From the webpage Diamond Theory Bibliography —
Golomb, Solomon W.
Shift register sequences (Revised edition)
Aegean Park Press, Laguna Hills, CA, 1982
The fifteen "stencils" in Golomb's Fig. VIII-8, page 219,
are the same as the fifteen affine hyperplanes that
account for patterns' symmetry in diamond theory.
This figure occurs in a discussion of Rademacher-
Walsh functions.
Material related to the previous post, "Symmetry" —
This is the group of "8 rigid motions
generated by reflections in midplanes"
of "Solomon's Cube."
Material from this journal on May 1, the date of Golomb's death —
"Weitere Informationen zu diesem Themenkreis
finden sich unter http://www.encyclopediaofmath.org/
index.php/Cullinane_diamond_theorem und
http://finitegeometry.org/sc/gen/coord.html ."
Ben Affleck sings "Aquellos Ojos Verdes"
at the end of "Hollywoodland."
This post was suggested by the death on a Beltane
of Solomon Golomb and by the appearance of
Prof. Colva M. Roney-Dougal in a CV linked to
here on yesterday's MIchaelmas morning.
Where's Y?
"Stencils" from a 1959 paper by Golomb —
These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).
This journal on Beltane (May 1), 2016 —
See "Cube Space" + Lovasz.
This search was suggested by . . .
The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , May-June 1982 —
The previous post included an illustration by Solomon Golomb
from his 1959 paper "On the Classification of Boolean Functions."
This suggests a review of some later work in this area —
This post was suggested by the word "Boolean" in a May 10
ChatGPT response —
In the above, "Boolean algebras" should be "Boolean functions,"
as indicated by Harrison's 1964 remarks.
"Stencils" from a 1959 paper by Golomb —
These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).
This journal on Beltane (May 1), 2016 —
"Rubik's Cube, and the simpler [2x2x2] Super Cube, represent
one form of mathematical and physical reality."
— Solomon W. Golomb, "Rubik's Cube and Quarks:
Twists on the eight corner cells of Rubik's Cube
provide a model for many aspects of quark behavior,"
American Scientist , Vol. 70, No. 3 (May-June 1982), pp. 257-259
From the last (Nov. 14, 2022) of the Log24 posts now tagged Groups and Spaces —
From the first (June 21, 2010) of the Log24 posts now tagged Groups and Spaces —
"Apart from its great antiquity the picture-story mode of presentation
favored by the unconscious has the appeal of its simple utility.
A picture can be recalled in its entirety whereas an essay cannot."
— Cormac McCarthy, essay on language and the unconscious,
April 17, 2017, quoted in a post of November 9, 2022.
See also Soifer in this journal and . . .
Related philosophical remarks —
Related entertainment —
See as well . . .
Three-color patterns from 1964,
rendered as shades of grey —
A rather different approach —
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)
"Solomon Golomb’s classic book Shift Register Sequences,
published in 1967—based on his work in the 1950s—
went out of print long ago. But its content lives on. . . ."
For part of that content, see Stencils .
A :Log24 post from the date of Golomb's death —
See as well other posts on Mathematics and Narrative.
The new URL supercube.space forwards to http://box759.wordpress.com/.
The term supercube is from a 1982 article by Solomon W. Golomb.
The related new URL supercube.group forwards to a page that
describes how the 2x2x2 (or eightfold, or "super") cube's natural
underlying automorphism group is Klein's simple group of order 168.
For further context, see the new URL supercube.art.
For some background, see the phrase Cube Space in this journal.
Geometry for Jews continues.
The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , May-June 1982 —
Related geometric meditation —
Archimedes at Hiroshima
in posts tagged Aitchison.
* As opposed to Solomon's Cube .
For "the Trojan family" —
Related material on the late Solomon W. Golomb —
"While at JPL, Sol had also been teaching some classes
at the nearby universities: Caltech, USC and UCLA. In
the fall of 1962, following some changes at JPL—and
perhaps because he wanted to spend more time with
his young children— he decided to become a full-time
professor. He got offers from all three schools. He
wanted to go somewhere where he could 'make
a difference'. He was told that at Caltech 'no one has
any influence if they don’t at least have a Nobel Prize',
while at UCLA 'the UC bureaucracy is such that no one
ever has any ability to affect anything'. The result was
that—despite its much-inferior reputation at the time—
Sol chose USC. He went there in the spring of 1963 as
a Professor of Electrical Engineering—and ended up
staying for 53 years." — Stephen Wolfram, 5/25/16
See also Priority (Nov. 25) and "What's in a Name" (Dec. 1).
Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:
Hudson in 1905 —
Golomb in 1967 —
It is also likely that some figures illustrating Walsh functions as
two-color square arrays were published prior to 1976.
Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.
The three Solomons of the previous post (LeWitt,
Marcus, and Golomb) suggest the three figures
-1, 0, and 1 … symbols for the three elements
of the Galois field GF(3). This in turn suggests a
Search for The Lost Theorem. Some cross-cultural
context: The First of May, 2010.
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 —
Random thoughts suggested by the reference in the
previous post to Aegean Park Press and to stencils —
"Stencil’s entire existence is focused on the hunt for V.,
a classic novelistic quest-without-resolution (in fact, V.
might be fiction’s greatest example of a MacGuffin). V.
may be a person, or may be a place, though it could
also be neither: Pynchon calls it, at one point,
'a remarkably scattered concept' and, at another,
'the ultimate Plot Which Has No Name.' "
— Alexander Nazaryan in The New Yorker ,
article dated March 29, 2013
How about a date ?
From this journal on Good Friday, March 29, 2013 —
"Rep-tiles Revisited," by Viorel Nitica, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics , American Mathematical Society,
"The goal of this note is to take a new look at some of the most amazing objects discovered in recreational mathematics. These objects, having the curious property of making larger copies of themselves, were introduced in 1962 by Solomon W. Golomb [2], and soon afterwards were popularized by Martin Gardner [3] in Scientific American…."
2. S. W. Golomb: "Replicating Figures in the Plane," Mathematical Gazette 48, 1964, 403-412
3. M. Gardner: "On 'Rep-tiles,' Polygons That Can Make Larger and Smaller Copies of Themselves," Scientific American 208, 1963, 154-157
Two such "amazing objects"—
|
Triangle |
Square |
For a different approach to the replicating properties of these objects, see the square-triangle theorem.
For related earlier material citing Golomb, see Not Quite Obvious (July 8, 2012; scroll down to see the update of July 15.).
Golomb's 1964 Gazette article may now be purchased at JSTOR for $14.
National…
International…
Click medal for some background. The medal may be regarded
as illustrating the 16-point Galois space. (See previous post.)
Related material: Jews in Hyperspace.
"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."
— Steven H. Cullinane, letter in the
American Mathematical Monthly 1985 June-July issue
If the ancient Greeks had not been distracted by
investigations of triangular (as opposed to square )
numbers, they might have done something with this fact.
A search for occurrences of the phrase
"n2 [i.e., n 2 ] congruent triangles"
indicates only fairly recent (i.e., later than 1984) results.*
Some related material, updated this morning—
|
This suggests a problem—
What mappings of a square array of n 2 points to
In the figure above, whether |
* Update of July 15, 2012 (11:07 PM ET)—
Theorem on " rep-n 2 " (Golomb's terminology)
triangles from a 1982 book—
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