Wednesday, May 4, 2016

Golomb and Symmetry

Filed under: General,Geometry — m759 @ 12:00 PM

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.


Solomon Golomb, 1932-2016

Filed under: General,Geometry — m759 @ 4:00 AM

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.​encyclopediaofma​th.​org/
http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

Saturday, December 3, 2016

SIAM Publication

Filed under: General — m759 @ 10:01 AM

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).

Friday, November 25, 2016


Filed under: General,Geometry — Tags: , — m759 @ 12:00 AM

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.

Monday, May 9, 2016

Search for the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

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.

Sunday, May 8, 2016

The Three Solomons

Filed under: General,Geometry — Tags: , , , , — m759 @ 11:00 PM

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:

by Sol LeWitt

“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

Some small Galois spaces (the Cullinane models)

Friday, May 6, 2016


Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

 Some small Galois spaces (the Cullinane models)

Wednesday, May 4, 2016

Hail Affleck

Filed under: General — m759 @ 1:00 PM

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

Friday, January 18, 2013

Solomon’s Rep-tiles

Filed under: General,Geometry — m759 @ 1:04 PM

"Rep-tiles Revisited," by Viorel Nitica, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics ,  American Mathematical Society, 2003—

"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"—



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.

Wednesday, January 16, 2013


Filed under: General,Geometry — m759 @ 11:00 AM


IMAGE- Golomb and Mazur awarded National Medals of Science


IMAGE- The Leibniz medal

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.

Sunday, July 8, 2012

Not Quite Obvious

Filed under: General,Geometry — m759 @ 11:00 AM

"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
a triangular  array of n 2 points are "natural"?


In the figure above, whether
the 322,560 natural permutations
of the square's 16 points
map in any natural way to
  permutations of the triangle's 16 points
is not immediately apparent.


* Update of July 15, 2012 (11:07 PM ET)—

Theorem on " rep-" (Golomb's terminology)
triangles from a 1982 book—

IMAGE- Theorem (12.3) on Golomb and 'rep-k^2' triangles in book published in 1982-- 'Transformation Geometry,' by George Edward Martin

Tuesday, January 17, 2012


Filed under: General,Geometry — m759 @ 8:48 PM

In memory of Bach interpreter
Gustav Leonhardt



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