"Galois space" is now a domain name: galois.space.
Friday, April 3, 2020
Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
Tuesday, May 31, 2016
Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
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
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 K 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
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
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."
Illustrations—
Thursday, July 12, 2012
Galois Space
An example of lines in a Galois space * —
The 35 lines in the 3-dimensional Galois projective space PG(3,2)—
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
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 —
from the Sept Tours Foundation
Sunday, December 29, 2024
For Harlan Kane: Husserl vs. Verhexung
"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 . . .
Tuesday, August 27, 2024
For Rubik Worshippers
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
Related art —
(For some backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
Sunday, August 6, 2023
Contra Gombrich
A search in this journal for Cornell + Warburg suggests
a review of the concept "iconology of the interval " . . .
Ikonologie des Zwischenraums —
"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
Monday, February 27, 2023
Friday, December 30, 2022
Bullshit Studies: The View from East Lansing
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.)
Friday, May 27, 2022
Great Escapes
The above scene from "Hanna" comes from a webpage
dated August 29, 2011. See also …
Tuesday, October 20, 2020
The Leibniz Methods
Click medal for some background. The medal may be regarded
as illustrating the 16-point Galois space.
Thursday, March 5, 2020
“Generated by Reflections”
See the title in this journal.
Such generation occurs both in Euclidean space …
… and in some Galois spaces —
In Galois spaces, some care must be taken in defining "reflection."
Monday, December 2, 2019
Aesthetics at Harvard
"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 —
Context: The Trinity Stone (Log24, June 4, 2018).
Monday, October 15, 2018
For Zingari Shoolerim*
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”
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.
Monday, August 27, 2018
Geometry and Simplicity
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.
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 .
Sunday, March 4, 2018
The Square Inch Space: A Brief History
Friday, September 15, 2017
Space Art
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 —
See also Galois Space in this journal.
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
Saturday, June 18, 2016
Midnight in Herald Square
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 "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 —






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