Friday, August 16, 2013
From April 23, 2013, in
"Classical Geometry in Light of Galois Geometry"—
Click above image for some background from 1986.
Related material on six-set geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 1-2, pp 161-176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160
Comments Off on Six-Set Geometry
Monday, October 26, 2020
Wednesday, September 23, 2020
Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
Comments Off on Geometry of Even Subsets
Thursday, April 30, 2020
A version more explicitly connected to finite geometry —
For the six synthematic totals , see The Joy of Six.
Comments Off on Walpurgisnacht Geometry
Sunday, December 8, 2019
Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*
Selah.
* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.
Update of 11:02 AM ET the same day:
See also Eightfold Geometry, a note from 2010.
Comments Off on Geometry of 6 and 8
Wednesday, October 9, 2019
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."
Comments Off on The Joy of Six
Saturday, September 21, 2019
See other posts now tagged Six Dots.
Related narrative: Raiders of the Lost Ring .
Comments Off on Six Dots
Tuesday, July 2, 2019
An illustration from the previous post may be interpreted
as an attempt to unbokeh an inscape —
The 15 lines above are Euclidean lines based on pairs within a six-set.
For examples of Galois lines so based, see Six-Set Geometry:
Comments Off on Depth Psychology Meets Inscape Geometry
Monday, April 29, 2019
Comments Off on Geometry Review
Monday, February 25, 2019
". . . this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it. . . ."
— G. H. Hardy, A Mathematician's Apology
See Six-Set in this journal.
Comments Off on The Deep Six
Monday, February 18, 2019
__________________________________________________________________________
See also the previous post.
I prefer the work of Josefine Lyche on the smallest perfect number/universe.
Context —
Lyche's Lynx760 installations and Vigeland's nearby Norwegian clusterfuck.
Comments Off on The Joy of Six
Monday, August 27, 2018
From the former date above —
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Saturday, September 17, 2016
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From the latter date above —
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Tuesday, October 18, 2016
The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space nature.
Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —
“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
Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space coordinates. He describes it as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)
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From March 2018 —
Comments Off on Children of the Six Sides
Thursday, June 21, 2018
"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.
Comments Off on Dirac and Geometry (continued)
Sunday, December 10, 2017
See also Symplectic in this journal.
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
The above symplectic figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of linear (or line ) complex
in the finite projective space PG(3,2) —
Comments Off on Geometry
Monday, June 26, 2017
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.
Comments Off on Upgrading to Six
Thursday, May 25, 2017
A post of March 22, 2017, was titled "The Story of Six."
Related material from that date —
"I meant… a larger map." — Number Six in "The Prisoner"
Comments Off on The Story of Six Continues
Friday, April 14, 2017
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 —
Comments Off on Hudson and Finite Geometry
Wednesday, March 22, 2017
On a psychotherapist who died at 86 on Monday —
"He studied mathematics and statistics at the Courant Institute,
a part of New York University — he would later write … a
mathematical fable, Numberland (1987)."
— The New York Times online this evening
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This wry parable by a psychotherapist contains one basic message: though death is inevitable, each moment in life is to be cherished. In the orderly but sterile kingdom of Numberland, digits live together harmoniously under a rigid president called The Professor. Their stable society is held intact by the firm conviction that they are immortal: When has a number ever died? This placid universe is plunged into chaos when the inquisitive hero SIX crosses over into the human world and converses with a young mathematician. This supposedly impossible transition convinces the ruling hierarchy that if SIX can talk to a mortal, then the rest of the numbers are, after all, mortal. The digits conclude that any effort or achievement is pointless in the face of inevitable death, and the cipher society breaks down completely. The solution? Banish SIX to the farthest corners of kingdom. Weinberg (The Heart of Psychotherapy ) uses his fable to gently satirize the military, academics, politicians and, above all, psychiatrists. But his tale is basically inspirational; a triumphant SIX miraculously returns from exile and quells the turmoil by showing his fellow digits that knowledge of one's mortality should enrich all other experiences and that death ultimately provides a frame for the magnificent picture that is life.
Copyright 1987 Reed Business Information, Inc.
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See also The Prisoner in this journal.
Comments Off on The Story of Six
Friday, December 23, 2016
(Continued)
Code Blue
Update of 7:04 PM ET —
The source of the 404 message in the browsing history above
was the footnote below:
Comments Off on Memory, History, Geometry
Friday, December 16, 2016
These are Rothko's Swamps .
See a Log24 search for related meditations.
For all three topics combined, see Coxeter —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The Non-Euclidean Revolution
Update of 10 AM ET — Related material, with an elementary example:
Posts tagged "Defining Form." The example —
Comments Off on Memory, History, Geometry
Thursday, March 24, 2016
Comments Off on The Nervous Set*
Monday, December 14, 2015
(Continued)
See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.
The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent. Some related remarks from a novel —
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From Broken Symmetries by Paul Preuss
(first published by Simon and Schuster in 1983) —
"He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination.
He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations.
The classical fox knows many things, but the quantum-mechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods.
The psychic investigators, on the other hand, demanded to know how the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius .
'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content.
One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s then-new wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later.
Those so-called crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry."
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Particularly relevant …
"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."
Some related pure mathematics —
Comments Off on Dirac and Geometry
Tuesday, December 1, 2015
See a search for "large Desargues configuration" in this journal.
The 6 Jan. 2015 preprint "Danzer's Configuration Revisited,"
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the
interesting context of Pascal's Hexagrammum Mysticum .
They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .
This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.
The duality was, as they note, previously described in 1898.
Related material on six-set geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 1-2, pp 161-176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160
Related material on six-set geometry from a more recent source —
Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage
Comments Off on Pascal’s Finite Geometry
Friday, November 27, 2015
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69-100.
The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355-385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods."
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132
Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —
"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung
in our older language."
— Heidegger's Basic Writings ,
edited by David Farrell Krell,
Harper Collins paperback, 1993, page 441
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Comments Off on Einstein and Geometry
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 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.
Comments Off on Dirac and Line Geometry
Tuesday, July 14, 2015
Comments Off on Go Set a Structure
Saturday, January 24, 2015
Wikipedia on a 1953 novel by Theodore Sturgeon:
"The novel concerns the coming together of
six extraordinary people with strange powers…."
Review of the novel by Ted Gioia at Conceptual Fiction :
"If there is a flaw to More Than Human , it comes from
the writer’s desire to achieve more than fiction. If you
think that sci-fi books don’t pay attention to deep
inner meanings, you will be surprised by the conclusion
to this work, in which Sturgeon reaches for something
bigger than a story of this scale can deliver."
Background: Sturgeon's novella Baby is Three (1952).
See also 6! in this journal.
Comments Off on Now We Are Six (continued)
Saturday, December 6, 2014
On the feast of Saint Nicholas
See also the six posts on this year's feast of Saint Andrew
and the following from the University of St. Andrews —
Comments Off on Six-Point Theology
Friday, April 25, 2014
Comments Off on Quilt Geometry
Tuesday, April 23, 2013
The configurations recently discussed in
Classical Geometry in Light of Galois Geometry
are not unrelated to the 27 "Solomon's Seal Lines"
extensively studied in the 19th century.
See, in particular—
The following figures supply the connection of Henderson's six-set
to the Galois geometry previously discussed in "Classical Geometry…"—
Comments Off on The Six-Set
Saturday, November 10, 2012
Comments Off on Battlefield Geometry
Friday, November 9, 2012
Comments Off on Battlefield Geometry
Wednesday, April 28, 2010
Related web pages:
Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square
Related folklore:
“It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common.” –“Generalized Polygons and Semipartial Geometries,” by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6
The Miracle Octad Generator may be regarded as illustrating the folklore.
Update of August 20, 2010–
For facts rather than folklore about the above bijection, see The Moore Correspondence.
Comments Off on Eightfold Geometry
Thursday, April 22, 2010
Stanford Encyclopedia of Philosophy —
“Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”
A non-Euclidean* approach to parts–
Corresponding non-Euclidean*
projective points —
Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–
“… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:
(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.
Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”
Trudeau’s book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”
Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.
* “Non-Euclidean” here means merely “other than Euclidean.” No violation of Euclid’s parallel postulate is implied.
Comments Off on Mere Geometry
Tuesday, October 27, 2020
Monday, October 26, 2020
See also LeWitt in this journal.
Wednesday, April 8, 2020
A screenshot from 10:07 PM EDT —
See also this journal on Sunset Boulevard.
Comments Off on For LA Boulevardiers
Friday, February 7, 2020
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).
Comments Off on Correspondences
Sunday, January 26, 2020
Comments Off on Harmonic-Analysis Building Blocks
The Wall Street Journal Jan. 24 on a Duke University professor —
"Dr. Daubechies is best known for her work on mathematical structures
called wavelets; her discoveries have been so influential, in fact, that
these are referred to in the field as Daubechies wavelets. She describes
them as 'mathematical building blocks' that can be used to extract the
essential elements of images or signals without losing their quality—
in effect, a new universal language for scientists and researchers."
See also this journal on January 20-21, and …
Comments Off on Duke Blocks
Tuesday, January 21, 2020
"it remains only to choose a pleasing arrangement of {1, 2, … 7}
to label the eye-in-the-pyramid points.
there are, as it’ll turn out, 168 of ’em that’ll work."
— Comment at a weblog on November 27, 2010.
See also Log24 on that date.
The 11/27/2010 comment was on a post dated November 23, 2010.
See also Log24 on that date.
Precisely.
Comments Off on Eye-in-the-Pyramid Points
Monday, January 20, 2020
The Fourfold Square and Eightfold Cube
Related material: A Google image search for "field dream" + log24.
Comments Off on Dyadic Harmonic Analysis:
Saturday, December 28, 2019
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
-actions-500w.jpg)
“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight
“The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”
— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.
“It must be remarked that these 8 heptads are the key to an elegant proof….”
— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.
Comments Off on Caballo Blanco
Sunday, December 22, 2019
Exercise: Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
Comments Off on M24 from the Eightfold Cube
Friday, December 20, 2019
Continued.
An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:
Comments Off on Triangles, Spreads, Mathieu…
Wednesday, December 11, 2019
(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)
Comments Off on Miracle Octad Generator Structure
Friday, November 22, 2019
Continued from October 29, 2019.
More illustrations (click to enlarge) —
Comments Off on Triangles, Spreads, Mathieu …
Thursday, October 31, 2019
The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.
Related material — A search in this journal for “56 Triangles.”
Comments Off on 56 Triangles
Tuesday, October 29, 2019
There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is well-known that …
There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group An is generated by 3-cycles.
The alternating group A8 is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M24.
Some background —
A Log24 post of May 19, 2013, cites …
Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.
See also a Google search for “56 triangles” “56 spreads” Mathieu.
Update of October 31, 2019 — A related illustration —
Update of November 2, 2019 —
See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.
Update of December 20, 2019 —
Comments Off on Triangles, Spreads, Mathieu
Monday, October 21, 2019
Related entertainment —
Detail:
George Steiner —
"Perhaps an insane conceit."
Perhaps.
See Quantum Tesseract Theorem .
Perhaps Not.
See Dirac and Geometry .
Comments Off on Algebra and Space… Illustrated.
Friday, August 16, 2019
(Continued)
A revision of the above diagram showing
the Galois-addition-table structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
Comments Off on Nocciolo
Wednesday, July 31, 2019
The geometry of the 15 point-pairs in the previous post suggests a review:
From "Exploring Schoolgirl Space," July 8 —
The date in the previous post — Oct. 9, 2018 — also suggests a review
of posts from that date now tagged Gen-Z:
Comments Off on The Epstein Chronicles, or: Z is for Zorro
Sunday, July 14, 2019
The Quantum Tesseract Theorem Revisited
From page 274 —
"The secret is that the super-mathematician expresses by the anticommutation
of his operators the property which the geometer conceives as perpendicularity
of displacements. That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.
Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."
A related illustration, from posts tagged Dirac and Geometry —
Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.
Comments Off on Old Pathways in Science:
Friday, June 21, 2019
See also "Six-Set" in this journal
and "Cube Geometry Continues."
Comments Off on Cube Tales for Solstice Day
Friday, May 10, 2019
For fans of Resonance Science —
“When the men on the chessboard
get up and tell you where to go ….”
Comments Off on I Ching g6
Continues.
Also from Fall Equinox 2018 — Looney Tune for Physicists —
Comments Off on Desperately Seeking Resonance
Thursday, May 9, 2019
(For other posts on the continuing triumph of entertainment
over truth, see a Log24 search for "Night at the Museum.")
See also yesterday's post When the Men and today's previous post.
Comments Off on Blade and Chalice at the Museum
Comments Off on Defense Against the Dark Arts
Wednesday, May 8, 2019
In Memoriam . . .
"When the men on the chessboard
get up and tell you where to go …."
"The I Ching encodes the geometry of the fabric of spacetime."
Sure it does.
Comments Off on When the Men
Wednesday, May 1, 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 in 1973
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also Ultron Cube.
Comments Off on For the First of May
Thursday, February 28, 2019
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of six-set geometry.
Comments Off on Fooling
Wednesday, February 27, 2019
From this journal on April 23, 2013 —
From this journal in 2003 —
From Wikipedia on Groundhog Day, 2019 —
Comments Off on Construction of PG(3,2) from K6
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 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.
Comments Off on Citation
Monday, February 25, 2019
See posts tagged depth.
See as well Eddington Song and the previous post.
Comments Off on “Far from the shallow now”
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 Gian-Carlo 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, Six-Set Geometry in this journal.
Comments Off on April 18, 2003 (Good Friday), Continued
Saturday, December 22, 2018
The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005 —
A more recent appearance of the configuration —
Comments Off on Cremona-Richmond
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, December 6, 2018
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Related material —
The 56 triangles of the eightfold cube . . .
Image from Christmas Day 2005.
Comments Off on The Mathieu Cube of Iain Aitchison
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
Sunday, October 21, 2018
Yesterday afternoon's post "Study in Blue and Pink" featured
an image related to the "Blade and Chalice" of Dan Brown …
Requiem for a comics character known as "The Blue Blade" —
"We all float down here."
About the corresponding "Pink Chalice," the less said the better.
Comments Off on For Connoisseurs of Bad Art
Saturday, October 20, 2018
Related Log24 posts — See Blade + Chalice.
Comments Off on Study in Blue and Pink
Tuesday, September 25, 2018
See some posts related to three names
associated with Trinity College, Cambridge —
Atiyah + Shaw + Eddington .
Comments Off on Trinity
Saturday, September 1, 2018
The date of Ron Shaw's 2016 death appears to be June 21:
All other Internet sources I have seen omit the June 21 date.
This journal on that date —
Comments Off on Ron Shaw — D. 21 June 2016
Sunday, July 1, 2018
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
| Abstract for a talk at the City University of New York:
The Experience of Meaning
Jan Zwicky, University of Victoria
09:00-09:40 Friday, April 5, 2013
Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. |
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
Comments Off on Deutsche Ordnung
Friday, June 29, 2018
From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
Related material from 1975 —
More recently …
Comments Off on Triangles in the Eightfold Cube
Thursday, March 29, 2018
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
“Is this Time Cube?”
Notes toward an answer —
And from Six-Set Geometry in this journal . . .
Comments Off on “Before Creation Itself . . .”
Monday, March 12, 2018
Remarks related to a recent film and a not-so-recent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The Saniga-Planat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Comments Off on “Quantum Tesseract Theorem?”
Thursday, March 1, 2018
Hume, from posts tagged "four-set" in this journal —
"The mind is a kind of theatre, where several perceptions
successively make their appearance; pass, repass, glide away,
and mingle in an infinite variety of postures and situations.
There is properly no simplicity in it at one time, nor identity
in different, whatever natural propension we may have
to imagine that simplicity and identity."
Paz, from a search for Paz + Identity in this journal —
|
"At the point of convergence
the play of similarities and differences
cancels itself out in order that
identity alone may shine forth.
The illusion of motionlessness,
the play of mirrors of the one:
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy
begins all over once again."
— The Monkey Grammarian
by Octavio Paz, translated by Helen Lane
|
Comments Off on The Movement of Analogy: Hume vs. Paz
Saturday, February 17, 2018
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —
- “The digital revolution based on the 2 symbols (0,1)”
- “The algebra of George Boole”
- “Binary codes”
- “Dirac’s spinors, with their up/down dichotomy”
These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
- The 2×2 square
- The 2x2x2 cube
- The 4×4 square
- The 4x4x4 cube
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on The Binary Revolution
On a Trinity classmate of Ian Macdonald (see previous post)—
Atiyah's eulogy of Shaw in Trinity Annual Record 2017
is on pages 137 through 146. The conclusion —
Comments Off on God’s Dice
Friday, February 2, 2018
"Plato's allegory of the cave describes prisoners,
inhabiting the cave since childhood, immobile,
facing an interior wall. A large fire burns behind
the prisoners, and as people pass this fire their
shadows are cast upon the cave's wall, and
these shadows of the activity being played out
behind the prisoner become the only version of
reality that the prisoner knows."
— From the Occupy Space gallery in Ireland
Comments Off on For Plato’s Cave
Monday, October 30, 2017
Location, Location, Location
From a Los Angeles Times piece on Epiphany (Jan. 6), 1988 —
“Some 30 paces east of the spooky old Chateau Marmont is
the intersection of Selma and Sunset Boulevard.” . . . .
“Though it is not much of an intersection, the owner of
the liquor store on that corner might resent that you have
slotted his parking lot in the Twilight Zone. . . .
And directly across Sunset from Selma looking south is
where the infamous Garden of Allah used to stand. . . .”
Comments Off on For Devil’s Night
Tuesday, October 10, 2017
The title refers to today's earlier post "The 35-Year Wait."
A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.
The Lévi-Strauss formula —
My "inscape" formula, from a note of Sept. 22, 1982 —
S = f ( f ( X ) ) .
Some mathematics from last year related to the 1982 formula —
See also Inscape in this journal and posts tagged Dirac and Geometry.
Comments Off on Another 35-Year Wait
Tuesday, September 12, 2017
"Truth and clarity remained his paramount goals…"
— Benedict Nightingale in today's online New York TImes on an
English theatre director, founder of the Royal Shakespeare Company,
who reportedly died yesterday at 86.
See also Paramount in this journal.
Comments Off on Goals
Monday, September 11, 2017
A sentence from the New York Times Wire discussed in the previous post —
"Through characters like Wolverine and Swamp Thing,
he helped bring a new depth to his art form."
For Wolverine and Swamp Thing in posts related to a different
art form — geometry — see …
Comments Off on New Depth
Wednesday, August 9, 2017
"Fire and Fury is one of the most popular
historical military miniatures wargames . . . .
For the new player it is an easy game
to learn and enjoy."
— fireandfury.com
Comments Off on War Game: 8/09
Thursday, May 25, 2017
Comments Off on Presidential Address of November 19, 1976
Comments Off on Deep Problems in the Faculty Lounge
Tuesday, May 2, 2017
Comments Off on Image Albums
Wednesday, April 26, 2017
A sketch, adapted tonight from Girl Scouts of Palo Alto —
From the April 14 noon post High Concept —
From the April 14 3 AM post Hudson and Finite Geometry —
From the April 24 evening post The Trials of Device —
Note that Hudson's 1905 "unfolding" of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.
Comments Off on A Tale Unfolded
Sunday, March 26, 2017
From a search in this journal for Seagram + Tradition —
Related art: Saturday afternoon's Twin Pillars of Symmetry.
Comments Off on Seagram Studies
Saturday, March 25, 2017
The phrase "twin pillars" in a New York Times Fashion & Style
article today suggests a look at another pair of pillars —
This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as "parodies of ideas (which are ghosts)." (See the previous post.)
For a relationship between a 3-dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.
Comments Off on Twin Pillars of Symmetry
… Continued from April 11, 2016, and from …
A tribute to Rothko suggested by the previous post —
For the idea of Rothko's obstacles, see Hexagram 39 in this journal.
Comments Off on Like Decorations in a Cartoon Graveyard
Wednesday, March 8, 2017
"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
Illustration —
Commentary —
|
The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):
1.  ,  ,  ,  ,  ,
2.  ,  ,  ,  ,  ,
3.  ,  ,  ,  ,  ,
4.  ,  ,  ,  ,  ,
5.  ,  ,  ,  ,  ,
6.  ,  ,  ,  ,  .
SEE ALSO: Pauli Matrices
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211-217, 1985.
Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, pp. 80-84, 1982.
Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms. New York: Plenum, pp. 47-48, 1977.
Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGraw-Hill, 1964.
Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980.
Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.
Referenced on Wolfram|Alpha: Dirac Matrices
CITE THIS AS:
Weisstein, Eric W. "Dirac Matrices."
From MathWorld— A Wolfram Web Resource.
http://mathworld.wolfram.com/DiracMatrices.html
|
Comments Off on Inscapes
Friday, February 24, 2017
Hollywood, from the Alto Nido Apartments
to Sunset Boulevard —
See also the Jan. 31 post "Sunset Passion."
Comments Off on For Your Consideration
Friday, February 3, 2017
Personally, I prefer
the religious symbolism
of Hudson Hawk .
Comments Off on Raiders of the Lost Chalice
Sunday, December 25, 2016
See also Robert M. Pirsig in this journal on Dec. 26, 2012.
Comments Off on Credit Where Due
Saturday, December 24, 2016
In memory of an American artist whose work resembles that of
the Soviet constructivist Karl Ioganson (c. 1890-1929).
The American artist reportedly died on Thursday, Dec. 22, 2016.
|
"In fact, the (re-)discovery of this novel structural principle was made in 1948-49 by a young American artist whom Koleichuk also mentions, Kenneth Snelson. In the summer of 1948, Snelson had gone to study with Joseph Albers who was then teaching at Black Mountain College. . . . One of the first works he made upon his return home was Early X Piece which he dates to December 1948 . . . . "
— "In the Laboratory of Constructivism:
Karl Ioganson's Cold Structures,"
by Maria Gough, OCTOBER Magazine, MIT,
Issue 84, Spring 1998, pp. 91-117
|
The word "constructivism" also refers to a philosophy of mathematics.
See a Log24 post, "Constructivist Witness," of 1 AM ET on the above
date of death.
Comments Off on Early X Piece
Thursday, December 22, 2016
See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.
Comments Off on The Laugh-Hospital
The title refers to a philosophy of mathematics.
For those who prefer metaphor… Folk Etymology.
See also Stages of Math at Princeton's
Institute for Advanced Study in March 2013 —
— and in this journal starting in August 2014.
Comments Off on Constructivist Witness
Monday, December 19, 2016
See also all posts now tagged Memory, History, Geometry.
Comments Off on ART WARS
The figure below is one approach to the exercise
posted here on December 10, 2016.
Some background from earlier posts —
Click the image below to enlarge it.
Comments Off on Tetrahedral Cayley-Salmon Model
Sunday, December 18, 2016
Click image to enlarge.
See also the large Desargues configuration in this journal.
Comments Off on Two Models of the Small Desargues Configuration
Saturday, December 17, 2016
Continuing the "Memory, History, Geometry" theme
from yesterday …
See Tetrahedral, Oblivion, and Tetrahedral Oblivion.
"Welcome home, Jack."
Comments Off on Tetrahedral Death Star
Friday, December 16, 2016
Comments Off on Read Something That Means Something
"… you don’t write off an aging loved one
just because he or she becomes cranky."
— Peter Schjeldahl on Rothko in The New Yorker ,
issue dated December 19 & 26, 2016, page 27
He was cranky in his forties too —
See Rothko + Swamp in this journal.
Related attitude —
From Subway Art for Times Square Church , Nov. 7
Comments Off on Rothko’s Swamps
Tuesday, December 13, 2016
John Updike on Don DeLillo's thirteenth novel, Cosmopolis —
" DeLillo’s post-Christian search for 'an order at some deep level'
has brought him to global computerization:
'the zero-oneness of the world, the digital imperative . . . . ' "
— The New Yorker , issue dated March 31, 2003
On that date ….
Related remark —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The Non-Euclidean Revolution
Comments Off on The Thirteenth Novel
Saturday, December 10, 2016
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the small-configuration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
Comments Off on Folk Etymology
Wednesday, December 7, 2016
For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of …
Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).
This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .
Update, morning of the following day (7:07 ET) — related material:
See also “56 spreads” in this journal.
Comments Off on Spreads and Conwell’s Heptads
Tuesday, October 18, 2016
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by
a blank plus the 15 line diagrams of the diamond theorem.
Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —
“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
Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vector-space
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)
Comments Off on Parametrization
Thursday, September 15, 2016
The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).
* For the definition of "perfect number," see any introductory
number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
the projective 3-space over the 2-element Galois field GF(2),
was coined by math writer Burkard Polster. Cullinane's square
model of PG(3,2) differs from the earlier tetrahedral model
discussed by Polster.
Comments Off on The Smallest Perfect Number/Universe
Tuesday, September 13, 2016
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.
Comments Off on Parametrizing the 4×4 Array
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 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.
Comments Off on The Kummer Lattice
Friday, September 9, 2016
The New York Times interviews Alan Moore —
"A version of this article appears in print on September 11, 2016,
on page BR9 of the Sunday Book Review …."
"What genres do you prefer? And which do you avoid?"
"To be honest, having worked in genre for so long, I’m happiest
when I’m outside it altogether, or perhaps more accurately,
when I can conjure multiple genres all at once, in accordance
with my theory (now available, I believe, as a greeting card and
fridge magnet) that human life as we experience it is a
simultaneous multiplicity of genres. I put it much more elegantly
on the magnet."
Comments Off on Notation
Tuesday, August 9, 2016
Tuesday, July 12, 2016
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 —
Comments Off on Group Elements and Skew Lines
Monday, June 20, 2016
See …
At the Still Point … (February 12, 2008)
For Balanchine's Birthday (January 9, 2007)
Go Set a Structure (Various dates)
and …
Comments Off on Plan 9 Continues
Friday, June 3, 2016
A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:
The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:
Comments Off on Bruins and van Dam
Monday, May 30, 2016
(A sequel to the previous post, Perfect Number)
Since antiquity, six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its
proper divisors — in the case of six: 1, 2, and 3.
The properties of a six-element set (a "6-set")
divided into three 2-sets and divided into two 3-sets
are those of what Burkard Polster, using the same
adjective in a different sense, has called
"the smallest perfect universe" — PG(3,2), the projective
3-dimensional space over the 2-element Galois field.
A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance,
that might please Yahoo CEO Marissa Mayer on her birthday —
The semantic turnaround here in the meaning of "perfect"
is accompanied by a model turnaround in the picture of PG(3,2) as
Polster's tetrahedral model is replaced by Cullinane's square model.
Further background from the previous post —
See also Kirkman's Schoolgirl Problem.
Comments Off on Perfect Universe
"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—
For the birthday of Marissa Mayer, who turns 41 today —
VOGUE Magazine,
AUGUST 16, 2013 12:01 AM
by JACOB WEISBERG —
"As she works to reverse the fortunes of a failing Silicon Valley
giant, Yahoo’s Marissa Mayer has fueled a national debate
about the office life, motherhood, and what it takes to be the
CEO of the moment.
'I really like even numbers, and
I like heavily divisible numbers.
Twelve is my lucky number—
I just love how divisible it is.
I don’t like odd numbers, and
I really don’t like primes.
When I turned 37,
I put on a strong face, but
I was not looking forward to 37.
But 37 turned out to be a pretty amazing year.
Especially considering that
36 is divisible by twelve!'
A few things may strike you while listening to Marissa Mayer
deliver this riff . . . . "
Yes, they may.
A smaller number for Marissa's meditations:
Six has been known since antiquity as the first "perfect" number.
Why it was so called is of little interest to anyone but historians
of number theory (a discipline that is not, as Wikipedia notes,
to be confused with numerology .)
What part geometry , on the other hand, played in Marissa's education,
I do not know.
Here, for what it's worth, is a figure from a review of posts in this journal
on the key role played by the number six in geometry —
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