Saturday, July 27, 2024
Sunday, February 4, 2024
Doily vs. Inscape: Same Abstract Structure, Different Models
-Thas-2020.jpg)
My own term "inscape" names a square incarnation of what is also
known as the "Cremona-Richmond configuration," the "generalized
quadrangle of order (2, 2)," and the "doily." —
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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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Wednesday, December 12, 2018
An Inscape for Douthat
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."
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Wednesday, March 8, 2017
Inscapes
"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
|
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. |
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Monday, October 3, 2016
Hudson’s Inscape
Yesterday evening's post Some Old Philosophy from Rome
(a reference, of course, to a Wallace Stevens poem)
had a link to posts now tagged Wittgenstein's Pentagram.
For a sequel to those posts, see posts with the term Inscape ,
a mathematical concept related to a pentagram-like shape.
The inscape concept is also, as shown by R. W. H. T. Hudson
in 1904, related to the square array of points I use to picture
PG(3,2), the projective 3-space over the 2-element field.
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Thursday, April 24, 2014
The Inscape of 24
“The more intellectual, less physical, the spell of contemplation
the more complex must be the object, the more close and elaborate
must be the comparison the mind has to keep making between
the whole and the parts, the parts and the whole.”
— The Journals and Papers of Gerard Manley Hopkins ,
edited by Humphry House, 2nd ed. (London: Oxford
University Press, 1959), p. 126, as quoted by Philip A.
Ballinger in The Poem as Sacrament
Related material from All Saints’ Day in 2012:
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Sunday, April 7, 2013
Pascal Inscape
Background: Inscapes and The 2-subsets of a 6-set are the points of a PG(3,2).
Related remarks: Classical Geometry in Light of Galois Geometry.
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Monday, March 9, 2026
The Sixteen Stone: A Hollywood Version
For an (imaginary) audience of mathematicians . . .
The 4×4 array of squares or dots that has been called
the Galois Tesseract might also be called
the Sixteen Stone. An example of such an array —
The points and lines of an "inscape", which may be identified
with those of the Cremona-Richmond configuration:
For an entirely different audience, a Hollywood 4×4 array . . .
The Sixteen Stone*
The points and lines of an "inscape"
may be identified with those of the
Cremona-Richmond configuration.
* Alternate name for a 4×4 array of unit squares — sometimes called the
"Galois tesseract" — that some fans of the rock band "Bush" may prefer.
Monday, February 9, 2026
Preprogrammed “Tiny, Pitiful Words”
from a Human Language Model
"Shawn’s characters ponder the preprogrammed compulsions
to fall in and out of love, to be overwhelmed by and then lose
all desire,
'to use the tiny, pitiful words that the creature uses
to point to invisible parts of itself, invisible parts
that grow so vast that they turn us inside out and
then swallow us up and eat us.' "
— www.nytimes.com/2026/02/08/t-magazine/wallace-shawn.html
A less "tiny, pitiful" word . . . "inscape" in this journal.
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from a Human Language Model
from a Human Language Model
Wednesday, April 2, 2025
Dramarama: The SIX Train
From a post of August 31, 2022 — "Release Dates: The Iceman Goeth" —
Also in May 1986 —
|
86-05-08… A linear complex related to M24 . Anatomy of the polarity pictured in the 86-04-26 note. 86-05-26… The 2-subsets of a 6-set are the points of a PG(3,2).
Beutelspacher's model of the 15 points of PG(3,2) |
More recently, Harrison Ford in the New York subway,
reportedly on Monday, March 31 —
See as well Agent Smith in Brick Space.
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Wednesday, August 14, 2024
For Anne Carson
From an essay by Carson in London Review of Books,
Vol. 46 No. 16 · 15 August 2024 —
"Handwriting is a mark from inside me
that I put outside me, often with a view
to showing, telling, communicating.
It carries what Gerard Manley Hopkins
calls ‘the inscape’ out.
(Note: Hopkins meant several different things
by ‘inscape’, which I don’t know enough
about his psyche or his poetics to represent here,
but those Dublin notebooks – wow!)"
For a rather different use of "inscape," see a Log24 search.
Some related mathematics, via a beta version of ChatGPT Search —
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Sunday, April 28, 2024
Minding the Gap
A design note from April 24 ten years ago —
A rather different design note from the same date ten years ago —
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Wednesday, March 13, 2024
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, October 3, 2023
Tuesday… Belgium.
The inscape in the previous post suggests a review of
work by the Belgian mathematician Koen Thas on what
might be called the "quantum tesseract theorem."
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Babes in Tweeland
The New Yorker yesterday on a film director —
"Lest viewers become even briefly comfortable with
the enchantments of his staging and of his actors’
performances, Anderson jolts them alert with
ever more audacious contrivances."
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
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Sunday, June 11, 2023
The Dreaming*
Ask Mark Wahlberg.
* See as well the September 1982 Kate Bush album.
Addendum of 10:50 AM June 11 —
My own concerns in September 1982 were
rather different —
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Thursday, May 25, 2023
Woo for Burton (Tara Isabella Burton, that is)
"William Blake's statement in The Marriage of Heaven and Hell
'Eternity is in love with the productions of time' is an adumbration
of the paradoxology of the game of hide-and-seek that Non-duality
is playing with and in celebration of itself in Ia divina commedia of
this night of its dream."
— Joseph Campbell in "The Inner Reaches of Outer Space" (©1986)
Related material from a Log24 search for "inscapes4"—
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Wednesday, February 8, 2023
Local-Global lnduced Actions
See "Two Approaches to Local-Global Symmetry"
(this journal, Jan. 19, 2023), which discusses
local group actions on plane and solid graphic
patterns that induce global group actions.
See also local and global group actions of a different sort in
the July 11, 1986, note "Inner and Outer Group Actions."
This post was suggested by some remarks of Barry Mazur,
quoted in the previous post, on " Wittgenstein's 'language game,' "
Grothendieck, global views, local views and "locales."
Further reading on "locales" — Wikipedia, Pointless topology.
The word "locale" in mathematics was apparently* introduced by Isbell —
ISBELL, JOHN R. “ATOMLESS PARTS OF SPACES.”
Mathematica Scandinavica, vol. 31, no. 1, 1972, pp. 5–32.
JSTOR, http://www.jstor.org/stable/24490585.
* According to page 841 of . . .
Johnstone, P. (2001). "Elements of the History of Locale Theory."
Pp. 835–851 in: Aull, C.E., Lowen, R. (eds) Handbook of the
History of General Topology, Vol 3. Springer, Dordrecht.
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Thursday, June 9, 2022
Some Like It Hotter
The "inscape.club" of the previous post suggests Princeton's
"Triangle Club." Related material —
From the December 14, 2021, post Notes on Lines —
The triangle, a percussion instrument that was
featured prominently in the Tom Stoppard play
"Every Good Boy Deserves Favour."
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Correspondence* Club
The new URL "inscape.club" forwards to …
http://m759.net/wordpress/?s=Inscape .
* For the "correspondences" of the above title, see …
http://m759.net/wordpress/?s=Correspondences+Ninth .
"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
– The Club Dumas , 1993
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Friday, May 6, 2022
Interality and the Bead Game
WIkipedia on the URL suffix ".io" —
"In computer science, "IO" or "I/O" is commonly used
as an abbreviation for input/output, which makes the
.io domain desirable for services that want to be
associated with technology. .io domains are often used
for open source projects, application programming
interfaces ("APIs"), startup companies, browser games,
and other online services."
An association with the Bead Game from a post of April 7, 2018 —
|
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate A less poetic meditation on the above 4x4x4 design cube —
"I saw that in the alternation between front and back, See also a related remark by Lévi-Strauss in 1955:
"…three different readings become possible: |
The recent use by a startup company of the URL "interality.io" suggests
a fourth reading for the 1955 list of Lévi-Strauss — in and out —
i.e., inner and outer group automorphisms — from a 2011 post
on the birthday of T. S. Eliot :
A transformation:
Click on the picture for details.
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Wednesday, February 23, 2022
Datascape
Another illustration of the difference between data and metadata —
A Google image search for Göpel inscape —
Data :
(Click for the full 11.7 MB version
of this data .)
Meta data :
Related film dialogue —
"Media is nothing but neuro-trigger response and viral conditioning."
"Wait, what are you two talking about?"
"Cat videos."
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Monday, December 27, 2021
Thursday, October 21, 2021
SIX — The Musical!
From an Instagram post today:
As for SIX — the non-musical —
For further details, see Lost in the Matrix.
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Tuesday, March 30, 2021
Focus
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Saturday, January 16, 2021
Dark City . . .
“Dark City is an action movie — and like all good sci-fi movies,
it has aliens in it, too. The aliens have the problem that they
do not possess individual identities or souls, and for that reason,
their race is on the brink of extinction. To prevent this from
happening they perform experiments on the inhabitants of the
city to learn the secret of individuality and to eventually acquire it.
The key ingredient is memory.”
— Chapter 13 of SHELL BEACH: The search for the final theory,
by Jesper Møller Grimstrup, published on January 10, 2021.
“She did not ask herself if the Shorter Way was really there,
did not wonder if she was easing into a delusion.
The issue was settled. Here it was.”
— Joe Hill, NOS4A2 (p. 680). William Morrow, April 30, 2013.
A different “shorter way” —
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Saturday, October 24, 2020
Grids
Wikipedia on what has been called “the doily” —
“The smallest non-trivial generalized quadrangle
is GQ(2,2), whose representation* has been dubbed
‘the doily’ by Stan Payne in 1973.”
A later publication relates the doily to grids.
From Finite Generalized Quadrangles , by Stanley E. Payne
and J. A. Thas, December 1983, at researchgate.net, pp. 81-82—
“Then the lines … define a 3×3 grid G (i.e. a grid
consisting of 9 points and 6 lines).”
. . . .
“So we have shown that the grid G can completed [sic ]
in a unique way to a grid with 8 lines and 16 points.”
. . . .
“A 4×4 grid defines a linear subspace
of the 2−(64,4,1) design, i.e. a 4×4 grid
together with the affine lines on it is AG(2,4).”
A more graphic approach from this journal —
Click the image for further details.
* This wording implies that GQ(2,2) has a unique
visual representation. It does not. See inscape .
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