Note that in the above illustration, there are four sets of points
that, with the exception of the top front corner point h0, form
four equilateral triangles . . .
z1, z2, z3
y1– , y2–, y3–
y1+, y2+, y3+
h0, h1, h2. h3 .
Enthusiasts of sacred geometry may investigate the mystical properties
of this four-triangle (plus h0) labeling.
For a less mystical approach to the 2011 Yu-Oh illustration, see . . .
Two posts from the date below in an image from today's
previous post have now also been tagged Congregated Light.
See as well Conway and Congregated Light . . .
and, for Hotel New Hampshire fans,
Conway Scenic Railroad . . .
An instance of T. S. Eliot's poetic "still point" is the
center of a 3x3x3 Galois cube made up of 27 subcubes …
Not Rubik's puzzle, whose center is a mere mechanical contrivance.
Associated with that Galois cube is the set of
13 symmetry axes of its central subcube.
The figure above is not unrelated to the so-called "free will theorem."
Mathematician Peter J. Cameron's recent quotation of St. Bernard*
on free will and grace, while not impressive as a philosophical
statement, is at least preferable to the TV sitcom "Will and Grace."
See also the notion of free will in other posts tagged "Congregated Light."
Some context: Tom Wolfe, below, on the word "clerisy." It seems that the
word applies to many academics besides those in areas named by Wolfe.
* Vide http://www.catholictradition.org/Tradition/efficacious-grace3.htm#67 —
"De gratia et Libero arbitrio, chaps. 1 and 14."
Sometimes the word "preform" is not a misspelling.
"… there are present in every psyche forms which are unconscious
but nonetheless active — living dispositions, ideas in the Platonic sense,
that preform and continually influence our thoughts and feelings and actions."
The Source: Jung on a facultas praeformandi . . .
Illustration —
"A primordial image . . . .
the axial system of a crystal"
For those who prefer a Jewish approach to these matters —
(Post last updated at about 2:10 PM ET on Jan. 23, 2023.)
See as well . . .
April 11, 2020, was the dies natalis ,
in the Catholic sense,
of John Horton Conway.
See also this journal on November 29, 2011 —The Flight from Ennui.
Related illustration from earlier in 2011 —
See also this journal on 20 Sept. 2011 — Relativity Problem Revisited —
as well as other posts tagged Congregated Light.
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
"John Horton Conway is a cross between
Archimedes, Mick Jagger and Salvador Dalí."
— The Guardian paraphrasing Siobhan Roberts,
John Horton Conway and his Leech lattice doodle
in The Guardian . Photo: Hollandse Hoogte/Eyevine.
. . . .
"In junior school, one of Conway’s teachers had nicknamed him 'Mary'.
He was a delicate, effeminate creature. Being Mary made his life
absolute hell until he moved on to secondary school, at Liverpool’s
Holt High School for Boys. Soon after term began, the headmaster
called each boy into his office and asked what he planned to do with
his life. John said he wanted to read mathematics at Cambridge.
Instead of 'Mary' he became known as 'The Prof'. These nicknames
confirmed Conway as a terribly introverted adolescent, painfully aware
of his own suffering." — Siobhan Roberts, loc. cit.
From the previous post —
See as well this journal on the above Guardian date —
Тут Аполлон — идеал, там Ниобея — печаль….
The source: The link from St. Lucia's Day to "Congregated Light,"
and the link from there to the Pushkin verse.
See also St. Lucia's Day ten years ago in posts tagged Meadow-down.
Review:
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Related literary remarks: Congregated Light.
Hermann Weyl's 1949 remarks in this morning's post
were made at an event on March 19 in honor of Einstein's
70th birthday five days earlier.
Somehow the conclusion of Margaret Atwood's 1988 novel
Cat's Eye seems appropriate:
A footnote was added to Finite Relativity—
Background:
Weyl on what he calls the relativity problem—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
– Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"
"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, 1946, The Classical Groups , Princeton University Press, p. 16
…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array. There is no obvious solution to Weyl's relativity problem for M 24. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….
Footnote of Sept. 20, 2011:
* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction "The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters…."
See "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.
Some related articles by Curtis:
R.T. Curtis, "Natural Constructions of the Mathieu groups," Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423-429
R.T. Curtis. "Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12 and M 24" In Proceedings of 1990 LMS Durham Conference 'Groups, Combinatorics and Geometry' (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396
R.T. Curtis, "A Survey of Symmetric Generation of Sporadic Simple Groups," in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57
From Nabokov's The Gift —
Click for more about the Pushkin verse.
See also Trevanian + meadow and Congregated Light.
Margaret Soltan on a summer's-day poem by D.A. Powell—
first, a congregated light, the brilliance of a meadowland in bloom
and then the image must fail, as we must fail, as we
graceless creatures that we are, unmake and befoul our beds
don’t tell me deluge. don’t tell me heat, too damned much heat
"Specifically, your trope is the trope of every life:
the organizing of the disparate parts of a personality
into a self (a congregated light), blazing youth
(a meadowland in bloom), and then the failure
of that image, the failure of that self to sustain itself."
Alternate title for Soltan's commentary, suggested by yesterday's Portrait:
Smart Jewish Girl Fwows Up.
Midrash on Soltan—
Wert thou my enemy, O thou my friend,
How wouldst thou worse, I wonder, than thou dost
Defeat, thwart me?
"…meadow-down is not distressed
For a rainbow footing…."
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