Vide The Maxwell Enticement . . .
The Sept. 15, 1984, date in the image below was possibly
the death date of Hawkline Monster author Richard Brautigan.
Quote adapted from a famous young-adult novel . . .
"There is such a thing as affine group generation
by permutation of parallel subspaces."
"A Nice Way to See"
Logos
"Me — Thank you very much, Claude, for your perceptive remarks.
(By the way, I had in mind only the affine block actions
that make the conjecture work.) That's all for now.
Claude — Thinking about acknowledging previous mathematical
discussion clarification
Claude — You're welcome — and with the affine block actions,
your conjecture holds exactly as stated: those coset-permuting maps
across all choices of W generate precisely AGL(n,2). It's a nice way
to see the affine group built up from many overlapping S4's.
Glad I could help!"
Background reading . . .
"Binary Coordinate Systems," by Steven H. Cullinane —
http://finitegeometry.org/sc/gen/coord.html .
Quote adapted from a famous young-adult novel . . .
"There is such a thing as affine group generation
by permutation of parallel subspaces."
Compare Google AI Mode with Grok . . .
AI Mode says a conjecture on affine group generation in binary spaces is false,
but Grok says it is true. Which conclusion is correct? Does either AI actually
supply a valid proof for its conclusion?
https://x.com/i/grok/share/e6c7759afbe9437da3be55d6cafa8269
For Steiner's "language animals" . . . and now language models . . .
Logos
|
Thursday, August 21, 2014 Nox — m759 @ 1:00 AM ( A sequel to Lux )
“By groping toward the light we are made to realize
— Arthur Koestler, The Call Girls: A Tragi-Comedy ,
Robin Williams and the Stages of Math
i) shock & denial A related description of the process —
“You know how sometimes someone tells you a theorem, — Tom Leinster yesterday at The n-Category Café |
See as well yesterday's post on a
possibly true mathematical theorem,
Checking the AI response . . .
Examples from dimensions 2, 3, and 4 indicate that the AI Mode response
may be wrong, and that the above affine-group-generation conjecture
may in fact be true. See a Log24 post from Monday, May 18.
Update . . .
See the blackboard list of "The Stages of Math" and remarks apropos of those
stages by Tom Leinster in the May 21, 2026, post "Returning to Bunker Hill
Community College."
In the square . . .
The affine group over GF(2) of order 24 acting on
the 4 vertices of a square
is generated by (and is) S4 actions
on the set of 4 vertices.
In the cube . . .
The affine group over GF(2) of order 1344 acting on
the 8 vertices of a cube
is generated by combining S4 actions
on each of the 3 sets of 4 parallel edges.
In the hypercube . . .
The affine group over GF(2) of order 322,560 acting on
the 16 vertices of a hypercube
is generated by combining S4 actions
on each of the 6 sets of 4 parallel faces.
Exercise . . . To what extent can these results be generalized?
(Specifically, is it true or false that the general n-dimensional
vector space over GF(2) is made up of sets of 4 parallel
(n-2)-dimensional subspaces, and that combining arbitrary
S4 permutations within these subspace-sets yields the
affine group on the full n-dimensional space?)
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