This post was suggested by a film release date and by a
synchronology check in this journal.

See The Righteous Gemstone and Boole vs. Galois.
Update of 9:48 AM EDT Oct. 28 . . .
Related material —
Markdown version uploaded Oct. 28, 2025, to NotebookLM.
Galois's birthday, 1993 —
The title rectangle is featured in a recent sequel to The Galois Tesseract —
"… Évariste was born on October 25, 1811."
— Eric Temple Bell, Men of Mathematics
Related material —
But seriously . . .

Finesse —
|
Sunday December 10, 2006 m759 @ 9:00 PM
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Geometrie —
This post was suggested by a review in the Jan. 2022
Notices of the American Mathematical Society :
My own sympathies are with Veblen.
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
Stanley E. Payne and J. A. Thas in 1983* (previous post) —
“… a 4×4 grid together with
the affine lines on it is AG(2,4).”
Payne and Thas of course use their own definition
of affine lines on a grid.
Actually, a 4×4 grid together with the affine lines on it
is, viewed in a different way, not AG(2,4) but rather AG(4,2).
For AG(4,2) in the proper context, see
Affine Groups on Small Binary Spaces and
The Galois Tesseract.
* And 26 years later, in 2009.
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" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "
— A footnote from page 229 of Sydney Padua's |
A related passage —
|
From The French Mathematician 0
I had foreseen it all in precise detail. i = an imaginary being
Here, on this complex space, |
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
This is a sequel to yesterday's post Cube Space Continued.
Related material now available online —
A less business-oriented sort of virtual reality —
For example, "A very important configuration is obtained by
taking the plane section of a complete space five-point."
(Veblen and Young, 1910, p. 39)—
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —

The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Continued from earlier posts on Boole vs. Galois.

From a Google image search today for “Galois Boole.”
Click the image to enlarge it.
From Psychoanalytic Aesthetics: The British School ,
by Nicola Glover, Chapter 4 —
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In his last theoretical book, Attention and Interpretation (1970), Bion has clearly cast off the mathematical and scientific scaffolding of his earlier writings and moved into the aesthetic and mystical domain. He builds upon the central role of aesthetic intuition and the Keats's notion of the 'Language of Achievement', which
… includes language that is both Bion distinguishes it from the kind of language which is a substitute for thought and action, a blocking of achievement which is lies [sic ] in the realm of 'preconception' – mindlessness as opposed to mindfulness. The articulation of this language is possible only through love and gratitude; the forces of envy and greed are inimical to it.. This language is expressed only by one who has cast off the 'bondage of memory and desire'. He advised analysts (and this has caused a certain amount of controversy) to free themselves from the tyranny of the past and the future; for Bion believed that in order to make deep contact with the patient's unconscious the analyst must rid himself of all preconceptions about his patient – this superhuman task means abandoning even the desire to cure . The analyst should suspend memories of past experiences with his patient which could act as restricting the evolution of truth. The task of the analyst is to patiently 'wait for a pattern to emerge'. For as T.S. Eliot recognised in Four Quartets , 'only by the form, the pattern / Can words or music reach/ The stillness'.30. The poet also understood that 'knowledge' (in Bion's sense of it designating a 'preconception' which blocks thought, as opposed to his designation of a 'pre -conception' which awaits its sensory realisation), 'imposes a pattern and falsifies'
For the pattern is new in every moment The analyst, by freeing himself from the 'enchainment to past and future', casts off the arbitrary pattern and waits for new aesthetic form to emerge, which will (it is hoped) transform the content of the analytic encounter. 29. Attention and Interpretation (Tavistock, 1970), p. 125 30. Collected Poems (Faber, 1985), p. 194. 31. Ibid., p. 199. |
See also the previous posts now tagged Bion.
Preconception as mindlessness is illustrated by Rubik's cube, and
"pre -conception" as mindfulness is illustrated by n×n×n Froebel cubes
for n= 1, 2, 3, 4.
Suitably coordinatized, the Froebel cubes become Galois cubes,
and illustrate a new approach to the mathematics of space .
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.
The title is a new URL.
Midrash on the URL suffix —
" 'I/O' is a computer term of very long standing
that means 'input/output,' i.e. the means by which
a computer communicates with the outside world.
In a domain name, it's a shibboleth that implies
that the intended audience for a site is other
programmers."
— Phil Darnowsky on Dec. 18, 2014
Remarks for a wider audience —
See some Log24 posts related to Dec. 18, 2014.
Yesterday's post suggests a review of the following —
|
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups" Pages 8-9: Substructures of S(5, 8, 24) An octad is a block of S(5, 8, 24). Theorem 5.1
Let B0 be a fixed octad. The 30 octads disjoint from B0
the design of the points and affine hyperplanes in AG(4, 2), Proof…. … (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2) An explicit construction of the vector space is also easy….) |
Related material: Posts tagged Priority.
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
“Symmetry invariance in a diamond ring.”
See also related historical remarks by Weyl and by Atiyah.
The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.
Update of Aug. 7, 2013: See also an expanded PDF version.
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