Earlier . . .
Today . . .
* For a connection with I Ching geometry, see a different 8×8 array.
Saturday, May 10, 2025
Six-Set and Eight-Set:
I Ching* and Brick Space
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I Ching* and Brick Space
I Ching* and Brick Space
Thursday, November 21, 2024
Six-Set Geometry and Witt Designs
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 —
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Friday, August 16, 2013
Six-Set Geometry
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
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Tuesday, April 23, 2013
The Six-Set
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…"—
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Wednesday, September 24, 2025
The Mind Trick in the Attic
"In my experience, every kind of writing requires
some kind of self-soothing Jedi mind trick, and,
when it comes to essay composition,
this rectangle is mine."
— Zadie Smith in The New Yorker, Sept. 22, 2025.
A mind trick that is perhaps less self-soothing —
The dimensional reduction above, from six affine dimensions over
GF(2) to four dimensions, is, like a similar reduction in the previous post,
done by considering only even-sized subsets, then considering as elements
only the boundaries between these subsets and their complements . . .
and the Galois (XOR) sums of those boundaties.
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Friday, October 25, 2024
The Space Structures Underlying M24
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) —
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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.
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Friday, May 31, 2024
The Geometry of Hexads and Duads
Not unrelated: Six-set Geometry.
For some historical background for the first (1984)
result above, see the second (2013) result.
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Saturday, April 6, 2024
Annals of Inexcusable Pythagorism
"To enlarge this contemplation unto all the mysteries and secrets,
accomodable unto this number, were inexcusable Pythagorism…."
— Sir Thomas Browne, Hydriotaphia: Urn Burial
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Monday, February 5, 2024
Quantum Kernel Incarnate
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.
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Tuesday, August 9, 2022
The Yosemite Six
"Jurassic World: Maisie Lockwood Adventures 2: The Yosemite Six
will be released on September 27, 2022."
Thanks for the warning.
Of greater interest to some: The Number Six.
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Monday, August 1, 2022
Interality Again: The Art of the Gefüge
"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.
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Sunday, January 2, 2022
Annals of Modernism: UR–Grid
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:
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Monday, November 23, 2020
In the Garden of Adding
“In the garden of Adding,
Live Even and Odd….”
— The Midrash Jazz Quartet in
City of God , by E. L. Doctorow
Related material — Schoolgirls and Six-Set Geometry.
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Friday, November 13, 2020
A Dyad for Ariana
From The New Yorker yesterday —
“The professor and the politician are a dyad of perpetual myth.”
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Thursday, April 30, 2020
Walpurgisnacht Geometry
A version more explicitly connected to finite geometry —
For the six synthematic totals , see The Joy of Six.
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Sunday, December 8, 2019
Geometry of 6 and 8
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.
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Wednesday, October 9, 2019
The Joy of Six
Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
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Sunday, September 22, 2019
Whitehead and the Relativity Problem
"This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them."
— Hermann Weyl, The Classical Groups,
Princeton University Press, 1946, p. 16
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Wednesday, July 31, 2019
The Epstein Chronicles, or: Z is for Zorro
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:
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Tuesday, July 2, 2019
Depth Psychology Meets Inscape Geometry
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:
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Friday, June 21, 2019
Cube Tales for Solstice Day
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Wednesday, May 1, 2019
For the First of May
"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.
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Thursday, February 28, 2019
Fooling
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.
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Wednesday, February 27, 2019
Construction of PG(3,2) from K6
From this journal on April 23, 2013 —
From this journal in 2003 —
From Wikipedia on Groundhog Day, 2019 —
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Monday, February 25, 2019
Wednesday, February 13, 2019
April 18, 2003 (Good Friday), Continued
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
— Igor Shafarevitch, 1973 remark published as above in 1982.
"Perhaps."
— Steven H. Cullinane, February 13, 2019
|
From Log24 on Good Friday, April 18, 2003 — . . . What, indeed, is truth? I doubt that the best answer can be learned from either the Communist sympathizers of MIT or the “Red Mass” leftists of Georgetown. For a better starting point than either of these institutions, see my note of April 6, 2001, Wag the Dogma. See, too, In Principio Erat Verbum , which notes that “numbers go to heaven who know no more of God on earth than, as it were, of sun in forest gloom.” Since today is the anniversary of the death of MIT mathematics professor Gian-Carlo Rota, an example of “sun in forest gloom” seems the best answer to Pilate’s question on this holy day. See
“Examples are the stained glass windows Motto of Plato’s Academy † The Exorcist, 1973 |
Detail from an image linked to in the above footnote —
"And the darkness comprehended it not."
Id est :
A Good Friday, 2003, article by
a student of Shafarevitch —
"… there are 25 planes in W . . . . Of course,
replacing {a,b,c} by the complementary set
does not change the plane. . . ."
Of course.
See. however, Six-Set Geometry in this journal.
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Monday, August 27, 2018
Children of the Six Sides
From the former date above —
|
Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
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Thursday, March 29, 2018
“Before Creation Itself . . .”
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 . . .
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Tuesday, May 2, 2017
Image Albums
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
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Wednesday, April 26, 2017
A Tale Unfolded
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.
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