Tuesday, May 11, 2021

The Triangle Induction (Attn: Harlan Kane)

Filed under: General — Tags: , , — m759 @ 7:23 AM

Related material from Wikipedia

Keith A. Gessen (born January 9, 1975) is a Russian-born
American novelist, journalist, and literary translator.
He is co-founder and co-editor of American literary magazine


and an assistant professor of journalism at the Columbia University
Graduate School of Journalism.
Early life and education
Born Konstantin Alexandrovich Gessen into a Jewish family in Moscow….

Some related images —

The logo of a news site that yesterday
covered a Colorado Springs story:

Monday, May 10, 2021

The Party

Filed under: General — Tags: , , — m759 @ 1:12 AM

A completely unrelated book from Colorado Springs —

Thursday, February 13, 2020

Square-Triangle Mappings: The Continuous Case

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

On Feb. 11, Christian Lawson-Perfect posed an interesting question
about mappings between square and triangular grids:

For the same question posed about non -continuous bijections,
see "Triangles are Square."

I posed the related non– continuous question in correspondence in
the 1980's, and later online in 2012. Naturally, I wondered in the
1980's about the continuous  question and conformal  mappings, 
but didn't follow up that line of thought.

Perfect last appeared in this journal on May 20, 2014,
in the HTML title line for the link "offensive."

Wednesday, November 27, 2019

A Companion-Piece for the Circular Rectangle:

Filed under: General — Tags: , , — m759 @ 4:30 PM

For the circular rectangle, see today’s earlier post “Enter Jonathan Miller….”

The Square Triangle

Triangles are Square

A recent view of the above address —

Tuesday, October 8, 2019

Also* in 1984

Filed under: General — Tags: — m759 @ 11:32 AM

American Mathematical Monthly , June-July 1984 


Triangles are square

"Every triangle consists of  n congruent copies of itself"
is true if and only if  n is a square. (The proof is trivial.)
— Steven H. Cullinane

* See Cube Bricks 1984  in previous post.

Tuesday, April 17, 2018

A Necessary Possibility*

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

"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

IMAGE- Harvard University Press, 1986 - A page on Derrida's 'inscription'

An inscription from 2010 —

An inscription from 1984 —

American Mathematical Monthly, June-July 1984, p. 382


Triangles are square

"Every triangle consists of  n congruent copies of itself"
is true if and only if  n is a square. (The proof is trivial.) 
— Steven H. Cullinane

* See also other Log24 posts mentioning this phrase.

Wednesday, September 13, 2017

Summer of 1984

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Sunday, November 1, 2015

Sermon for All Saints’ Day

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

From St. Patrick's Day this year —

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

Tuesday, March 17, 2015


Filed under: General,Geometry — Tags: , , — m759 @ 11:30 AM

A sequel to Dude!

See also "Triangles are Square."

Monday, July 7, 2014

Tricky Task

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM

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.

Saturday, January 18, 2014

The Triangle Relativity Problem

Filed under: General,Geometry — Tags: , — m759 @ 5:01 PM

A sequel to last night’s post The 4×4 Relativity Problem —

IMAGE- Triangle Coordinatization

In other words, how should the triangle corresponding to
the above square be coordinatized ?

See also a post of July 8, 2012 — “Not Quite Obvious.”

Context — “Triangles Are Square,” a webpage stemming
from an American Mathematical Monthly  item published
in 1984.

Wednesday, January 8, 2014

Not Subversive, Not Fantasy

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

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

IMAGE- Internet Archive, 'Notes on Groups and Geometry, 1978-1986'

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.)

Sunday, November 24, 2013

Logic for Jews*

Filed under: General,Geometry — Tags: , , , — m759 @ 7:20 AM

The search for 1984 at the end of last evening’s post
suggests the following Sunday meditation.

My own contribution to this genre—

A triangle-decomposition result from 1984:

American Mathematical Monthly ,  June-July 1984, p. 382


Triangles are square

“Every triangle consists of n  congruent copies of itself”
is true if and only if  is a square. (The proof is trivial.)
— Steven H. Cullinane

The Orwell slogans are false. My own is not.

* The “for Jews” of the title applies to some readers of Edward Frenkel.

Wednesday, January 2, 2013

PlanetMath link

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

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:

IMAGE- Miscellanea, 129: 'Triangles are square'- Amer. Math. Monthly, Vol. 91, No. 6, June-July 1984, p. 382

Click on the Monthly  page for some background.

Sunday, July 15, 2012

Squares Are Triangular

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

"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—

IMAGE- 16 points in a square array and in a triangular array

Call a convex polytope P  an n-replica  if  P  consists of
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.


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.

Saturday, July 14, 2012


Filed under: General,Geometry — Tags: — m759 @ 2:25 PM

IMAGE- 'Lemma (mathematics)' in Wikipedia

For example—

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

… a "square-triangle" lemma:

   ( t ∈ T , t  is an  -replica )
    if and only if  
n  is a square.

  [I.e., "Every triangle is an -replica"
   is true if and only if n  is a square.]

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—

"Suppose that one side of a triangle
has length . Then it can be cut
into n  2 congruent triangles which
are similar to the original one and
whose corresponding sides to the
side of length  have lengths 1."

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.

Thursday, March 22, 2012

Square-Triangle Theorem continued

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

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 


       March 2, 2012

Triangle Tiling II 


       February 18, 2012

Triangle Tiling III 


       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 .

Wednesday, March 21, 2012

Square-Triangle Theorem

Filed under: General,Geometry — Tags: — m759 @ 10:30 PM

(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.

Thursday, January 19, 2012

Square Triangles

Filed under: General,Geometry — Tags: — m759 @ 1:26 PM

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.

Monday, January 16, 2012

Mapping Problem

Filed under: General,Geometry — Tags: , — m759 @ 5:10 PM

Thursday's post Triangles Are Square posed the problem of
finding "natural" maps from the 16 subsquares of a 4×4 square
to the 16 equilateral subtriangles of an edge-4 equilateral triangle.


Here is a trial solution of the inverse problem—


(Click for larger version.)

Exercise— Devise a test for "naturality" of
such mappings and apply it to the above.

Thursday, January 12, 2012

Triangles Are Square

Filed under: General,Geometry — Tags: , — m759 @ 11:30 AM

Coming across John H. Conway's 1991*
pinwheel  triangle decomposition this morning—


— suggested a review of a triangle decomposition result from 1984:

IMAGE- Triangle and square, each with 16 parts

Figure A

(Click the below image to enlarge.)

IMAGE- 'Triangles Are Square,' by Steven H. Cullinane (American Mathematical Monthly, 1985)

The above 1985 note immediately suggests a problem—

What mappings of a square  with c 2 congruent parts
to a triangle  with c 2 congruent parts are "natural"?**

(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)

* Communicated to Charles Radin in January 1991. The Conway
  decomposition may, of course, have been discovered much earlier.

** Update of Jan. 18, 2012— For a trial solution to the inverse
    problem, see the "Triangles are Square" page at finitegeometry.org.

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