Log24

Friday, April 3, 2020

Galois Space

Filed under: General — Tags: — m759 @ 7:58 pm

"Galois space" is now a domain name:  galois.space.

Sunday, November 19, 2017

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 8:00 pm

This is a sequel to yesterday's post Cube Space Continued.

Tuesday, May 31, 2016

Galois Space —

Filed under: General,Geometry — Tags: — m759 @ 7:00 pm

A very brief introduction:

Seven is Heaven...

Tuesday, January 12, 2016

Harmonic Analysis and Galois Spaces

Filed under: General,Geometry — Tags: — m759 @ 7:59 am

The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.

For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.

Sunday, March 10, 2013

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 5:30 pm

(Continued)

The 16-point affine Galois space:

Further properties of this space:

In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.

Some closely related material:

  • Wolfgang Kühnel,
    "Minimal Triangulations of Kummer Varieties,"
    Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).

     

    For the first two pages, click here.

  • Jonathan Spreer and Wolfgang Kühnel,
    "Combinatorial Properties of the 3 Surface:
    Simplicial Blowups and Slicings,"
    preprint, 26 pages. (2009/10) (pdf).
    (Published in Experimental Math. 20,
    issue 2, 201–216 (2011).)

Monday, March 4, 2013

Occupy Galois Space

Filed under: General,Geometry — Tags: — m759 @ 3:00 am

Continued from February 27, the day Joseph Frank died

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review  in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times  (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

See also Galois Space and Occupy Space in this journal.

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Frank is survived by, among others, his wife, a mathematician.

Thursday, February 21, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 6:00 pm

(Continued)

The previous post suggests two sayings:

"There is  such a thing as a Galois space."

— Adapted from Madeleine L'Engle

"For every kind of vampire, there is a kind of cross."

Thomas Pynchon

Illustrations—

(Click to enlarge.)

Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 6:01 pm

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams
represents the structure of one of the 35  4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958  
[Edinburgh].
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Friday, August 15, 2025

Castle in the Air
from the Sept Tours Foundation

Filed under: General — Tags: — m759 @ 12:39 pm

Pinterest suggests that an image of the CGI castle Sept Tours
should be saved to the board "Binary Galois Spaces" —

Already there.

  Partitions  of an 8-set into four 2-sets are related to
lines in projective geometry as follows . . .

A related castle from a Groundhog Day Depth Haiku  post —

Sunday, December 29, 2024

For Harlan Kane: Husserl vs. Verhexung

Filed under: General — Tags: , , , — m759 @ 1:17 pm

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

— Wittgenstein, Philosophical Investigations  (1953),  
Section 109
 

"The newly redesigned Museum of Modern art
bracketed a rectangular open space."

— Photo caption in a Dec. 23 New York Times  obituary
 

"The literature is replete with explanations of the benefits of
bracketing
, not only in phenomenological studies but in other
types of qualitative research."

— Thomas, S. P., & Sohn, B. K. (2023).
From Uncomfortable Squirm to Self-Discovery:
A Phenomenological Analysis of the Bracketing Experience.
International Journal of Qualitative Methods, 22.
https://doi.org/10.1177/16094069231191635 

 

An application of the Husserl approach to Verhexung

Bracketing the phrase "Galois space" in the literature yields different
mathematical concepts, some derived from "Galois geometry," some
from "topological space."

The former relates to structures with a finite number of points, the latter
to structures with an infinite number of points. Sometimes the two sorts
of structure are related to one another.  For example . . .

Square and Rectangle, 16 and 24

Tuesday, August 27, 2024

For Rubik Worshippers

Filed under: General — Tags: , , , — m759 @ 2:37 pm

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

The above is six-dimensional as an affine  space, but only five-dimensional
as a  projective  space . . . the space PG(5, 2).

As the domain of the smallest model of the Klein correspondence and the
Klein quadric, PG (5,2) is not without mathematical importance.

See Chess Bricks and Ovid.group.

This post was suggested by the date July 6, 2024 in a Warren, PA obituary
and by that date in this  journal.

Friday, November 10, 2023

Logos

Filed under: General — Tags: , , , , — m759 @ 12:08 pm

Related art —

(For some backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Sunday, August 6, 2023

Contra Gombrich

Filed under: General — Tags: — m759 @ 10:42 am

A search in this journal for Cornell + Warburg suggests
a review of the concept "iconology of the interval " . . .
Ikonologie des Zwischenraums —

Some small Galois spaces (the Cullinane models)

"Yet if this Denkraum ,  this 'twilight region,'  is where the artist and
emblem-maker invent, then, as Gombrich well knew, Warburg also
constantly regrets the 'loss' of this 'thought-space,' which he also
dubs the Zwischenraum  and Wunschraum ."

— Memory, Metaphor, and Aby Warburg's Atlas of Images ,
     Christopher D. Johnson, Cornell University Press, 2012, p. 56

Sunday, June 4, 2023

The Galois Core

Filed under: General — Tags: , , — m759 @ 9:24 pm
 

  Rubik core:

 

Swarthmore Cube Project, 2008


Non- Rubik core:

Illustration for weblog post 'The Galois Core'

Central structure from a Galois plane

    (See image below.)

Some small Galois spaces (the Cullinane models)

Monday, February 27, 2023

For Gen Z: The Mark of Zorro

Filed under: General — m759 @ 12:50 pm

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Friday, December 30, 2022

Bullshit Studies: The View from East Lansing

Filed under: General — Tags: — m759 @ 1:40 pm

Detail of the above screen (click to enlarge) —

See also this  journal on the above date  — June 10, 2021.

From this journal on May 6, 2009

A related picture of images that "reappear metamorphosed
in the coordinate system of the high region" —

(For the backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Friday, May 27, 2022

Great Escapes

Filed under: General — Tags: — m759 @ 2:12 pm

The above scene from "Hanna" comes from a webpage
dated August 29, 2011. See also

this journal on that date  —

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

and today's previous "Escape" post.

Tuesday, October 20, 2020

The Leibniz Methods

Filed under: General — m759 @ 7:20 pm

IMAGE- The Leibniz medal

Click medal for some background. The medal may be regarded
as illustrating the 16-point Galois space.

Thursday, March 5, 2020

“Generated by Reflections”

Filed under: General — Tags: — m759 @ 8:42 pm

See the title in this journal.

Such generation occurs both in Euclidean space 

Order-8 group generated by reflections in midplanes of cube parallel to faces

… and in some Galois spaces —

Generating permutations for the Klein simple group of order 168 acting on the eightfold cube .

In Galois spaces, some care must be taken in defining "reflection."

Monday, December 2, 2019

Aesthetics at Harvard

Filed under: General — Tags: , , , , — m759 @ 4:05 pm

"What the piece of art is about is the gray space in the middle."

— David Bowie, as quoted in the above Crimson  piece.

Bowie's "gray space" is the space between the art and the beholder.

I prefer the gray space in the following figure —

Some small Galois spaces (the Cullinane models)

Context:  The Trinity Stone  (Log24, June 4, 2018).

Monday, October 15, 2018

For Zingari Shoolerim*

Filed under: General,Geometry — Tags: , — m759 @ 12:19 pm

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

The structure at top right is that of the
ROMA-ORAM-MARO-AMOR square
in the previous post.

* "Zingari shoolerim" is from
    Finnegans Wake .

Saturday, September 29, 2018

“Ikonologie des Zwischenraums”

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 9:29 am

The title is from Warburg. The Zwischenraum  lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
set-theoretic symmetric difference  operator.

Some small Galois spaces (the Cullinane models)

Monday, August 27, 2018

Geometry and Simplicity

Filed under: General,Geometry — Tags: , — m759 @ 9:27 pm

From

Thinking in Four Dimensions
By Dusa McDuff

"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.

I’m a geometer and topologist, which means that
I study the structure of space
. . . .

In each dimension there is a simplest space
called Euclidean space … "

— In Roman Kossak, ed.,
Simplicity:  Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705-710, 735). Kindle Edition.

For some much simpler spaces of various
dimensions, see Galois Space in this journal.

Some small Galois spaces (the Cullinane models)

http://www.log24.com/log/pix18/180827-Simplicity-Springer-April_2013_conference.jpg

Monday, June 4, 2018

The Trinity Stone Defined

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Sunday, March 4, 2018

The Square Inch Space: A Brief History

Filed under: General,Geometry — Tags: , , — m759 @ 11:21 am

1955  ("Blackboard Jungle") —

1976 —

2009 —

2016 —

 Some small Galois spaces (the Cullinane models)

Friday, September 15, 2017

Space Art

Filed under: General,Geometry — Tags: , — m759 @ 2:05 pm

Silas in "Equals" (2015) —

Ever since we were kids it's been drilled into us that 
Our purpose is to explore the universe, you know.
Outer space is where we'll find 
…  the answers to why we're here and 
…  and where we come from.

Related material — 

'The Art of Space Art' in The Paris Review, Sept. 14, 2017

See also Galois Space  in this  journal.

Tuesday, May 2, 2017

Image Albums

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Saturday, June 18, 2016

Midnight in Herald Square

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

In memory of New Yorker  artist Anatol Kovarsky,
who reportedly died at 97 on June 1.

Note the Santa, a figure associated with Macy's at Herald Square.

See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.

A note related both to Galois space and to
the "Herald Square"-tagged posts —

"There is  such a thing as a length-16 sequence."
— Saying adapted from a young-adult novel.

Sunday, May 8, 2016

The Three Solomons

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

 

THE SQUARE AND THE CUBE
by Sol LeWitt

"The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed."

"Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America  55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"

See also the Cullinane models of some small Galois spaces

Some small Galois spaces (the Cullinane models)

Friday, May 6, 2016

Review

Filed under: General,Geometry — Tags: — m759 @ 9:48 pm

 Some small Galois spaces (the Cullinane models)

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