From a 1964 recreational-mathematics essay —
Note that the first two triangle-dissections above are analogous to
mutually orthogonal Latin squares . This implies a connection to
affine transformations within Galois geometry. See triangle graphics
in this journal.
Update of 4:40 AM ET —
Other mystical figures —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
Cicero, In Verrem II. 1. 46 —
He reached Delos. There one night he secretly 46 carried off, from the much-revered sanctuary of Apollo, several ancient and beautiful statues, and had them put on board his own transport. Next day, when the inhabitants of Delos saw their sanc- tuary stripped of its treasures, they were much distressed . . . .
Delum venit. Ibi ex fano Apollinis religiosissimo noctu clam sustulit signa pulcherrima atque anti- quissima, eaque in onerariam navem suam conicienda curavit. Postridie cum fanum spoliatum viderent ii
See also "Ex Fano" in this journal.
For more crazed gravitas, vide . . .
Addendum:
The above New Yorker passage is dated Sept. 26, 2024.
Also on that date . . .
Embedded in the Sept. 26 New Yorker review of Coppola's
Megalopolis is a ghostly transparent pyramidal figure . . .
The pyramidal figure is not unrelated to Scandia.tech —
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American Mathematical Monthly, Vol. 92, No. 6 LETTERS TO THE EDITOR Material for this department should be prepared exactly the same way as submitted manuscripts (see the inside front cover) and sent to Professor P. R. Halmos, Department of Mathematics, University of Santa Clara, Santa Clara, CA 95053 Editor: Miscellaneum 129 ("Triangles are square," June-July 1984 Monthly ) may have misled many readers. Here is some background on the item. That n2 points fall naturally into a triangular array is a not-quite-obvious fact which may have applications (e.g., to symmetries of Latin-square "k-nets") and seems worth stating more formally. To this end, call a convex polytope P an n-replica if P consists of n mutually congruent polytopes similar to P packed together. Thus, for n ∈ ℕ, (A) An equilateral triangle is an n-replica if and only if n is a square. Does this generalize to tetrahedra, or to other triangles? A regular tetrahedron is not a (23)-replica, but a tetrahedron ABCD with edges AB, BC, and CD equal and mutually orthogonal is an n-replica if and only if n is a cube. Every triangle satisfies the "if" in (A), so, letting T be the set of triangles, one might surmise that (B) ∀ t ∈ T (t is an n-replica if and only if n is a square). This, however, is false. A. J. Schwenk has pointed out that for any m ∈ ℕ, the 30°-60°-90° triangle is a (3m2)-replica, and that a right triangle with legs of integer lengths a and b is an ((a2 + b2)m2)-replica. As Schwenk notes, it does not seem obvious which other values of n can occur in counterexamples to (B). Shifting parentheses to fix (B), we get a "square-triangle" lemma:
(C) (∀ t ∈ T, t is an n-replica) if and only if n is a square.
Steven H. Cullinane
501 Follett Run Road Warren, PA 16365 |
| Name Tag | .Space | .Group | .Art |
|---|---|---|---|
| Box4 |
2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space) |
S4 = AGL(2,2) (Box4.group) |
(Box4.art) |
| Box6 |
3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set. |
S6 | |
| Box8 | 2x2x2 cube or 4×2 (4-row, 2-column) array. | S8 or A8 or AGL(3,2) of order 1344, or GL(3,2) of order 168 | |
| Box9 | The 3×3 square. | AGL(2,3) or GL(2,3) | |
| Box12 | The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M12. | Symmetries of the cube or elements of the group M12 | |
| Box13 | The 13 symmetry axes of the cube. | Symmetries of the cube. | |
| Box15 |
The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field. |
Collineations of PG(3,2) | |
| Box16 |
The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field. |
AGL(4,2), the affine group of |
|
| Box20 | The configuration representing Desargues's theorem. | ||
| Box21 | The 21 points and 21 lines of PG(2,4). | ||
| Box24 | The 24 points of the Steiner system S(5, 8, 24). | ||
| Box25 | A 5×5 array representing PG(2,5). | ||
| Box27 |
The 3-dimensional Galois affine space over the 3-element Galois field GF(3). |
||
| Box28 | The 28 bitangents of a plane quartic curve. | ||
| Box32 |
Pair of 4×4 arrays representing orthogonal Latin squares. |
Used to represent elements of AGL(4,2) |
|
| Box35 |
A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2) |
PGL(3,2), order 20,160 | |
| Box36 | Eurler's 36-officer problem. | ||
| Box45 | The 45 Pascal points of the Pascal configuration. | ||
| Box48 | The 48 elements of the group AGL(2,3). | AGL(2,3). | |
| Box56 |
The 56 three-sets within an 8-set or |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
Fermat’s Sombrero
Mexican singer Vincente Fernandez holds up the Latin Grammy award (L) for Best Ranchero Album he won for “Mas Con El Numero Uno” and the Latin Grammy Legend award at the third annual Latin Grammy Awards September 18, 2002 in Hollywood. REUTERS/Adrees Latif
From a (paper) journal note of January 5, 2002:
Princeton Alumni Weekly
January 24, 2001
The Sound of Math:
Turning a mathematical theorem
and proof into a musical
|
How do you make a musical about a bunch of dead mathematicians and one very alive, very famous, Princeton math professor? |
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Wallace Stevens:
Poet of the American Imagination
Consider these lines from
“Six Significant Landscapes” part VI:
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses-
As, for example, the ellipse of the half-moon-
Rationalists would wear sombreros.Addendum of 9/19/02: See also footnote 25 in
Theological Method and Imagination
by Julian N. Hartt
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