From other posts tagged Six-Set —
The six-set also underlies the 21-point projective plane PG(2,4) —
PG(2,4) may be used to construct S(5,8,24), also known as
the large Witt design. Some related research . . .
On four codes with automorphism group PΣL(3,4)
and pseudo-embeddings of the large Witt designs,
by Bart De Bruyn (UGent) and Mou Gao (UGent),
(2020) DESIGNS CODES AND CRYPTOGRAPHY. 88(2), pp. 429-452.
Some background reading —
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
Not unrelated: Six-set Geometry.
For some historical background for the first (1984)
result above, see the second (2013) result.
A version more explicitly connected to finite geometry —
For the six synthematic totals , see The Joy of Six.
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.
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:
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 —
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
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…"—
“In the garden of Adding,
Live Even and Odd….”
— The Midrash Jazz Quartet in
City of God , by E. L. Doctorow
Related material — Schoolgirls and Six-Set Geometry.
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.
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
— Igor Shafarevitch, 1973 remark published as above in 1982.
"Perhaps."
— Steven H. Cullinane, February 13, 2019
|
From Log24 on Good Friday, April 18, 2003 — . . . What, indeed, is truth? I doubt that the best answer can be learned from either the Communist sympathizers of MIT or the “Red Mass” leftists of Georgetown. For a better starting point than either of these institutions, see my note of April 6, 2001, Wag the Dogma. See, too, In Principio Erat Verbum , which notes that “numbers go to heaven who know no more of God on earth than, as it were, of sun in forest gloom.” Since today is the anniversary of the death of MIT mathematics professor Gian-Carlo Rota, an example of “sun in forest gloom” seems the best answer to Pilate’s question on this holy day. See
“Examples are the stained glass windows Motto of Plato’s Academy † The Exorcist, 1973 |
Detail from an image linked to in the above footnote —
"And the darkness comprehended it not."
Id est :
A Good Friday, 2003, article by
a student of Shafarevitch —
"… there are 25 planes in W . . . . Of course,
replacing {a,b,c} by the complementary set
does not change the plane. . . ."
Of course.
See. however, Six-Set Geometry in this journal.
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 . . .
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
(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.
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 —
260731-Fano-1892-PG(3,2)…AI_Overview.jpg
...AI_Overview.jpg)
260731-Square_Model_of_PG(3,2)-by-Cullinane…AI_Overview.jpg
-by-Cullinane...AI_Overview.jpg)
* See Kirkman's schoolgirl problem in Wikipedia, esp.
the sections "Galois Geometry" and "Spreads and Packing."
The geometry of what might be called "Schoolgirl Space"
is, as the above image "Geometry of the Six Element Set"
indicates, the same as the geometry of the "Hexastigm"
described by Richmond (King's College, Cambridge,
March 30, 1899) —
"On the Figure of Six Points in Space of Four Dimensions,"
by H. W. Richmond, Quarterly Journal of Pure and Applied
Mathematics, Vol. 31 (1900), pp. 125-160.
This is available on the Web at Scribd.com . . .
https://www.scribd.com/document/424182244/
Richmond-QuarterlyJournal-1900-SixPoints .
Richmond's excellent historical survey, together with the Cullinane
square model of PG(3,2), provide the connection of finite geometry
to earlier, non-finite geometry that is lacking in some later treatments
of the subject . . . in particular, the connection to finite geometry of
studies of the classical sixty Pascal lines of the Hexagrammum.
The structures of the title are the even subsets of a six-set and of
an eight-set, viewed modulo set complementation.
The "Brick Space" model of PG(5,2) —
.gif)
For the M24 relationship between these spaces, of 15 and of 63 points,
see G. M. Conwell's 1910 paper "The 3-Space PG (3,2) and Its Group,"
as well as Conwell heptads in this journal.
"To enlarge this contemplation unto all the mysteries and secrets,
accomodable unto this number, were inexcusable Pythagorism…."
— Sir Thomas Browne, Hydriotaphia: Urn Burial
The "quantum kernel" of Koen Thas is a version of the incidence
structure — the Cremona-Richmond configuration — discussed
in the previous post, Doily vs. Inscape .
That post's inscape is, as noted there, an incarnation of the
abstract incidence structure. More generally, see incarnation
in this journal . . . In particular, from Michaelmas last year,
Annals of Mathematical Theology.
A somewhat more sophisticated "incarnation" example
related to the "inscape" concept —
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also Numberland in this journal.
"Schufreider shows that a network of linguistic relations
is set up between Gestalt, Ge-stell, and Gefüge, on the
one hand, and Streit, Riß, and Fuge, on the other . . . ."
— From p. 14 of French Interpretations of Heidegger ,
edited by David Pettigrew and François Raffoul.
State U. of New York Press, Albany, 2008. (Links added.)
One such "network of linguistic relations" might arise from
a non-mathematician's attempt to describe the diamond theorem.
(The phrase "network of linguistic relations" appears also in
Derrida's remarks on Husserl's Origin of Geometry .)
For more about "a system of slots," see interality in this journal.
The source of the above prefatory remarks by editors Pettigrew and Raffoul —
"If there is a specific network that is set up in 'The Origin of the Work of Art,'
a set of structural relations framed in linguistic terms, it is between
Gestalt, Ge-stell and Gefüge, on the one hand, and Streit, Riß and Fuge,
on the other; between (as we might try to translate it)
configuration, frame-work and structure (system), on the one hand, and
strife, split (slit) and slot, on the other. On our view, these two sets go
hand in hand; which means, to connect them to one another, we will
have to think of the configuration of the rift (Gestalt/Riß) as taking place
in a frame-work of strife (Ge-stell/Streit) that is composed through a system
of slots (Gefüge/Fuge) or structured openings."
— Quotation from page 197 of Schufreider, Gregory (2008):
"Sticking Heidegger with a Stela: Lacoue-Labarthe, art and politics."
Pp. 187-214 in David Pettigrew & François Raffoul (eds.),
French Interpretations of Heidegger: An Exceptional Reception.
State University of New York Press, 2008.
Update at 5:14 AM ET Wednesday, August 3, 2022 —
See also "six-set" in this journal.
"There is such a thing as a six-set."
— Saying adapted from a 1962 young-adult novel.
The above New Yorker art illustrates the 2×4 structure of
an octad in the Miracle Octad Generator of R. T. Curtis.
Enthusiasts of simplicity may note how properties of this eight-cell
2×4 grid are related to those of the smaller six-cell 3×2 grid:
See Nocciolo in this journal and . . .
Further reading on the six-set – eight-set relationship:
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."
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:
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also Ultron Cube.
From this journal on April 23, 2013 —
From this journal in 2003 —
From Wikipedia on Groundhog Day, 2019 —
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.
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.
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