Tuesday, December 15, 2015
Click image for some background.
Exercise: Note that, modulo colorinterchange, the set of 15 twocolor
patterns above is invariant under the group of six symmetries of the
equilateral triangle. Are there any other such sets of 15 twocolor triangular
patterns that are closed as sets , modulo colorinterchange, 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 "SquareTriangle Theorem."
An article of my own, whose HTML title was previously "Triangles are Square," has been retitled accordingly.
Comments Off on Square 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 16part square map in any natural way to transformations
of the 16part 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
Friday, December 18, 2020
(A sequel to the previous post, Square Space at Wikipedia)
Related remarks: A Dec. 16 Wikipedia revision by Quack5quack,
and posts in this journal tagged Helsinki Math.
Comments Off on Square Space at Athens
Thursday, December 17, 2020
The State of SquareSpace Art at Wikipedia as of December 16, 2020,
after a revision by an anonymous user on that date:
See also Square Space at Squarespace.
Comments Off on Square Space at Wikipedia
Thursday, February 13, 2020
On Feb. 11, Christian LawsonPerfect 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 SquareTriangle Mappings: The Continuous Case
Friday, December 20, 2019
Continued.
An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:
Comments Off on Triangles, Spreads, Mathieu…
Friday, November 22, 2019
Continued from October 29, 2019.
More illustrations (click to enlarge) —
Comments Off on Triangles, Spreads, Mathieu …
Thursday, October 31, 2019
The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.
Related material — A search in this journal for “56 Triangles.”
Comments Off on 56 Triangles
Tuesday, October 29, 2019
There are many approaches to constructing the Mathieu
group M_{24}. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is wellknown that …
There are 56 triangles in an 8set.
There are 56 spreads in PG(3,2).
The alternating group A_{n }is generated by 3cycles.
The alternating group A_{8 }is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M_{24}.
Some background —
A Log24 post of May 19, 2013, cites …
Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.
See also a Google search for “56 triangles” “56 spreads” Mathieu.
Update of October 31, 2019 — A related illustration —
Update of November 2, 2019 —
See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.
Update of December 20, 2019 —
Comments Off on Triangles, Spreads, Mathieu
Thursday, August 15, 2019
The exercise in the previous post was suggested by a passage
purporting to "use standard block design theory" that was written
by some anonymous author at Wikipedia on March 1, 2019:
Here "rm OR" apparently means "remove original research."
Before the March 1 revision . . .
The "original research" objected to and removed was the paragraph
beginning "To explain this further." That paragraph was put into the
article earlier on Feb. 28 by yet another anonymous author (not by me).
An account of my own (1976 and later) original research on this subject
is pictured below, in a note from Feb. 20, 1986 —
Comments Off on Schoolgirl Space — Tetrahedron or Square?
Friday, June 29, 2018
From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eighttriangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eighttriangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
Related material from 1975 —
More recently …
Comments Off on Triangles in the Eightfold Cube
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 nreplica if P consists of n
mutually congruent polytopes similar to P packed together.
The squaretriangle theorem (or lemma) says that
"Every triangle is an nreplica"
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 nreplica.
(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 ,
JuneJuly 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 SquareTriangle 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 "squaretriangle" result I published in a letter to the
editor of the JuneJuly 1985 American Mathematical Monthly
(Vol. 92, No. 6, p. 443). See SquareTriangle 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 SquareTriangle Theorem
Sunday, March 18, 2012
The diamond shape of yesterday's noon post
is not wholly without mathematical interest …
The squaretriangle 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 16part square array. The same group may be
viewed as permuting the centers of the 16 subtriangles
in a 16part triangular array.
(Updated March 29, 2012, to correct wording and add Weyl link.)
Comments Off on SquareTriangle Diamond
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.
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Tuesday, December 29, 2020
Variations on the title theme —
Comments Off on Raiders of the Lost Coordinates
Saturday, December 19, 2020
An Internet search for the anonymous author Quack5quack yields . . .
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Friday, December 18, 2020
In the altered headline above, ” Q******* ” may, if you like,
be interpreted as ” Quellers ,” an invented term for scholars
who investigate the origins of Christianity.
See the Log24 post “Q is for Quelle ” (November 7, 2020).
Dan Brown, like the earlier novelist who wrote The Source ,
is such an investigator (of sorts), though not a scholar .
(For an example of actual scholarship , see the webpage
https://quod.lib.umich.edu/m/
middleenglishdictionary/dictionary/MED35525.
That page may be interpreted as putting the “hit” in “s***.”)
Comments Off on Notes Towards the Redefinition of Culture
Tuesday, October 6, 2020
(An error in Fig. 4 was corrected at about
10:25 AM ET on Tuesday, Oct. 6, 2020.)
Comments Off on Spreads via the Knight Cycle
Sunday, December 22, 2019
Exercise: Use the Guitart 7cycles below to relate the 56 triples
in an 8set (such as the eightfold cube) to the 56 triangles in
a wellknown Kleinquartic hyperbolicplane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M_{24}.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
Comments Off on M_{24} from the Eightfold Cube
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 CompanionPiece for the Circular Rectangle:
Wednesday, October 23, 2019
"Leonardo was something like what we now call a Conceptual artist,
maybe the original one. Ideas — experiments, theories — were
creative ends in themselves."
— Holland Cotter in the online New York TImes this evening
From other Log24 posts tagged Tetrahedron vs. Square —
* Phrase from the previous post, "Overarching Narrative."
Comments Off on ArtHistorical Narrative*
In memory of a retired codirector of Galerie St. Etienne
who reportedly died on October 17 . . .
"It is… difficult to mount encyclopedic exhibitions
without an overarching arthistorical narrative…."
— Jane Kallir, director of Galerie St. Etienne, in
https://www.tabletmag.com/jewishartsandculture/
visualartanddesign/269564/theendofmiddleclassart
An overarching narrative from the above death date —
See as well the previous post
and "Dancing at Lughnasa."
Comments Off on Overarching Narrative
Thursday, October 17, 2019
The previous post, Tetrahedron Dance, suggests a review of . . .
A figure from St. Patrick's Day 2004 that might
represent a domed roof …
Inscribed Carpenter's Square:
In Latin, NORMA
… and a cinematic "Fire Temple" from 2019 —
In related news . . .
Related background: "e. e. cummings" in this journal.
Comments Off on Dance of the Fire Temple
Comments Off on Tetrahedron Dance
Wednesday, October 9, 2019
Note that in the pictures below of the 15 twosubsets of a sixset,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
Comments Off on The Joy of Six
Tuesday, October 8, 2019
* See Cube Bricks 1984 in previous post.
Comments Off on Also* in 1984
Monday, October 7, 2019
(A sequel to Simplex Sigillum Veri and
Rabbit Hole Meets Memory Hole)
" Wittgenstein does not, however, relegate all that is not inside the bounds
of sense to oblivion. He makes a distinction between saying and showing
which is made to do additional crucial work. 'What can be shown cannot
be said,' that is, what cannot be formulated in sayable (sensical)
propositions can only be shown. This applies, for example, to the logical
form of the world, the pictorial form, etc., which show themselves in the
form of (contingent) propositions, in the symbolism, and in logical
propositions. Even the unsayable (metaphysical, ethical, aesthetic)
propositions of philosophy belong in this group — which Wittgenstein
finally describes as 'things that cannot be put into words. They make
themselves manifest. They are what is mystical' " (Tractatus 6.522).
— Stanford Encyclopedia of Philosophy , "Ludwig Wittgenstein"
From Tractatus LogicoPhilosophicus by Ludwig Wittgenstein.
(First published in Annalen der Naturphilosophie ,1921.
English edition first published 1922 by Kegan Paul, Trench and Trübner. This translation first published 1961 by Routledge & Kegan Paul. Revised edition 1974.)
5.4541
The solutions of the problems of logic must be simple, since they set the standard of simplicity.
Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined — a priori — to form a selfcontained system.
A realm subject to the law: Simplex sigillum veri.

Somehow, the old Harvard seal, with its motto "Christo et Ecclesiae ,"
was deleted from a bookplate in an archived Harvard copy of Whitehead's
The Axioms of Projective Geometry (Cambridge U. Press, 1906).
In accordance with Wittgenstein's remarks above, here is a new
bookplate seal for Whitehead, based on a simplex —
Comments Off on Oblivion
Saturday, October 5, 2019
Comments Off on Midnight Landmarks
Friday, September 27, 2019
The 15 points of the finite projective 3space PG(3,2)
arranged in tetrahedral form:
The letter labels, but not the tetrahedral form,
are from The Axioms of Projective Geometry , by
Alfred North Whitehead (Cambridge U. Press, 1906).
The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."
Screen Rant on July 31, 2019:
A Google Search sidebar this morning:
Apocalypse Soon! —
Comments Off on Algebra for Schoolgirls
Thursday, September 26, 2019
For Dan Brown
“It’s a combination of elation and fear, a certain kind of terror,”
Dr. ScottWarren, a lecturer at Cambridge University, said
Thursday [Sept. 19] in an interview, describing his feelings.
“As a scholar, you get a sense of the fixed landmarks,” he said.
“Suddenly to have a new landmark to come right up through
the ground is quite disconcerting; there’s something alarming
about that.”
Comments Off on Personalities …
Wednesday, September 25, 2019
Previous posts now tagged Pyramid Game suggest …
A possible New Yorker caption: " e . . . (ab) . . . (cd) . "
Caption Origins —
Playing with shapes related to some 1906 work of Whitehead:
Comments Off on Language Game
Tuesday, September 24, 2019
Playing with shapes related to some 1906 work of Whitehead:
Comments Off on Emissary
Saturday, September 21, 2019
For Dan Brown fans …
… and, for fans of The Matrix, another tale
from the above death date: May 16, 2019 —
An illustration from the above
Miracle Octad Generator post:
Related mathematics — Tetrahedron vs. Square.
Comments Off on Annals of Random Fandom
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 rightangled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses —
As, for example, the ellipse of the halfmoon —
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
Comments Off on The Inappropriate Capstone
Comments Off on The AllNight Record Player
Friday, September 13, 2019
Comments Off on Schoolgirl Space…
Thursday, September 12, 2019
In memory of a Church emissary who reportedly died on September 4,
here is a Log24 flashback reposted on that date —
Related poetry —
"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"
— Macaulay, quoted in the April 2013 film "Oblivion"
Related fiction —
Comments Off on Tetrahedral Structures
Thursday, August 15, 2019
An image from a Log24 post of March 5, 2019 —
The following paragraph from the above image remains unchanged
as of this morning at Wikipedia:
"A 3(16,4,1) block design has 140 blocks of size 4 on 16 points,
such that each triplet of points is covered exactly once. Pick any
single point, take only the 35 blocks containing that point, and
delete that point. The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points."
Exercise —
Prove or disprove the above assertion about a general "3(16,4,1)
block design," a structure also known as a Steiner quadruple system
(as I pointed out in the March 5 post).
Relevant literature —
A paper from Helsinki in 2005* says there are more than a million
3(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4space over GF(2),
from which PG(3,2) can be derived using the wellknown process
from finite geometry described in the above Wikipedia paragraph.
* "The Steiner quadruple systems of order 16," by Kaski et al.,
Journal of Combinatorial Theory Series A Volume 113, Issue 8,
November 2006, pages 17641770.
Comments Off on On Steiner Quadruple Systems of Order 16
Saturday, August 10, 2019
The Square "Inscape" Model of
the Generalized Quadrangle W(2)
Click image to enlarge.
* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.
Comments Off on Schoolgirl Space* Revisited:
Wednesday, July 17, 2019
arXiv.org > quantph > arXiv:1905.06914
Quantum Physics
Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones
J. P. Marceaux, A. R. P. Rau
(Submitted on 14 May 2019)
A recreational problem from nearly two centuries ago has featured prominently in recent times in the mathematics of designs, codes, and signal processing. The number 15 that is central to the problem coincidentally features in areas of physics, especially in today's field of quantum information, as the number of basic operators of two quantum spins ("qubits"). This affords a 1:1 correspondence that we exploit to use the wellknown Pauli spin or LieClifford algebra of those fifteen operators to provide specific constructions as posed in the recreational problem. An algorithm is set up that, working with four basic objects, generates alternative solutions or designs. The choice of four base colors or four basic chords can thus lead to color diagrams or acoustic patterns that correspond to realizations of each design. The Fano Plane of finite projective geometry involving seven points and lines and the tetrahedral threedimensional simplex of 15 points are key objects that feature in this study.
Comments:16 pages, 10 figures
Subjects:Quantum Physics (quantph)
Cite as:arXiv:1905.06914 [quantph]
(or arXiv:1905.06914v1 [quantph] for this version)
Submission history
From: A. R. P. Rau [view email]
[v1] Tue, 14 May 2019 19:11:49 UTC (263 KB)

See also other posts tagged Tetrahedron vs. Square.
Comments Off on The Artsy Quantum Realm
See also other posts tagged Tetrahedron vs. Square, and a related
Log24 search for "Schoolgirl + Space."
Comments Off on Life in Palermo
Saturday, July 13, 2019
Curse of the Fire Temple
"Power outages hit parts of Manhattan
plunging subways, Broadway, into darkness"
— New York Post this evening
Comments Off on Live from New York, It’s …
Related material — Tetrahedron vs. Square and Cézanne's Greetings.
Compare and contrast:
A figure from St. Patrick's Day 2004 that might represent a domed roof …
Inscribed Carpenter's Square:
In Latin, NORMA
… and a cinematic "Fire Temple" from 2019 —
Comments Off on Which Roof?
Friday, July 12, 2019
"The area is home to many artists and people who work in
the media, including many journalists, writers and professionals
working in film and television." — Wikipedia
Tusen takk to My Square Lady —
Comments Off on Holloway Today
Tuesday, July 9, 2019
(Continued)
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
Comments Off on Perception of Space
“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”
— John Malkovich at squarespace.com, January 10, 2017
Also on that date . . .
.
Comments Off on Dreamtimes
Monday, July 8, 2019
Comments Off on Exploring Schoolgirl Space
Sunday, July 7, 2019
Anonymous remarks on the schoolgirl problem at Wikipedia —
"This solution has a geometric interpretation in connection with
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing."
See also Polster + Tetrahedron in this journal.
There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square model rather
than a tetrahedral model. The square model of PG(3,2) last
appeared in the schoolgirlproblem article on Feb. 11, 2017, just
before a revision that removed it.
Comments Off on Schoolgirl Problem
Tuesday, June 4, 2019
Or: Burning Bright
A post in memory of Chicago architect Stanley Tigerman,
who reportedly died at 88 on Monday.
Comments Off on Zen and the Art
The title is a quotation from the 2015 film "Mojave."
Comments Off on “Hello the Camp”
Wednesday, March 20, 2019
"Cell 461" quote from Curzio Malaparte superimposed on a scene from
the 1963 Godard film "Le Mépris " ("Contempt") —
"The architecture… beomes closely linked to the script…."
Malaparte's cell number , 461, is somewhat less closely linked
to the phrase "eternal blazon" —
Irving was quoted here on Dec. 22, 2008 —
The Tale of
the Eternal Blazon
by Washington Irving
“Blazon meant originally a shield , and then
the heraldic bearings on a shield .
Later it was applied to the art of describing
or depicting heraldic bearings in the proper
manner; and finally the term came to signify
ostentatious display and also description or
record by words or other means . In Hamlet ,
Act I Sc. 5, the Ghost, while talking with
Prince Hamlet, says:
‘But this eternal blazon must not be
To ears of flesh and blood.’
Eternal blazon signifies revelation or description
of things pertaining to eternity .”
— Irving’s Sketch Book , p. 461
Update of 6:25 PM ET —
"SelfBlazon… of Edenic Plenitude"
(The Issuu text is taken from Speaking about Godard , by Kaja Silverman
and Harun Farocki, New York University Press, 1998, page 34.)
Comments Off on Secret Characters
Thursday, December 6, 2018
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Related material —
The 56 triangles of the eightfold cube . . .
Image from Christmas Day 2005.
Comments Off on The Mathieu Cube of Iain Aitchison
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 GianCarlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183196. 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.
Sunday, July 1, 2018
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
Abstract for a talk at the City University of New York:
The Experience of Meaning
Jan Zwicky, University of Victoria
09:0009:40 Friday, April 5, 2013
Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. 
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
Comments Off on Deutsche Ordnung
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*
Saturday, April 14, 2018
The title refers to the previous two posts.
Related literature —
Plato’s Ghost: The Modernist Transformation of Mathematics
(Princeton University Press, 2008) and . . .
Plato’s diamondinamatrix:
Comments Off on Immanentizing the Transcendence
Saturday, October 28, 2017
The New Yorker on the recent film "The Square" —
"It’s an aesthetic that presents,
so to speak, just the facts,
as if the facts themselves weren’t
deeply layered with living history
and crisscrossed with vectors
of divergent ideas and ideals."
— Richard Brody, Thursday, Oct. 26, 2017
For other images deeply layered and crisscrossed ,
see Geometry of the I Ching.
Comments Off on Just the Facts
Comments Off on Dating Harvard
Wednesday, October 18, 2017
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Twopart decomposition of base4 array
as two (nonLatin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Fourpart decomposition of base2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
Comments Off on Dürer for St. Luke’s Day
Tuesday, October 17, 2017
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a wellknown construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
Comments Off on Plan 9 Continues
Monday, October 16, 2017
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and ConwayNortonRyba in this journal.
Some remarks on an orderfive magic square over GF(5^{2}):
"Ultra Super Magic Square"
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2space over GF(5)
(or the vector 1space over GF(5^{2})) …
All vector row sums = (0, 0) (or 0, over GF(5^{2})).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
Comments Off on Highway 61 Revisited
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
Thursday, May 11, 2017
Dialogue from the film "Interstellar" —
Cooper: Did it work?
TARS: I think it might have.
Cooper: How do you know?
TARS: Because the bulk beings
are closing the tesseract.
Related material — "Bulk apperception"
in this journal, and …
Comments Off on Reopening the Tesseract
Wednesday, May 10, 2017
In memory of an art dealer who
reportedly died on Sunday, May 7—
Decorations for a Cartoon Graveyard
Comments Off on In the Park with Yin and Yang
Tuesday, May 9, 2017
Comments Off on Text and Context
Monday, May 8, 2017
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Sunday, May 7, 2017
Detail of an image in the previous post —
This suggests a review of a post on a work of art by fashion photographer
Peter Lindbergh, made when he was younger and known as "Sultan."
The balls in the foreground relate Sultan's work to my own.
Linguistic backstory —
The art space where the pieces by Talman and by Lindbergh
were displayed is Museum Tinguely in Basel.
As the previous post notes, the etymology of "glamour" (as in
fashion photography) has been linked to "grammar" (as in
George Steiner's Grammars of Creation ). A sculpture by
Tinguely (fancifully representing Heidegger) adorns one edition
of Grammars .
Yale University Press, 2001:
Tinguely, "Martin Heidegger,
Philosopher," sculpture, 1988
Comments Off on Art Space
Saturday, December 17, 2016
Continuing the "Memory, History, Geometry" theme
from yesterday …
See Tetrahedral, Oblivion, and Tetrahedral Oblivion.
"Welcome home, Jack."
Comments Off on Tetrahedral Death Star
Saturday, December 10, 2016
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the smallconfiguration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 3839 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
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Thursday, September 15, 2016
The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).
* For the definition of "perfect number," see any introductory
numbertheory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
the projective 3space over the 2element Galois field GF(2),
was coined by math writer Burkard Polster. Cullinane's square
model of PG(3,2) differs from the earlier tetrahedral model
discussed by Polster.
Comments Off on The Smallest Perfect Number/Universe
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 magicsquares.net of
"perimetermagic 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!
Thursday, August 21, 2014
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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 , JuneJuly 1984)
Insight: The SquareTriangle 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
Friday, April 25, 2014
Quoted here on April 11 —
“…direct access to the godhead, which
in this case was Creativity.”
— Tom Wolfe, From Bauhaus to Our House
From “Today in History: April 25, 2014,” by The Associated Press:
“Five years ago… University of Georgia professor
George Zinkhan, 57, shot and killed his wife
and two men outside a community theater in Athens
before taking his own life.”
Related material:
A Google Scholar search for Zinkhan’s 1993 paper,
“Creativity in Advertising,” Journal of Advertising 22,2: 13 —
Obiter Dicta:
“Dour wit” — Obituary of a Scots herald who died on Palm Sunday
“Remember me to Herald Square.” — Song lyric
“Welcome to Scotland.” — Kincade in Skyfall
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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 halfsquares
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).
Comments Off on Figurate Numbers
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 triangledecomposition 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.
Comments Off on Logic for Jews*
Monday, October 28, 2013
From the AP Today in History page
for October 28, 2013 —
From this journal seven years ago:
Recommended.
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Sunday, August 11, 2013
"Welcome home, Jack."
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Wednesday, June 26, 2013
“I could a tale unfold whose lightest word
Would harrow up thy soul….“
— Hamlet’s Father’s Ghost
The results of a search in this journal for “a tale unfold” suggest
a review of the following passage from Donna Tartt’s Secret History…
A math weblog discussed this passage on January 24, 2013.
For related alliances, see this weblog on that same date.
Comments Off on Tale
Monday, June 24, 2013
“For in that sleep of death what dreams may come
When we have shuffled off this mortal coil,
Must give us pause.” — Hamlet
Sleep well, Mr. Matheson.
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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 threetriangle 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:
Comments Off on Caution: Slow Art
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.
Comments Off on Two Poems and Some Images
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.
Comments Off on PlanetMath link
Thursday, November 22, 2012
(Continued from 1986)
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 ntuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such ntuples? 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 4space 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 8sets.
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 4tuples over GF(2).

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 ntuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such ntuples? 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 4space 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 8sets.
A representative coordinatization:
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4tuples over GF(2). 
For some background on the triangular version,
see the SquareTriangle Theorem,
noting particularly the linkedto coordinatization picture.
Comments Off on Finite Relativity
Saturday, July 14, 2012
For example—
A letter to the editor of the American Mathematical Monthly
from the JuneJuly 1985 issue has—
… a "squaretriangle" 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.]

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 squaretriangle 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."

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.
Comments Off on Lemma
Sunday, July 8, 2012
"That n^{ 2} points fall naturally into a triangular array
is a notquiteobvious fact which may have applications…
and seems worth stating more formally."
— Steven H. Cullinane, letter in the
American Mathematical Monthly 1985 JuneJuly 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 " repn ^{2 }" (Golomb's terminology)
triangles from a 1982 book—
Comments Off on Not Quite Obvious
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
Comments Off on Quartet
For remarks related by logic, see the squaretriangle 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 4set."
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).
Comments Off on Étude
Saturday, March 24, 2012
In Turing's Cathedral
"At the still point…" — T. S. Eliot
In memory of David L. Waltz, artificialintelligence pioneer,
who died Thursday, March 22, 2012—
 The Log24 post of March 22 on the squaretriangle theorem
 The March 18 post, SquareTriangle 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 edge4 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.
Comments Off on Mapping Problem
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 colorinterchange 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.
Comments Off on Diamond Theorem Exposition
Saturday, February 5, 2011
Click to enlarge
This updates a webpage on the 4×4 Latin squares.
Comments Off on Cover Art
Friday, October 8, 2010
… and Finishing Up at Noon
This post was suggested by last evening’s post on mathematics and narrative
and by Michiko Kakutani on Vargas Llosa in this morning’s New York Times.
“One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage.”
— “Is Fiction the Art of Lying?”* by Mario Vargas Llosa, New York Times essay of October 7, 1984
My own adventures in that realm— as reader, not author— may illustrate Llosa’s remark.
A nearby stack of paperbacks I haven’t touched for some months (in order from bottom to top)—
 Pale Rider by Alan Dean Foster
 Franny and Zooey by J. D. Salinger
 The Hobbit by J. R. R. Tolkien
 Le Petit Prince by Antoine de Saint Exupéry
 Literary Reflections by James A. Michener
 The Ninth Configuration by William Peter Blatty
 A Streetcar Named Desire by Tennessee Williams
 Nine Stories by J. D. Salinger
 A Midsummer Night’s Dream by William Shakespeare
 The Tempest by William Shakespeare
 Being There by Jerzy Kosinski
 What Dreams May Come by Richard Matheson
 Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig
 A Gathering of Spies by John Altman
 Selected Poems by Robinson Jeffers
 Hook— Tinkerbell’s Challenge by Tristar Pictures
 Rising Sun by Michael Crichton
 Changewar by Fritz Leiber
 The Painted Word by Tom Wolfe
 The Hustler by Walter Tevis
 The Natural by Bernard Malamud
 Truly Tasteless Jokes by Blanche Knott
 The Man Who Was Thursday by G. K. Chesterton
 Under the Volcano by Malcolm Lowry
What moral Vargas Llosa might draw from the above stack I do not know.
Generally, I prefer the sorts of books in a different nearby stack. See Sisteen, from May 25. That post the fanciful reader may view as related to number 16 in the above list. The reader may also relate numbers 24 and 22 above (an odd couple) to By Chance, from Thursday, July 22.
* The Web version’s title has a misprint— “living” instead of “lying.”
Comments Off on Starting Out in the Evening
Friday, April 17, 2009
Begettings of
the Broken Bold
Thanks for the following
quotation (“Non deve…
nella testa“) go to the
weblog writer who signs
himself “Conrad H. Roth.”
… Yesterday I took leave of my Captain, with a promise of visiting him at Bologna on my return. He is a true
A PAPAL SOLDIER’S IDEAS OF PROTESTANTS 339
representative of the majority of his countrymen. Here, however, I would record a peculiarity which personally distinguished him. As I often sat quiet and lost in thought he once exclaimed “Che pensa? non deve mai pensar l’uomo, pensando s’invecchia;” which being interpreted is as much as to say, “What are you thinking about: a man ought never to think; thinking makes one old.” And now for another apophthegm of his; “Non deve fermarsi l’uomo in una sola cosa, perche allora divien matto; bisogna aver mille cose, una confusione nella testa;” in plain English, “A man ought not to rivet his thoughts exclusively on any one thing, otherwise he is sure to go mad; he ought to have in his head a thousand things, a regular medley.”
Certainly the good man could not know that the very thing that made me so thoughtful was my having my head mazed by a regular confusion of things, old and new. The following anecdote will serve to elucidate still more clearly the mental character of an Italian of this class. Having soon discovered that I was a Protestant, he observed after some circumlocution, that he hoped I would allow him to ask me a few questions, for he had heard such strange things about us Protestants that he wished to know for a certainty what to think of us.

Notes for Roth:
The title of this entry,
“Begettings of the Broken Bold,”
is from Wallace Stevens’s
“The Owl in the Sarcophagus”–
This was peace after death, the brother of sleep,
The inhuman brother so much like, so near,
Yet vested in a foreign absolute,
Adorned with cryptic stones and sliding shines,
An immaculate personage in nothingness,
With the whole spirit sparkling in its cloth,
Generations of the imagination piled
In the manner of its stitchings, of its thread,
In the weaving round the wonder of its need,
And the first flowers upon it, an alphabet
By which to spell out holy doom and end,
A bee for the remembering of happiness.
Peace stood with our last blood adorned, last mind,
Damasked in the originals of green,
A thousand begettings of the broken bold.
This is that figure stationed at our end,
Always, in brilliance, fatal, final, formed
Out of our lives to keep us in our death....

Related material:
Some further context:
Roth’s entry of Nov. 3, 2006–
“Why blog, sinners?“–
and Log24 on that date:
“First to Illuminate.”
Comments Off on Friday April 17, 2009
Monday, December 22, 2008
The Folding
Hamlet, Act 1, Scene 5 —
Ghost:
“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combined locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood. List, list, O, list!”
This recalls the title of a piece in this week’s New Yorker:”The Book of Lists:
Susan Sontag’s early journals.” (See Log24 on Thursday, Dec. 18.)
In the rather grim holiday spirit of that piece, here are some journal notes for Sontag, whom we may imagine as the ghost of Hanukkah past.
There are at least two ways of folding a list (or tale) to fit a rectangular frame.The normal way, used in typesetting English prose and poetry, starts at the top, runs from left to right, jumps down a line, then again runs left to right, and so on until the passage is done or the bottom right corner of the frame is reached.
The boustrophedonic way again goes from top to bottom, with the first line running from left to right, the next from right to left, the next from left to right, and so on, with the lines’ directions alternating.
The word “boustrophedon” is from the Greek words describing the turning, at the end of each row, of an ox plowing (or “harrowing”) a field.
The Tale of
the Eternal Blazon
by Washington Irving
“Blazon meant originally a shield, and then the heraldic bearings on a shield.
Later it was applied to the art of describing or depicting heraldic bearings
in the proper manner; and finally the term came to signify ostentatious display
and also description or record by words or other means. In Hamlet, Act I. Sc. 5,
the Ghost, while talking with Prince Hamlet, says:
‘But this eternal blazon
must not be
To ears of flesh and blood.’
Eternal blazon signifies revelation or description of things pertaining to eternity.”
— Irving’s Sketch Book, p. 461
By Washington Irving and Mary Elizabeth Litchfield, Ginn & Company, 1901
Related material:
Folding (and harrowing up)
some eternal blazons —
These are the foldings
described above.
They are two of the 322,560
natural ways to fit
the list (or tale)
“1, 2, 3, … 15, 16”
into a 4×4 frame.
For further details, see
The Diamond 16 Puzzle.
Moral of the tale:
Cynthia Zarin in The New Yorker, issue dated April 12, 2004–
“Time, for L’Engle, is accordionpleated. She elaborated, ‘When you bring a sheet off the line, you can’t handle it until it’s folded, and in a sense, I think, the universe can’t exist until it’s folded– or it’s a story without a book.'”
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Tuesday, December 16, 2008
The Square Wheel
(continued)
From The nCategory Cafe today:
David Corfield at 2:33 PM UTC quoting a chapter from a projected second volume of a biography:
"Grothendieck’s spontaneous reaction to whatever appeared to be causing a difficulty… was to adopt and embrace the very phenomenon that was problematic, weaving it in as an integral feature of the structure he was studying, and thus transforming it from a difficulty into a clarifying feature of the situation."
John Baez at 7:14 PM UTC on research:
"I just don’t want to reinvent a wheel, or waste my time inventing a square one."
For the adoption and embracing of such a problematic phenomenon, see The Square Wheel (this journal, Sept. 14, 2004).
For a connection of the square wheel with yesterday's entry for Julie Taymor's birthday, see a note from 2002:
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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 onediamond 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 twentyfour 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."
Comments Off on Monday November 10, 2008
Monday, August 18, 2008
Lotteries on August 17, 2008 
Pennsylvania (No revelation) 
New York (Revelation) 
Midday (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

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).” 
Comments Off on Monday August 18, 2008
Friday, July 25, 2008
56 Triangles
“This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It’s probably the best way for a nonmathematician to appreciate the symmetry of Klein’s quartic. It’s a 3holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron’s 4 corners, and 8 for each of its 6 edges.”
Exercise:
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
Baez’s discussion says that the Klein quartic’s 56
triangles can be partitioned into 7 eighttriangle Egan “cubes” that correspond to the 7 points of the Fano plane in such a way that automorphisms of the Klein quartic correspond to automorphisms of the Fano plane. Show that the 56
triangles within the eightfold cube can also be partitioned into 7 eighttriangle sets that correspond to the 7 points of the Fano plane in such a way that (affine) transformations of the eightfold cube induce (projective) automorphisms of the Fano plane.
Comments Off on Friday July 25, 2008
Wednesday, May 23, 2007
Strong Emergence Illustrated:
The Beauty Test
"There is no royal road
to geometry"
— Attributed to Euclid
There are, however, various nonroyal roads. One of these is indicated by yesterday's Pennsylvania lottery numbers:
The midday 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 nonroyal 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 / nonrelationship 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 nonsuperposable 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)

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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Tuesday, September 14, 2004
The Square Wheel
Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions. George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel. Harmonic analysis based on square functions– the "square wheel," as it were– is also not without its importance.
For some observations of Stephen Wolfram on squarewheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002). Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finitegeometry hyperplanes that I wrote in 1986. A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.
That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh functions– which have, in fact, properties very like those of the trig functions. For details, see Symmetry of Walsh Functions, updated today.
"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake. Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."
— From ErrorCorrecting Codes
Through Sphere Packings
To Simple Groups,
by Thomas M. Thompson,
Mathematical Association of America, 1983
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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:
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.”
© 19972003 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
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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
How do you make a musical about a bunch of dead mathematicians and one very alive, very famous, Princeton math professor?


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 rightangled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses
As, for example, the ellipse of the halfmoon
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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