Click image for some background.

Exercise:  Note that, modulo color-interchange, the set of 15 two-color
patterns above is invariant under the group of six symmetries of the
equilateral triangle. Are there any other such sets of 15 two-color triangular
patterns that are closed as sets , modulo color-interchange, under the six
triangle symmetries and  under the 322,560 permutations of the 16
subtriangles induced by actions of the affine group AGL(4,2)
on the 16 subtriangles' centers , given a suitable coordinatization?

MathWorld.Wolfram.com has an article titled "Square-Triangle Theorem."

An article of my own, whose HTML title was previously "Triangles are Square," has been retitled accordingly.

"A figurate number… is a number
that can be represented by
a regular geometrical arrangement
of equally spaced points."

— Eric W. Weisstein at Wolfram MathWorld

For example—

Call a convex polytope P  an n-replica  if  P  consists of n 
mutually congruent polytopes similar to P  packed together.

The square-triangle theorem (or lemma) says that

"Every triangle is an n-replica"
is true if and only if n  is a square.

Equivalently,

The positive integer n  is a square
if and only if every triangle is an n-replica.

(I.e., squares are triangular.)

This supplies the converse to the saying that

Triangles Are Square.

Last night's post described a book by Alexander Soifer
on questions closely related to— and possibly
suggested by— a Miscellanea  item and a letter to
the editor in the American Mathematical Monthly ,
June-July issues of 1984 and 1985.

Further search yields a series of three papers by
Michael Beeson on the same questions. These papers are
more mathematically  presentable than Soifer's book.

Triangle Tiling I —

http://www.michaelbeeson.com/research/papers/TriangleTiling1.pdf

       March 2, 2012

Triangle Tiling II —

http://www.michaelbeeson.com/research/papers/TriangleTiling2.pdf

       February 18, 2012

Triangle Tiling III —

http://www.michaelbeeson.com/research/papers/TriangleTiling3.pdf

       March 11, 2012 

These three recent preprints replace some 2010 drafts not now available.
Here are the abstracts of those drafts—

"Tiling triangle ABC with congruent triangles similar to ABC"
 (March 13, 2010),

"Tiling a triangle with congruent triangles"
(July 1, 2010).

Beeson, like Soifer, omits any reference to the "Triangles are square" item
of 1984 and the followup letter of 1985 in the Monthly .

(Continued from March 18, 2012)

Found in a search this evening—

How Does One Cut a Triangle?  by Alexander Soifer

(Second edition, Springer, 2009. First edition published
by Soifer's Center for Excellence in Mathematical Education,
Colorado Springs, CO, in 1990.)

This book, of xxx + 174 pages, covers questions closely related
to the "square-triangle" result I published in a letter to the 
editor of the June-July 1985 American Mathematical Monthly
(Vol. 92, No. 6, p. 443).  See Square-Triangle Theorem.

Soifer's four pages of references include neither that letter
nor the Monthly  item, "Miscellaneum 129: Triangles are square"
of a year earlier that prompted the letter.

"Without the possibility that an origin can be lost, forgotten, or
alienated into what springs forth from it, an origin could not be
an origin. The possibility of inscription is thus a necessary possibility,
one that must always be possible."

— Rodolphe Gasché, The Tain of the Mirror ,
     Harvard University Press, 1986

An inscription from 2010 —

An inscription from 1984 —

* See also other Log24 posts mentioning this phrase.

From St. Patrick's Day this year —

The March 17 post's title is a reference to a recent film.

Roger Cooke in the Notices of the American
Mathematical Society , April 2010 —

“Life on the Mathematical Frontier:
Legendary Figures and Their Adventures“

“In most cases involving the modern era, there
are enough documents to produce a clear picture
of mathematical developments, and conjectures
for which there is no eyewitness or documentary
evidence are not needed. Even so, legends do
arise. (Who has not heard the ‘explanation’ of
the absence of a Nobel Prize in mathematics?)
The situation is different regarding ancient math-
ematics, however, especially in the period before
Plato’s students began to study geometry. Much
of the prehistory involves allegations about the
mysterious Pythagoreans, and sorting out what is
reliable from what is not is a tricky task.

In this article, I will begin with some modern
anecdotes that have become either legend or
folklore, then work backward in time to take a
more detailed look at Greek mathematics, especially
the Pythagoreans, Plato, and Euclid. I hope at the
very least that the reader finds my examples
amusing, that being one of my goals. If readers
also take away some new insight or mathematical
aphorisms, expressing a sense of the worthiness of
our calling, that would be even better.”

Aphorism:  “Triangles are square.” 

(American Mathematical Monthly , June-July 1984)

Insight:  The Square-Triangle Theorem.

The title is both a legal phrase and a phrase
used by Tom Wolfe in his writings on art.

See, too, the pattern of nine triangular half-squares
arranged in a 3×3 square used in the logo of  the
Jean Stephen art galleries in Minneapolis…

… and in a print at the Tate in London  (click to enlarge)—

See as well an obit of the print’s artist, Justin Knowles, who reportedly died
on Feb. 24, 2004.

Some instances of that date in this journal are related to Knowles’s aesthetics.

The title refers to that of today's previous post, which linked to
a song from the June 1, 1983, album Synchronicity .
(Cf.  that term in this journal.)

For some work of my own from the following year, 1984, see…

… as well as the Orwellian dictum Triangles Are Square.

(The cubical figure at left above is from the same month,
if not the same day, as Synchronicity —  June 21, 1983.)

The title refers to a post from July 2012:

The above post, a new description of a class of figurate
numbers that has been studied at least since Pythagoras,
shows that the "triangular numbers" of tradition are not
the only  triangular numbers.

"Thus the theory of description matters most. 
It is the theory of the word for those 
For whom the word is the making of the world…." 

— Wallace Stevens, "Description Without Place"

See also Finite Relativity (St. Cecilia's Day, 2012).

From an obituary of singer Patti Page, who died on New Year's Day—

"Clara Ann Fowler was born Nov. 8, 1927, in Claremore, Okla., and grew up in Tulsa. She was one of 11 children and was raised during the Great Depression by a father who worked for the railroad.

She told the Times that her family often did not have enough money to buy shoes. To save on electricity bills, the Fowlers listened to only a few select radio programs. Among them was 'Grand Ole Opry.'"

See also two poems by Wallace Stevens and some images related to yesterday's Log24 post.

Update of May 27, 2013:
The post below is now outdated. See
http://planetmath.org/cullinanediamondtheorem .

__________________________________________________________________

The brief note on the diamond theorem at PlanetMath
disappeared some time ago. Here is a link to its
current URL: http://planetmath.org/?op=getobj;from=lec;id=49.

Update of 3 PM ET Jan. 2, 2013—

Another item recovered from Internet storage:

Click on the Monthly  page for some background.

(Continued from 1986)

For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.

For example—

A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—

For definitions, see the 1985 letter in Triangles Are Square.

(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)

A search today for related material yielded the following—

This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite  obvious.

At any rate, it served in Problem 3 as a lemma , in the sense
described above by Wikipedia. See related remarks by Doron Zeilberger.

"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—

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

Theorem on " rep-n ² " (Golomb's terminology)
triangles from a 1982 book—

"Euclid (Ancient Greek: Εὐκλείδης Eukleidēs), fl. 300 BC, 
also known as Euclid of Alexandria, was a Greek
mathematician, often referred to as the 'Father of Geometry.'"

— Wikipedia

A Euclidean quartet (see today's previous post)—

Image by Alexander Soifer

See also a link from June 28, 2012, to a University Diaries  post
discussing "a perfection of thought."

Perfect means, among other things, completed .

See, for instance, the life of another Alexandrian who reportedly
died on the above date—

"Gabriel Georges Nahas was born in Alexandria, Egypt, on
 March 4, 1920…."

 — This afternoon's online New York Times

In Turing's Cathedral

"At the still point…" — T. S. Eliot

In memory of David L. Waltz, artificial-intelligence pioneer,
who died Thursday, March 22, 2012—

1. The Log24 post of March 22 on the square-triangle theorem
2. The March 18 post, Square-Triangle Diamond
3. Remarks from the BBC on linguistic embedding
   that begin as follows—
        "If we draw a large triangle and embed smaller triangles in it,
         how does it look?"—
   and include discussion of a South American "tribe called Piranha" [sic ]
4. The result of a Cartoon Bank search suggested by no. 3 above—
   (Click image for some related material.)
   
5. A suggestion from the Cartoon Bank—
   

6. The following from the First of May, 2010—

   The Nine Divisions of Heaven–
   

   

   Some context–

   

   "This pattern is a square divided into nine equal parts.
   It has been called the 'Holy Field' division and
   was used throughout Chinese history for many
   different purposes, most of which were connected
   with things religious, political, or philosophical."

   – The Magic Square: Cities in Ancient China,
   by Alfred Schinz, Edition Axel Menges, 1996, p. 71

7. The phrase "embedding the stone" —
   • in an article on buying jewelry,
   • in this journal yesterday, and
   • in the Bible.

"THE DIAMOND THEOREM AND QUILT PATTERNS
Victoria Blumen, Mathematics, Junior, Benedictine University
Tim Comar, Benedictine University
Mathematics
Secondary Source Research
 
Let D be a 4 by 4 block quilt shape, where each of the 16 square blocks is consists of [sic ] two triangles, one of which is colored red and the other of which is colored blue.  Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D.  The diamond theorem states that G(D) = D_g has either ordinary or color-interchange symmetry.  In this talk, we will prove the diamond theorem and explore symmetries of quilt patterns of the form G(D)."

Exercise— Correct the above statement of the theorem.

Background— This is from a Google search result at about 10:55 PM ET Feb. 25, 2011—

[DOC] THE DIAMOND THEOREM AND QUILT PATTERNS – acca.elmhurst.edu
File Format: Microsoft Word – 14 hours ago –
Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either …
acca.elmhurst.edu/…/victoria_blumen9607_
THE%20DIAMOND%20THEOREM%20AND%20QUILT%20PATTERNS…

The document is from a list of mathematics abstracts for the annual student symposium of the ACCA (Associated Colleges of the Chicago Area) held on April 10, 2010.

Update of Feb. 26— For a related remark quoted here  on the date of the student symposium, see Geometry for Generations.

Click to enlarge

This updates a webpage on the 4×4 Latin squares.

Frame Tales

From June 30 —

("Will this be on the test?")

Frame Tale One:

Frame Tale Two:

Barry Sharples
on his version of the
  Kaleidoscope Puzzle —

Background:

"A possible origin of this puzzle is found in a dialogue
 between Socrates and Meno written by the Greek philosopher,
 Plato, where a square is drawn inside a square such that
the blue square is twice the area  of the yellow square.

Colouring the triangles produces a starting pattern
which is a one-diamond figure made up of four tiles
and there are 24 different possible arrangements."

The King and the Corpse  —

"The king asked, in compensation for his toils during this strangest
of all the nights he had ever known, that the twenty-four riddle tales
told him by the specter, together with the story of the night itself,
should be made known over the whole earth
and remain eternally famous among men."

Frame Tale Three:

Finnegans Wake —

"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."

There are, however, various non-royal roads.  One of these is indicated by yesterday's Pennsylvania lottery numbers:

The mid-day number 515 may be taken as a reference to 5/15. (See the previous entry, "Angel in the Details," and 5/15.)

Birthday Song

Today is the birthday of the late Jewish media magnate and art collector Walter H. Annenberg, whose name appears on a website that includes the following text:

Shape and Space in Geometry “Making quilt blocks is an excellent way to explore symmetry. A quilt block is made of 16 smaller squares. Each small square consists of two triangles. Study this example of a quilt block: This block has a certain symmetry. The right half is a mirror image of the left, and the top half is a mirror of the bottom.” © 1997-2003 Annenberg/CPB. All rights reserved. Legal Policy

Symmetries of patterns such as the above are the subject of my 1976 monograph “ Diamond Theory,” which also deals with “shape and space in geometry,” but in a much more sophisticated way.  For more on Annenberg, see my previous entry, “Daimon Theory.”  For more on the historical significance of March 13, see Neil Sedaka, who also has a birthday today, in “ Jews in the News.”

Sedaka is, of course, noted for the hit tune “Happy Birthday, Sweet Sixteen,” our site music for today.

See also Geometry for Jews and related entries.

For the phrase “diamond theory” in a religious and philosophical context, see

Pilate, Truth, and Friday the Thirteenth.

“It’s quarter to three….” — Frank Sinatra

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

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