Log24

Sunday, September 4, 2022

Dice and the Eightfold Cube

Filed under: General — Tags: , , — m759 @ 4:47 pm

At Hiroshima on March 9, 2018, Aitchison discussed another 
"hexagonal array" with two added points… not at the center, but
rather at the ends  of a cube's diagonal axis of symmetry.

See some related illustrations below. 

Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

The exceptional role of  0 and  in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a 
Miracle Octad Generator octad —

Transposition of  0 and  in the knight coordinatization 
induces the symplectic polarity of PG(3,2) discussed by 
(for instance) Anne Duncan in 1968.

Friday, September 22, 2023

Figurate Space

Filed under: General — Tags: , — m759 @ 11:01 am

For the purpose of defining figurate geometry , a figurate space  might be
loosely described as any space consisting of finitely many congruent figures  —
subsets of Euclidean space such as points, line segments, squares, 
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them. 

Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate  space, considered as a collection of figures —  vertices, edges,
faces —
seen in the nineteenth century as acted upon by a group  of symmetries .

More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator 
of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set for instance, the 759 octads  that are analogous
to the faces  of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.

Counting symmetries with the orbit-stabilizer theorem

Thursday, September 21, 2023

Monday, March 7, 2022

Saturday, May 23, 2020

Structure for Linguists

Filed under: General — Tags: , — m759 @ 11:34 am

"MIT professor of linguistics Wayne O’Neil died on March 22
at his home in Somerville, Massachusetts."

MIT Linguistics, May 1, 2020

The "deep  structure" above is the plane cutting the cube in a hexagon
(as in my note Diamonds and Whirls of September 1984).

See also . . .

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

Sunday, January 6, 2019

For Broom Bridge*

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

GL(2,3) is not unrelated to GL(3,2).

See Quaternion Automorphisms 
and Spinning in Infinity.

* See Wikipedia.

Thursday, June 7, 2018

For Dan Brown

Filed under: General,Geometry — Tags: , , , — m759 @ 1:09 pm

See also Eightfold Trinity in this  journal.

Symbologist Robert Langdon views a corner of Solomon's Cube

Friday, March 23, 2018

From the Personal to the Platonic

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 am

On the Oslo artist Josefine Lyche —

"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."

Ann Cathrin Andersen
    http://bryggmagasin.no/2017/behind-the-glitter/

Personal —

The Rushkoff Logo

— From a 2016 graphic novel by Douglas Rushkoff.

See also Rushkoff and Talisman in this journal.

Platonic —

The Diamond Cube.

Compare and contrast the shifting hexagon logo in the Rushkoff novel above 
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).

Saturday, August 6, 2016

Mystic Correspondence:

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

The Cube and the Hexagram

The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic 
hexagram, or Star of David, with the six cube faces 
mapped to the six hexagram lines and the twelve  
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large). 

Exercise:  Is this noteworthy mapping** of faces to lines, 
edges to points, and vertices to triangles an isolated 
phenomenon, or can it be viewed in a larger context?

* See the discussion at magic-squares.net of
   "perimeter-magic cubes"

** Apparently derived from the Cube + Hexagon figure
    discussed here in various earlier posts. See also
    "Diamonds and Whirls," a note from 1984.

Tuesday, February 9, 2016

Cubism

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

The hexagons above appear also in Gary W. Gibbons,
"The Kummer Configuration and the Geometry of Majorana Spinors," 
1993, in a cube model of the Kummer 166 configuration

From Gary W. Gibbons, 'The Kummer Configuration and the Geometry of Majorana Spinors,' 1993, a cube model of the Kummer 16_6 configuration

Related material — The Religion of Cubism (May 9, 2003).

Tuesday, October 20, 2015

Liminal

Filed under: General — m759 @ 3:33 pm

The New York Times  has a readable, if not informative,
review of a recent controversial account of history —

"For many, it exists in a kind of liminal state,
floating somewhere between fact and mythology."

Jonathan Mahler, online Times  on Oct. 15, 2015

[See Wikipedia on Liminality.]

Mahler begins his review with a statement by the President
on the night of May 1, 2011.

A more easily checked statement quoted here  on that date:

"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."

— Victor Turner, The Forest of Symbols , Ithaca, NY,
Cornell University Press, 1967, p. 51, quoted by
Beth Barrie in "Victor Turner."

A Gestalt  from "Verhexung ," the previous post —

Guitart's statement that the above figure is a "Boolean logical cube"
seems, in the words of the Times , to be "floating somewhere
between fact and mythology."  Discuss.

(My apologies to those who feel that attempting to make sense
of Guitart makes them feel like Vin Diesel in the Dreamworld.)

Verhexung

Filed under: General — Tags: — m759 @ 5:04 am

“Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache.”

— Philosophical Investigations  (1953),  Section 109

An example of Verhexung  from the René Guitart article in the previous post

See also Ein Kampf .

Monday, October 19, 2015

Symmetric Generation of the Simple Order-168 Group

Filed under: General,Geometry — Tags: , , , — m759 @ 8:48 pm

This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —

“Hexagonal Logic of the Field F8 as a Boolean Logic
with Three Involutive Modalities,” pp. 191-220 in

The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,

Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.

See also the eightfold cube in this journal.

Wednesday, January 22, 2014

A Riddle for Davos

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 pm

Hexagonale Unwesen

Einstein and Thomas Mann, Princeton, 1938


IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.


See also the life of Diogenes Allen, a professor at Princeton
Theological Seminary, a life that reportedly ended on the date—
January 13, 2013— of the above Log24 post.

January 13 was also the dies natalis  of St. James Joyce.

Some related reflections —

"Praeterit figura huius mundi  " — I Corinthians 7:31 —

Conclusion of of "The Dead," by James Joyce—

The air of the room chilled his shoulders. He stretched himself cautiously along under the sheets and lay down beside his wife. One by one, they were all becoming shades. Better pass boldly into that other world, in the full glory of some passion, than fade and wither dismally with age. He thought of how she who lay beside him had locked in her heart for so many years that image of her lover's eyes when he had told her that he did not wish to live.

Generous tears filled Gabriel's eyes. He had never felt like that himself towards any woman, but he knew that such a feeling must be love. The tears gathered more thickly in his eyes and in the partial darkness he imagined he saw the form of a young man standing under a dripping tree. Other forms were near. His soul had approached that region where dwell the vast hosts of the dead. He was conscious of, but could not apprehend, their wayward and flickering existence. His own identity was fading out into a grey impalpable world: the solid world itself, which these dead had one time reared and lived in, was dissolving and dwindling.

A few light taps upon the pane made him turn to the window. It had begun to snow again. He watched sleepily the flakes, silver and dark, falling obliquely against the lamplight. The time had come for him to set out on his journey westward. Yes, the newspapers were right: snow was general all over Ireland. It was falling on every part of the dark central plain, on the treeless hills, falling softly upon the Bog of Allen and, farther westward, softly falling into the dark mutinous Shannon waves. It was falling, too, upon every part of the lonely churchyard on the hill where Michael Furey lay buried. It lay thickly drifted on the crooked crosses and headstones, on the spears of the little gate, on the barren thorns. His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead.

Tuesday, February 19, 2013

Configurations

Filed under: General,Geometry — Tags: , , — m759 @ 12:24 pm

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Monday, February 18, 2013

Permanence

Filed under: General,Geometry — m759 @ 2:00 pm

Inscribed hexagon (1984)

The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry

Inscribed hexagon (2013)

A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

Related material

"Here is another way to present the deep question 1984  raises…."

— "The Quest for Permanent Novelty," by Michael W. Clune,
     The Chronicle of Higher Education , Feb. 11, 2013

“What we do may be small, but it has a certain character of permanence.”

— G. H. Hardy, A Mathematician’s Apology

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