Tuesday, December 15, 2015
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?
Comments Off on Square Triangles
Thursday, January 19, 2012
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.
Comments Off on Square Triangles
Friday, May 20, 2022
Comments Off on Squares to Triangles
Thursday, January 12, 2012
Coming across John H. Conway's 1991*
pinwheel triangle decomposition this morning—
— suggested a review of a triangle decomposition result from 1984:
Figure A
(Click the below image to enlarge.)
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.
Comments Off on Triangles Are Square
Saturday, September 10, 2022
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)
Comments Off on Orthogonal Latin Triangles
Friday, August 13, 2021
Compare and contrast —
Update of 2:25 PM ET on Friday, August 13th, 2021 —
Plato's alleged motto, "Let no one ignorant of geometry enter,"
seems to have been of little use to those attempting to make sense
of his "divided line" analogy in the Republic.
Some related geometry —
The Divided Square :
Three Similarly Divided Squares :
Scholium —
Comments Off on The Divided Square
Thursday, February 13, 2020
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."
Comments Off on Square-Triangle Mappings: The Continuous Case
Sunday, July 15, 2012
"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.
Comments Off on Squares Are Triangular
Thursday, March 22, 2012
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 .
Comments Off on Square-Triangle Theorem continued
Wednesday, March 21, 2012
(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.
Comments Off on Square-Triangle Theorem
Sunday, March 18, 2012
The diamond shape of yesterday's noon post
is not wholly without mathematical interest …
The square-triangle theorem—
"Every triangle is an n -replica" is true
if and only if n is a square.
The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.
(See the diamond lattice in Weyl's Symmetry .)
Similarly for any square n , not just 16.
There is a group of 322,560 natural transformations
that permute the centers of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers of the 16 subtriangles
in a 16-part triangular array.
(Updated March 29, 2012, to correct wording and add Weyl link.)
Comments Off on Square-Triangle Diamond
Tuesday, May 31, 2022
Last night's posts on triangles, and today's anniversary of the
death of Evariste Galois, suggest a review . . .
"Take triangles, perhaps" . . . as a category.
And then . . . take squares, perhaps, as another category,
and then . . . find a suitable "translation machine."
See "Square Triangles."
* Title adapted from a 2001 essay by Pierre Cartier.
Comments Off on A Mad Night’s Work*
Tuesday, September 10, 2019
A search for "congruent subarrays" yields few results. Hence this post.
Some relevant mathematics: the Cullinane diamond theorem, which
deals with permutations of congruent subarrays.
A related topic: Square Triangles (December 15, 2015).
Comments Off on Congruent Subarrays
Sunday, April 15, 2018
Or: Personalities Before Principles
Personalities —
Principles —
This journal on April 28, 2004 at 7:00 AM.
Backstory —
Square Triangles in this journal.
Comments Off on Colorado Olympiad
Monday, March 11, 2024
Click to enlarge.
See as well "Triangles are Square," at
http://finitegeometry.org/sc/16/trisquare.html.
(I happened to find the Basu-Owen paper tonight
via a Google image search for "congruent subsets" . . .
as opposed to the "congruent subarrays" of
the previous post.)
Update of 3:54 PM ET Monday, March 11, 2024 —
This Stanford version of my square-to-triangle mapping
is the first publication in a new Zenodo community —
Citation for the research note:
Cullinane, Steven H. (2024). Fundamental Figurate Geometry:
Triangle Subdivision (Version 2). Zenodo.
https://doi.org/10.5281/zenodo.10822848
(latest version as of March 15, 2024)
Comments Off on Fundamental Figurate Geometry: Triangle Subdivision
Wednesday, February 28, 2024
https://theosophy.wiki/en/Jirah_Dewey_Buck —
" Dr. Jirah Dewey Buck (November 20, 1838 – December 13, 1916)
was a physician who worked to establish one of the first Theosophical
lodges in the United States, the Cincinnati Theosophical Society, and
the American Section of the international Theosophical Society in 1886 . . . ."
"Buck was born in Fredonia, New York
on November 20, 1838 . . . .
[He was] 'a recognized leader of a definite school
of Masonic thought and propaganda'."
The above metadata was suggested by an image I happened to see today,
the "Tetragrammaton of Pythagoras" —
"Duck Soup" fans may recall the war between Freedonia and Sylvania.
For some images more in the spirit of Sylvania, see "Triangles Are Square."
Comments Off on A Definite School of Thought…
“The Buck Starts Here”
Friday, September 22, 2023
For the purpose of defining figurate geometry , a figurate space might be
loosely described as any space consisting of finitely many congruent figures —
subsets of Euclidean space such as points, line segments, squares,
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them.
Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate space, considered as a collection of figures — vertices, edges,
faces — seen in the nineteenth century as acted upon by a group of symmetries .
More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads that are analogous
to the faces of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.
Comments Off on Figurate Space
Tuesday, September 19, 2023
The above title for a new approach to finite geometry
was suggested by the old phrase "figurate numbers."
See other posts in this journal now tagged Figurate Geometry.
Update of 10 AM ET on Sept. 19, 2023 —
Related material from social media:
Update of 10:30 AM ET Sept. 19 —
A related topic from figurate geometry:
The square-to-triangle mapping problem.
Comments Off on Figurate Geometry
Friday, July 21, 2023
From https://clever-chat.ai/demo/32f6d9c9107/complete
at about 3 PM ET on Friday, July 21, 2023 —
|
Hi there 👋 I'm a self-service chatbot and I'm here to help you with anything related to Finite Geometry of the Square and Cube. Just ask away! 🙂
What is the diamond theorem?
The Diamond Theorem is a mathematical concept that was introduced by Steven H. Cullinane. It is closely related to finite geometry and explores the symmetries and patterns found in diamond-shaped arrays. The theorem has applications in various areas, including combinatorial designs, group theory, and quantum information theory. If you'd like more information or have any specific questions about the Diamond Theorem, feel free to ask! 😊
Is this website related to the work of Peter J. Cameron?
No, the website "Finite Geometry of the Square and Cube" is not directly related to the work of Peter J. Cameron. It is primarily focused on the study of finite geometry, particularly the symmetries and patterns found in square and cube arrays. The website is authored by Steven H. Cullinane, who has made significant contributions to the field. If you have any more questions or need further information, feel free to ask! 😊
What is the theorem of Desargues?
The theorem of Desargues, named after the French mathematician Gérard Desargues, is a fundamental result in projective geometry. It states that if two triangles are perspective from a point, then they are perspective from a line as well. In other words, if the three pairs of corresponding sides of two triangles intersect at three points that are collinear, then the three pairs of corresponding vertices of the triangles are collinear as well. This theorem has important implications in the study of projective geometry and has applications in various fields, including computer graphics and architecture. If you have any more questions or need further information, feel free to ask! 😊
|
Comments Off on “Terminator Zero: Rise of the Chatbots” … Continues.
Thursday, January 5, 2023
See also "Triangles Are Square" in 1984 —
Harvard Square:
Harvard Square, 1964
Comments Off on Logic and Geometry at Harvard
Sunday, June 26, 2022
For Monty Python —
"Glastonbury has been described as having a New Age community[6]
and possibly being where New Age beliefs originated at the turn of
the twentieth century.[7] It is notable for myths and legends often
related to Glastonbury Tor, concerning Joseph of Arimathea, the
Holy Grail and King Arthur." — Wikipedia
For American Democracy —
Related mockery from 2012 —
See also "Triangles Are Square" in 1984 —
Comments Off on Mockery Day
Wednesday, June 22, 2022
Alternate Title —
Types of Ambiguity:
The Circle in the Triangle,
the Singer in the Song.
From an excellent June 17 Wall Street Journal review of a new
Isaac Bashevis Singer book from Princeton University Press —
" 'Old Truths and New Clichés,' a collection of 19
prose articles, most appearing in English for the
first time, reveals that Singer was as consummate
an essayist as he was a teller of tales." — Benjamin Balint
From a search in this journal for Singer —
Related material —
From a post of June 2, "Self-Enclosing" —
|
"… the self-enclosing processes by which late 20th-century
American academics established and secured their status
(you painfully develop a thesis in competition with your peers,
then you keep on elaborating it until you die)."
— Colin Burrow in the June 9, 2022 issue
of London Review of Books
|
From the December 14, 2021, post Notes on Lines —
The triangle, a percussion instrument that was
featured prominently in the Tom Stoppard play
"Every Good Boy Deserves Favour."
Comments Off on Occupy Space Continues.
Tuesday, June 14, 2022
The title refers to this year's
Cannes Film Festival winner.
Related material:
From a post of June 2, "Self-Enclosing" —
|
"… the self-enclosing processes by which late 20th-century
American academics established and secured their status
(you painfully develop a thesis in competition with your peers,
then you keep on elaborating it until you die)."
— Colin Burrow in the June 9, 2022 issue
of London Review of Books
|
From a post of June 13, "The Theater Game" —
From a post of June 12, "Triangle.graphics, 2012-2022" —
Comments Off on A Triangle of Sadness
Sunday, June 12, 2022
Comments Off on Triangle.graphics, 2012-2022
Saturday, June 4, 2022
Comments Off on Old Dog, New Trick
Wednesday, June 1, 2022
"… every sufficiently good analogy is yearning to become a functor."
— John Baez, 2004: "Lessons from Topological Quantum Field Theory"
Related images — Quoting scripture at The Master's University
on Jan. 31, 2017 —
… and some images (re)posted here on that same date — Jan. 31, 2017:
A related analogy — Between (1) and (2) below —
-
A right triangle A as the sum of two smaller,
inner, right triangles B and C, each similar to A .
-
The square on the longest side of A as the sum
of the squares on the longest sides of B and C.
Exercise for Baez fans: Express the above analogy as a functor.
Comments Off on The Yearning
Thursday, May 26, 2022
Comments Off on Mystical Mathematicks
Sunday, May 22, 2022
Cannes Film Festival news in Variety yesterday —
"Östlund was last at Cannes with “The Square” in 2017, which
won the Palme d’Or. While there are many films left in competition
to screen, the reaction to “Triangle of Sadness” hints that
it could be a contender for one of the Cannes prizes."
“Triangle of Sadness” … takes its name from a fashion-world term for
the deep-V crease that appears between one’s eyebrows with stress
or age. Nothing a little Botox can’t fix."
For geometry of less social interest, see Friday's post "Squares to Triangles."
Comments Off on Social Geometry
Monday, May 16, 2022
Updates from later the same day —
Related affine structures —
See also "Square+Triangles" in this journal.
The fishlike shapes within three of the above
ninefold colored triangles suggest some . . .
Related Entertainment —
Comments Off on Sketch for a Magic Triangle
Friday, April 1, 2022
Update of 11:30 AM ET April 1, 2022 — A simpler version:
The above picture may be used to to introduce the concept of a "shape constant"
in similar figures — like the shape constant pi in a circle or the square root of 2
in a square. In each of the three similar figures at right above, the ratio of the
triangular area to the area of the attached square is a shape constant …
the same, because of their similarity, for each of the three shapes. Since the
areas of the top two triangles at right sum to that of the enclosed triangle at left,
their attached square areas sum to the area of the bottom square, Q.E.D.
The source of the proof —
Comments Off on Beauty Bare … ?
Saturday, March 26, 2022
Many structures of finite geometry can be modeled by
rectangular or cubical arrays ("boxes") —
of subsquares or subcubes (also "boxes").
Here is a draft for a table of related material, arranged
as internet URL labels.
Finite Geometry Notes — Summary Chart
|
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
322,560 permutations of the parts
of a 4×4 array (a Galois tesseract)
|
|
|
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
56 triangles in a model of Klein's quartic surface or
the 56 spreads in PG(3,2).
|
|
|
|
Box60
|
The Klein configuration.
|
|
|
|
Box64
|
Solomon's cube.
|
|
|
— Steven H. Cullinane, March 26-27, 2022
Comments Off on Box Geometry: Space, Group, Art (Work in Progress)
Saturday, September 18, 2021
Comments Off on Avoiding “Details That Might Scare You” . . .
Sunday, August 15, 2021
The following image (click to enlarge) is now the target of
a link on the phrase "similarly divided" in Friday's post
"The Divided Square."
Related material —
A version of the above Schroeder pages, dumbed down for
readers of The New Yorker —
Note that the proof under discussion has nothing to do with
the New Yorker 's rubric "Annals of Technology."
Note also the statement by Strogatz that
"Einstein’s proof reveals why the Pythagorean theorem
applies only to right triangles: they’re the only kind
made up of smaller copies of themselves."
Exercise: Discuss the truth or falsity of the Strogatz statement
after reviewing the webpage Triangles Are Square.
For approaches to geometry that are more advanced, see
this journal on the above New Yorker date — Nov. 19, 2015 —
Highlights of the Dirac-Mathieu Connection.
Comments Off on Simple Similarity
Wednesday, November 27, 2019
For the circular rectangle, see today's earlier post "Enter Jonathan Miller…."
The Square Triangle —
A recent view of the above address —
Comments Off on A Companion-Piece for the Circular Rectangle:
Tuesday, October 8, 2019
* See Cube Bricks 1984 in previous post.
Comments Off on Also* in 1984
Saturday, September 14, 2019
From "Six Significant Landscapes," by Wallace Stevens (1916) —
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.
But see "cones, waving lines, ellipses" in Kummer's Quartic Surface
(by R. W. H. T. Hudson, Cambridge University Press, 1905) and their
intimate connection with the geometry of the 4×4 square.
Comments Off on Landscape Art
Sunday, September 9, 2018
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
Comments Off on Plan 9 Continues.
Tuesday, April 17, 2018
"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.
Comments Off on A Necessary Possibility*
Wednesday, September 13, 2017
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
Comments Off on Summer of 1984
Monday, August 21, 2017
Saturday, August 6, 2016
The Cube and the Hexagram
The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic
hexagram, or Star of David, with the six cube faces
mapped to the six hexagram lines and the twelve
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large).
Exercise: Is this noteworthy mapping** of faces to lines,
edges to points, and vertices to triangles an isolated
phenomenon, or can it be viewed in a larger context?
* See the discussion at magic-squares.net of
"perimeter-magic cubes"
** Apparently derived from the Cube + Hexagon figure
discussed here in various earlier posts. See also
"Diamonds and Whirls," a note from 1984.
Comments Off on Mystic Correspondence:
Sunday, November 1, 2015
From St. Patrick's Day this year —
The March 17 post's title is a reference to a recent film.
Comments Off on Sermon for All Saints’ Day
Tuesday, March 17, 2015
A sequel to Dude!
See also "Triangles are Square."
Comments Off on Focus!
Monday, July 7, 2014
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.
Comments Off on Tricky Task
Thursday, June 26, 2014
The authors of the following offer an introduction to symmetry
in quilt blocks. They assume, perhaps rightly, that their audience
is intellectually impaired:
“A quilt block is made of 16 smaller squares.
Each small square consists of two triangles.”
Study this example of definition.
(It applies quite precisely to the sorts of square patterns
discussed in the 1976 monograph Diamond Theory , but
has little relevance for quilt blocks in general.)
Some background for those who are not intellectually impaired:
Robinson’s book Definition , in this journal and at Amazon.
Comments Off on Study This Example
Wednesday, March 12, 2014
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.
Comments Off on Obiter Dictum
Saturday, January 18, 2014
A sequel to last night's post The 4×4 Relativity Problem —
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.
Comments Off on The Triangle Relativity Problem
Wednesday, January 8, 2014
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.)
Comments Off on Not Subversive, Not Fantasy
Monday, November 25, 2013
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).
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Sunday, November 24, 2013
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:
The Orwell slogans are false. My own is not.
* The “for Jews” of the title applies to some readers of Edward Frenkel.
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Wednesday, April 10, 2013
"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."
— Geoff Dyer, review of Point Omega
Related material:
A phrase of critic Robert Hughes,
"slow art," in this journal.
A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:
The above 3×3 grid is embedded in a
somewhat more sophisticated example
of conceptual art from April 1, 2013:
Update of April 12, 2013
The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a three-triangle version of Desargues's theorem.
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
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Thursday, January 3, 2013
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.
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Wednesday, January 2, 2013
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.
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Thursday, November 22, 2012
(Continued from 1986)
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S. H. Cullinane
The relativity problem in finite geometry.
Feb. 20, 1986.
This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16
In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.
This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.
The frame: A 4×4 array.
The invariant structure:
The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
0000 0001 0010 0011
0100 0101 0110 0111
1000 1001 1010 1011
1100 1101 1110 1111
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).
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S. H. Cullinane
The relativity problem in finite geometry.
Nov. 22, 2012.
This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16
In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.
This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.
The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle.
The invariant structure:
The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).
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For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.
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Saturday, July 14, 2012
For example—
A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—
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… a "square-triangle" lemma:
(∀ t ∈ T , t is an n -replica )
if and only if n is a square.
[I.e., "Every triangle is an n -replica"
is true if and only if n is a square.]
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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 n . 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 n have lengths 1."
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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.
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Sunday, July 8, 2012
"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—
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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.
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* 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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Saturday, July 7, 2012
"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
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For remarks related by logic, see the square-triangle theorem.
For remarks related by synchronicity, see Log24 on
the above publication date, June 15, 2010.
According to Google (and Soifer's page xix), Soifer wants to captivate
young readers.
Whether young readers should be captivated is open to question.
"There is such a thing as a 4-set."
Update of 9:48 the same morning—
Amazon.com says Soifer's book was published not on June 15, but on
June 29 , 2010
(St. Peter's Day).
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Saturday, March 24, 2012
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—
- The Log24 post of March 22 on the square-triangle theorem
- The March 18 post, Square-Triangle Diamond
- 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 ]
- The result of a Cartoon Bank search suggested by no. 3 above—
(Click image for some related material.)
- A suggestion from the Cartoon Bank—
-
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
- The phrase "embedding the stone" —
Comments Off on The David Waltz…
Monday, January 16, 2012
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.
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Friday, February 25, 2011
"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.
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Saturday, February 5, 2011
Click to enlarge
This updates a webpage on the 4×4 Latin squares.
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Monday, November 10, 2008
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."
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Monday, August 18, 2008
Lotteries on
August 17,
2008 |
Pennsylvania
(No revelation) |
New York
(Revelation) |
Mid-day
(No belief) |
No belief,
no revelation
492
Chinese
Magic
Square:
4 9 2
3 5 7
8 1 6
(See below.)
|
Revelation
without belief
423
4/23:
Upscale
Realism:
Triangles
in Toronto
|
Evening
(Belief) |
Belief without
revelation
272
Rahner
on Grace
(See below.) |
Belief and
revelation
406
4/06:
Ideas
and Art
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No belief, no revelation:
An encounter with “492”–
“What is combinatorial mathematics? Combinatorial mathematics, also referred to as combinatorial analysis or combinatorics, is a mathematical discipline that began in ancient times. According to legend the Chinese Emperor Yu (c. 2200 B.C.) observed the magic square
4 9 2
3 5 7
8 1 6
on the shell of a divine turtle….”
— H.J. Ryser, Combinatorial Mathematics, Mathematical Association of America, Carus Mathematical Monographs 14 (1963) |
Belief without revelation:
Theology and human experience,
and the experience of “272”–
From Christian Tradition Today,
by Jeffrey C. K. Goh
(Peeters Publishers, 2004), p. 438:
“Insisting that theological statements are not simply deduced from human experience, Rahner nevertheless stresses the experience of grace as the ‘real, fundamental reality of Christianity itself.’ 272
272 ‘Grace’ is a key category in Rahner’s theology. He has expended a great deal of energy on this topic, earning himself the title, amongst others, of a ‘theologian of the graced search for meaning.’ See G. B. Kelly (ed.), Karl Rahner, in The Making of Modern Theology series (Edinburgh: T&T Clark, 1992).” |
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Wednesday, May 23, 2007
Strong Emergence Illustrated:
The Beauty Test
"There is no royal road
to geometry"
— Attributed to Euclid
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.)
The evening number 062, in the context of Monday's entry "No Royal Roads" and yesterday's "Jewel in the Crown," may be regarded as naming a non-royal road to geometry: either U. S. 62, a major route from Mexico to Canada (home of the late geometer H.S.M. Coxeter), or a road less traveled– namely, page 62 in Coxeter's classic Introduction to Geometry (2nd ed.):
The illustration (and definition) is
of
regular tessellations of the plane.
This topic Coxeter offers as an
illustration of remarks by G. H. Hardy
that he quotes on the preceding page:
One might argue that such beauty is
strongly emergent because of the "harmonious way" the parts fit together: the regularity (or fitting together) of the whole is not reducible to the regularity of the parts. (Regular triangles, squares, and hexagons fit together, but regular pentagons do not.)
The symmetries of these regular tessellations of the plane are less well suited as illustrations of emergence, since they are tied rather closely to symmetries of the component parts.
But the symmetries of regular tessellations of the
sphere— i.e., of the five Platonic solids–
do emerge strongly, being apparently independent of symmetries of the component parts.
Another example of strong emergence: a group of 322,560 transformations acting naturally on the 4×4 square grid— a much larger group than the group of 8 symmetries of each component (square) part.
The lottery numbers above also supply an example of strong emergence– one that nicely illustrates how it can be, in the words of Mark Bedau, "uncomfortably like magic."
(Those more comfortable with magic may note the resemblance of the central part of Coxeter's illustration to a magical counterpart– the Ojo de Dios of Mexico's Sierra Madre.)
Comments Off on Wednesday May 23, 2007
Tuesday, March 22, 2005
Make a Différance
From Frida Saal's
Lacan 
Derrida:
"Our proposal includes the lozenge (diamond) in between the names, because in the relationship / non-relationship that is established among them, a tension is created that implies simultaneously a union and a disjunction, in the perspective of a theoretical encounter that is at the same time necessary and impossible. That is the meaning of the lozenge that joins and separates the two proper names. For that reason their respective works become totally non-superposable and at the same time they were built with an awareness, or at least a partial awareness, of each other. What prevails between both of them is the différance, the Derridean signifier that will become one of the main issues in this presentation."
"Différance is that which all signs have, what constitutes them as signs, as signs are not that to which they refer: i) they differ, and hence open a space from that which they represent, and ii) they defer, and hence open up a temporal chain, or, participate in temporality. As well, following de Sassure's famous argument, signs 'mean' by differing from other signs. The coined word 'différance' refers to at once the differing and the deferring of signs. Taken to the ontological level†, the differing and deferring of signs from what they mean, means that every sign repeats the creation of space and time; and ultimately, that différance is the ultimate phenomenon in the universe, an operation that is not an operation, both active and passive, that which enables and results from Being itself."
From a text purchased on
Make a Difference Day, Oct. 23, 1999:
 22. Without using the Pythagorean Theorem prove that the hypotenuse of an isosceles right triangle will have the length  if the equal legs have the length 1. Suggestion: Consider the similar triangles in Fig. 39.
23. The ancient Greeks regarded the Pythagorean Theorem as involving areas, and they proved it by means of areas. We cannot do so now because we have not yet considered the idea of area. Assuming for the moment, however, the idea of the area of a square, use this idea instead of similar triangles and proportion in Ex. 22 above to show that x = .
— Page 98 of Basic Geometry, by George David Birkhoff, Professor of Mathematics at Harvard University, and Ralph Beatley, Associate Professor of Education at Harvard University (Scott, Foresman 1941)
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Though it may be true, as the president of Harvard recently surmised, that women are inherently inferior to men at abstract thought — in particular, pure mathematics* — they may in other respects be quite superior to men:
The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.
† For the diamond symbol at "the ontological level," see Modal Theology, Feb. 21, 2005. See also Socrates on the immortality of the soul in Plato's Meno, source of the above Basic Geometry diamond.
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Thursday, March 13, 2003
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:
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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
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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
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Thursday, September 19, 2002
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
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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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