we
can say that the four mahābhūtas are the structure
of extended things—or of coloured things, or of smells,
or of tastes, and so on. We can leave the scientists'
space (full of right angles and without reference to the
things in it) to the scientists. 'Space' (= ākāsa) is the
space or emptiness of the world we live in; and this,
when analyzed, is found to depend on a complex
superposition of different extendednesses (because
all these extendednesses define the visible world
'of the five senses'—which will include, notably,
tangible objects—and this world 'of the five
senses' is the four mahābhūtas [everyday space]
and ākāsa).
Your second letter seems to suggest that the space
of the world we live in—the set of patterns
(superimposed) in which “we” are—is scientific space.
This I quite disagree with—if you do suggest it—,
since scientific space is a pure abstraction, never
experienced by anybody, whereas the superimposed
set of patterns is exactly what I experience—the set
is different for each one of us—, but in all of these
sets 'space' is infinite and undifferentiable, since it is,
by definition, in each set, 'what the four mahābhūtas
are not'.

A simpler metaphysical system along the same lines—
The theory, he had explained, was that the persona
was a fourdimensional figure, a tessaract in space,
the elementals Fire, Earth, Air, and Water permutating
and pervolving upon themselves, making a cruciform
(in threespace projection) figure of equal lines and
ninety degree angles.
— The Gameplayers of Zan ,
a 1977 novel by M. A. Foster

"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, [Early Letters, 131] 17.vii.1958
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Sunday, June 5, 2011
"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."
— Wallace Stevens, "To an Old Philosopher in Rome"
The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…
"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology
The Kummer 16_{6} Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)
For some background, see Configurations and Squares.
For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes twothirds of the MOG as originally defined .)
Related material— The Schwartz Notes of June 1.
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Sunday, March 27, 2011
On this date 106 years ago…
Prefatory note from Hudson's classic Kummer's Quartic Surface ,
Cambridge University Press, 1905—
RONALD WILLIAM HENRY TURNBULL HUDSON would have
been twentynine years old in July of this year; educated at
St Paul's School, London, and at St John's College, Cambridge,
he obtained the highest honours in the public examinations of the
University, in 1898, 1899, 1900; was elected a Fellow of St John's
College in 1900; became a Lecturer in Mathematics at University
College, Liverpool, in 1902; was D.Sc. in the University of London
in 1903; and died, as the result of a fall while climbing in Wales,
in the early autumn of 1904….
A manysided theory such as that of this volume is
generally to be won only by the work of many lives;
one who held so firmly the faith that the time is well spent
could ill be spared.
— H. F. Baker, 27 March 1905
For some more recent remarks related to the theory, see
Defining Configurations and its updates, March 2027, 2011.
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Wednesday, June 8, 2005
Kernel of Eternity
Today is the feast day of Saint Gerard Manley Hopkins, “immortal diamond.”
“At that instant he saw, in one blaze of light, an image of unutterable conviction, the reason why the artist works and lives and has his being–the reward he seeks–the only reward he really cares about, without which there is nothing. It is to snare the spirits of mankind in nets of magic, to make his life prevail through his creation, to wreak the vision of his life, the rude and painful substance of his own experience, into the congruence of blazing and enchanted images that are themselves the core of life, the essential pattern whence all other things proceed, the kernel of eternity.”
— Thomas Wolfe, Of Time and the River
“… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A_{8} on the octad, the kernel of this action being the translation group of the affine space.)”
— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)
“… donc Dieu existe, réponse!”
— attributed, some say falsely, to Leonhard Euler
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Sunday, November 9, 2003
For Hermann Weyl's Birthday:
A StructureEndowed Entity
"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structureendowed entity S, try to determine its group of automorphisms, the group of those elementwise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way."
— Hermann Weyl in Symmetry
Exercise: Apply Weyl's lesson to the following "structureendowed entity."
What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array's structure you choose to examine. It could be in the hundreds, or in the hundreds of thousands.)
Wednesday, September 3, 2003
Reciprocity
From my entry of Sept. 1, 2003:
"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."
— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994
Last year's entry on this date:
The picture above is of the complete graph K_{6 }… Six points with an edge connecting every pair of points… Fifteen edges in all.
Diamond theory describes how the 15 twoelement subsets of a sixelement set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to grouptheoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M_{24}.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Shu: Reciprocity.
Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K_{6} graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of K_{6}, applied to the sixteen 4×4 Dirac matrices, in
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
The Jewel of Arithmetic and
The Golden Theorem.
Thursday, December 5, 2002
Sacerdotal Jargon
From the website
Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:
Paper: clfalg/good9601
From: David M. Goodmanson
Address: 2725 68th Avenue S.E., Mercer Island, Washington 98040
Title: A graphical representation of the Dirac Algebra
Abstract: The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….
Published: Am. J. Phys. 64, 870880 (1996)
The following is a picture of K_{6}, the complete graph on six points. It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.
From
"The Relations between Poetry and Painting,"
by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
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