Log24

Tuesday, December 15, 2015

Square Triangles

Filed under: General,Geometry — m759 @ 3:57 PM

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 subtrianglescenters , given a suitable coordinatization?

Thursday, January 19, 2012

Square Triangles

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

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

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

Thursday, January 12, 2012

Triangles Are Square

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

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

http://www.log24.com/log/pix12/120112-ConwayTriangleDecomposition.jpg

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

IMAGE- Triangle and square, each with 16 parts

Figure A

(Click the below image to enlarge.)

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

The above 1985 note immediately suggests a problem—

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

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

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

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

Thursday, February 13, 2020

Square-Triangle Mappings: The Continuous Case

Filed under: General — m759 @ 12:00 PM

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

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

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

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

Friday, December 20, 2019

Triangles, Spreads, Mathieu…

Filed under: General — Tags: , — m759 @ 1:38 AM

Continued.

An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:

Friday, November 22, 2019

Triangles, Spreads, Mathieu …

Filed under: General — Tags: , — m759 @ 4:39 PM

Continued from October 29, 2019.

More illustrations (click to enlarge) —

Thursday, October 31, 2019

56 Triangles

Filed under: General — Tags: , — m759 @ 8:09 AM

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.”

Tuesday, October 29, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: , — m759 @ 8:04 PM

There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.

Exercise:

It is well-known that

 There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group An is generated by 3-cycles.
The alternating group Ais isomorphic to GL(4,2).

Use the above facts, along with the correspondence
described below, to construct M24.

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 —

Thursday, August 15, 2019

Schoolgirl Space — Tetrahedron or Square?

Filed under: General — Tags: , — m759 @ 9:03 PM

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 —

'The relativity problem in finite geometry,' 1986

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — Tags: , — m759 @ 9:10 PM

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 eight-triangle 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 eight-triangle 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

Sunday, July 15, 2012

Squares Are Triangular

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

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

Eric W. Weisstein at Wolfram MathWorld

For example—

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

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

The square-triangle theorem (or lemma) says that

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

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.

Thursday, March 22, 2012

Square-Triangle Theorem continued

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

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

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

Triangle Tiling I 

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 .

Wednesday, March 21, 2012

Square-Triangle Theorem

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

(Continued from March 18, 2012)

Found in a search this evening—

How Does One Cut a Triangle?  by Alexander Soifer

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

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

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

Sunday, March 18, 2012

Square-Triangle Diamond

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

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.

IMAGE- Square-to-diamond (rhombus) shear in proof of square-triangle theorem

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.)

Tuesday, September 10, 2019

Congruent Subarrays

Filed under: General — m759 @ 10:10 PM

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).

Sunday, April 15, 2018

Colorado Olympiad

Filed under: General,Geometry — Tags: — m759 @ 1:01 PM

Or:  Personalities Before Principles

Personalities —

Principles —

This  journal on April 28, 2004 at 7:00 AM.

Backstory —

Square Triangles in this journal.

Tuesday, October 6, 2020

Spreads via the Knight Cycle

Filed under: General — Tags: — m759 @ 2:10 AM

A Graphic Construction of the 56 Spreads of PG(3,2)

(An error in Fig. 4 was corrected at about
10:25 AM ET on Tuesday, Oct. 6, 2020.)

Sunday, December 22, 2019

M24 from the Eightfold Cube

Filed under: General — Tags: , — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

Click image below to download a Guitart PowerPoint presentation.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

Wednesday, November 27, 2019

A Companion-Piece for the Circular Rectangle:

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

For the circular rectangle, see today's earlier post "Enter Jonathan Miller…."

The Square Triangle

Triangles are Square

A recent view of the above address —

Wednesday, October 23, 2019

Art-Historical Narrative*

Filed under: General — Tags: , — m759 @ 9:25 PM

"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."

Overarching Narrative

Filed under: General — Tags: , — m759 @ 8:13 PM

In memory of a retired co-director of Galerie St. Etienne
who reportedly died on October 17 . . .

"It is difficult to mount encyclopedic exhibitions
without an overarching art-historical narrative…."

—  Jane Kallir, director of Galerie St. Etienne, in
https://www.tabletmag.com/jewish-arts-and-culture/
visual-art-and-design/269564/the-end-of-middle-class-art

An overarching narrative from the above death date

See as well the previous post 
and "Dancing at Lughnasa."

Thursday, October 17, 2019

Dance of the Fire Temple

Filed under: General — Tags: , , — m759 @ 10:13 AM

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.

Tetrahedron Dance

Filed under: General — Tags: , , , — m759 @ 9:42 AM

John Lithgow in "The Tomorrow Man" (2019)

" connect the dots…."

IMAGE- 'The geometry of the dance' is that of a tetrahedron, according to Peter Pesic

Wednesday, October 9, 2019

The Joy of Six

Note that in the pictures below of the 15 two-subsets of a six-set,
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).

Anticommuting Dirac matrices as spreads of projective lines

Related narrative The "Quantum Tesseract Theorem."

Tuesday, October 8, 2019

Also* in 1984

Filed under: General — m759 @ 11:32 AM
 

American Mathematical Monthly , June-July 1984 

MISCELLANEA, 129

Triangles are square

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

* See Cube Bricks 1984  in previous post.

Monday, October 7, 2019

Oblivion

Filed under: General — Tags: , — m759 @ 1:09 PM

(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 Logico-Philosophicus  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 self-contained 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

Saturday, October 5, 2019

Midnight Landmarks

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

Friday, September 27, 2019

Algebra for Schoolgirls

Filed under: General — Tags: , — m759 @ 8:37 AM

The 15 points of the finite projective 3-space 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!

Thursday, September 26, 2019

Personalities …

Filed under: General — Tags: , — m759 @ 8:06 AM

For Dan Brown 

It’s a combination of elation and fear, a certain kind of terror,”
Dr. Scott-Warren, 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.”

Wednesday, September 25, 2019

Language Game

Filed under: General — Tags: , — m759 @ 9:28 AM

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:

Tuesday, September 24, 2019

Emissary

Filed under: General — Tags: , , — m759 @ 8:04 PM
 

Thursday, September 12, 2019

Tetrahedral Structures

Filed under: General — Tags:  —
m759 @ 8:11 PM 

In memory of a Church emissary  who reportedly died on  September 4 . . . .

Playing with shapes related to some 1906 work of Whitehead:

Saturday, September 21, 2019

Annals of Random Fandom

Filed under: General — Tags: , — m759 @ 5:46 PM

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.

Saturday, September 14, 2019

Landscape Art

Filed under: General — Tags: — m759 @ 11:18 AM

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.
 

The mysterious 'ellipse of the half-moon'?

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.

The Inappropriate Capstone

Filed under: General — Tags: — m759 @ 4:59 AM

The All-Night Record Player

Filed under: General — Tags: — m759 @ 4:00 AM

See "Politics of Experience" and "Blue Guitar."

IMAGE- Scene from 'Oblivion' (2013) 

Friday, September 13, 2019

Schoolgirl Space…

Filed under: General — Tags: — m759 @ 4:56 AM

According to Wikipedia

See also Schoolgirl Space in this journal.

Thursday, September 12, 2019

Tetrahedral Structures

Filed under: General — Tags: — m759 @ 8:11 PM

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 —

Thursday, August 15, 2019

On Steiner Quadruple Systems of Order 16

Filed under: General — Tags: , — m759 @ 4:11 AM

An image from a Log24 post of March 5, 2019

Cullinane's 1978  square model of PG(3,2)

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 4-space over GF(2),
from which PG(3,2) can be derived using the well-known 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 1764-1770.

Saturday, August 10, 2019

Schoolgirl Space* Revisited:

Filed under: General — Tags: , — m759 @ 10:51 PM

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.

Wednesday, July 17, 2019

The Artsy Quantum Realm

Filed under: General — Tags: — m759 @ 6:38 PM
 

arXiv.org > quant-ph > 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 well-known Pauli spin or Lie-Clifford 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 three-dimensional simplex of 15 points are key objects that feature in this study.

Comments:16 pages, 10 figures

Subjects:Quantum Physics (quant-ph)

Cite as:arXiv:1905.06914 [quant-ph]

 (or arXiv:1905.06914v1 [quant-ph] 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.

Life in Palermo

Filed under: General — Tags: — m759 @ 9:55 AM

See also other posts tagged Tetrahedron vs. Square, and a related
Log24 search for "Schoolgirl + Space."

Saturday, July 13, 2019

Live from New York, It’s …

Filed under: General — Tags: , , , — m759 @ 11:30 PM

Curse of the Fire Temple

"Power outages hit parts of Manhattan
plunging subways, Broadway, into darkness"

New York Post  this evening

Which Roof?

Filed under: General — Tags: , , — m759 @ 10:15 AM

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

Friday, July 12, 2019

Holloway Today

Filed under: General — Tags: — m759 @ 8:23 AM

"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 —

Tuesday, July 9, 2019

Perception of Space

Filed under: General — Tags: , , , — m759 @ 10:45 AM

(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 —

Dreamtimes

Filed under: General — Tags: , , — m759 @ 4:27 AM

“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 . . .

.

Monday, July 8, 2019

Exploring Schoolgirl Space

See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."

Sunday, July 7, 2019

Schoolgirl Problem

Filed under: General — Tags: , , — m759 @ 11:18 PM

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 schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.

Tuesday, June 4, 2019

Zen and the Art

Filed under: General — Tags: , , — m759 @ 6:13 PM

Or:  Burning Bright

A post in memory of Chicago architect Stanley Tigerman,
who reportedly died at 88 on Monday.

“Hello the Camp”

Filed under: General — Tags: , — m759 @ 12:20 AM

The title is a quotation from the 2015 film "Mojave."

Wednesday, March 20, 2019

Secret Characters

Filed under: General — Tags: , , , , — m759 @ 2:23 PM

"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 —

"Self-Blazon 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.)

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

Santa and a cube
From a talk by a Melbourne mathematician on March 9, 2018 —

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

The source — Talk II below —

Search Results

pdf of talk I  (March 8, 2018)

www.math.sci.hiroshima-u.ac.jp/branched/…/Aitchison-Hiroshima-2018-Talk1-2.pdf

Iain Aitchison. Hiroshima  University March 2018 … Immediate: Talk given last year at Hiroshima  (originally Caltech 2010).

pdf of talk II  (March 9, 2018)  (with model for M24)

www.math.sci.hiroshima-u.ac.jp/branched/files/…/Aitchison-Hiroshima-2-2018.pdf

Iain Aitchison. Hiroshima  University March 2018. (IRA: Hiroshima  03-2018). Highly symmetric objects II.

Abstract

www.math.sci.hiroshima-u.ac.jp/branched/files/2018/abstract/Aitchison.txt

Iain AITCHISON  Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some …

Related material — 

The 56 triangles of  the eightfold cube . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Sunday, September 9, 2018

Plan 9 Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 9:00 AM

"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

Sunday, July 1, 2018

Deutsche Ordnung

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

Steiner quadruple system in eightfold cube

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
09:00-09:40 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

Tuesday, April 17, 2018

A Necessary Possibility*

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

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

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

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

An inscription from 2010 —

An inscription from 1984 —

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

MISCELLANEA, 129

Triangles are square

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

* See also other Log24 posts mentioning this phrase.

Saturday, October 28, 2017

Just the Facts

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

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.

Dating Harvard

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

See also this journal on today's date four years ago.

Wednesday, October 18, 2017

Dürer for St. Luke’s Day

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 PM

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 .

Two-part decomposition of base-4 array
as two (non-Latin) 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 .

Four-part decomposition of base-2 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.

Tuesday, October 17, 2017

Plan 9 Continues

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

See also Holy Field in this journal.

Some related mathematics —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

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 well-known 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

Monday, October 16, 2017

Highway 61 Revisited

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 10:13 AM

"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"

Related material — 

See as well Charles Small, Harvard '64, 
"Magic Squares over Fields" —

— and Conway-Norton-Ryba in this  journal.

Some remarks on an order-five  magic square over GF(52):

"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

Base-5:

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 2-space over GF(5)
(or the vector 1-space over GF(52)) 

All vector row sums = (0, 0)  (or 0, over GF(52)).
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

Wednesday, September 13, 2017

Summer of 1984

Filed under: General,Geometry — Tags: , , , — m759 @ 9:11 AM

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

Group actions on partitions —

Cube Bricks 1984 —

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

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Monday, August 21, 2017

Hid

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

Thursday, May 11, 2017

Reopening the Tesseract

Filed under: General,Geometry — Tags: — m759 @ 11:27 AM

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

Wednesday, May 10, 2017

In the Park with Yin and Yang

Filed under: General,Geometry — Tags: , — m759 @ 10:35 PM

In memory of an art dealer who 
reportedly died on Sunday, May 7—

Decorations for a Cartoon Graveyard

Tuesday, May 9, 2017

Text and Context

Filed under: General,Geometry — Tags: — m759 @ 2:11 PM

Some context for the previous post, which was about
a new Art Space  Pinterest board

Monday, May 8, 2017

New Pinterest Board

Filed under: General,Geometry — Tags: — m759 @ 9:29 PM

https://www.pinterest.com/stevenhcullinane/art-space/

Sunday, May 7, 2017

Art Space

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

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

Saturday, December 17, 2016

Tetrahedral Death Star

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

Continuing the "Memory, History, Geometry" theme
from yesterday

See Tetrahedral,  Oblivion,  and Tetrahedral Oblivion.

IMAGE- From 'Oblivion' (2013), the Mother Ship

"Welcome home, Jack."

Saturday, December 10, 2016

Folk Etymology

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 small-configuration 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 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."

Cézanne's Greetings.

Thursday, September 15, 2016

The Smallest Perfect Number/Universe

Filed under: General,Geometry — Tags: , , — m759 @ 6:29 AM

The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

  * For the definition of "perfect number," see any introductory
    number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
     the projective 3-space over the 2-element 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.

Saturday, August 6, 2016

Mystic Correspondence:

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

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.

Sunday, November 1, 2015

Sermon for All Saints’ Day

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

From St. Patrick's Day this year —

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

Tuesday, March 17, 2015

Focus!

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

A sequel to Dude!

See also "Triangles are Square."

Thursday, August 21, 2014

Remember me to…

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

Herald Square.

Monday, July 7, 2014

Tricky Task

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

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

Life on the Mathematical Frontier:
Legendary Figures and Their Adventures

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

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

Aphorism:  “Triangles are square.” 

(American Mathematical Monthly , June-July 1984)

Insight:  The Square-Triangle Theorem.

Thursday, June 26, 2014

Study This Example

Filed under: General,Geometry — Tags: — m759 @ 10:09 AM

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.

Friday, April 25, 2014

Creativity

Filed under: General,Geometry — Tags: — m759 @ 2:45 AM

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: 1-3 —

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

Wednesday, March 12, 2014

Obiter Dictum

Filed under: General,Geometry — m759 @ 7:59 PM

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…

IMAGE - Former location of Jean Stephen art galleries

… 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.

Saturday, January 18, 2014

The Triangle Relativity Problem

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

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

IMAGE- Triangle Coordinatization

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

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

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

Wednesday, January 8, 2014

Not Subversive, Not Fantasy

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

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

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

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

as well as the Orwellian dictum Triangles Are Square.

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

Monday, November 25, 2013

Figurate Numbers

Filed under: General,Geometry — m759 @ 8:28 AM

The title refers to a post from July 2012:

IMAGE- Squares, triangles, and figurate numbers

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).

Sunday, November 24, 2013

Logic for Jews*

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

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

My own contribution to this genre—

A triangle-decomposition result from 1984:

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

MISCELLANEA, 129

Triangles are square

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

The Orwell slogans are false. My own is not.

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

Monday, October 28, 2013

Harvard Anniversary

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

From the AP Today in History  page
for October 28, 2013 —

IMAGE- Harvard founded Oct. 28.

From this journal seven years ago:

The Practical Cogitator

Recommended.

Sunday, August 11, 2013

Demonstrations

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM

IMAGE- Wolfram Demonstrations, '15 Point Projective Space'

IMAGE- From 'Oblivion' (2013), the Mother Ship

"Welcome home, Jack."

Wednesday, June 26, 2013

Tale

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

“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.

Monday, June 24, 2013

What Dreams

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

“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.

Wednesday, April 10, 2013

Caution: Slow Art

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

"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 3x3 square

The above 3×3 grid is embedded in a 
somewhat more sophisticated example
of conceptual art from April 1, 2013:

IMAGE- A Galois-geometry key to Desargues' theorem

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:

IMAGE- Desargues' theorem with three triangles (the large Desargues configuration) and Galois-geometry version

Thursday, January 3, 2013

Two Poems and Some Images

Filed under: General,Geometry — m759 @ 1:11 AM

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.

Wednesday, January 2, 2013

PlanetMath link

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

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

__________________________________________________________________

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

Update of 3 PM ET Jan. 2, 2013—

Another item recovered from Internet storage:

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

Click on the Monthly  page for some background.

Thursday, November 22, 2012

Finite Relativity

Filed under: General,Geometry — m759 @ 10:48 PM

(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 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.

Fifteen partitions of a 4x4 array 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).

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.


Fifteen partitions of an array of 16 triangles into two 8-sets


A representative coordinatization:

Coordinates for a triangular finite geometry

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

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

Saturday, July 14, 2012

Lemma

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

IMAGE- 'Lemma (mathematics)' in Wikipedia

For example—

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

… a "square-triangle" lemma:

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

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

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

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

A search today for related material yielded the following—

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

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

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

Sunday, July 8, 2012

Not Quite Obvious

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

"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."

— Steven H. Cullinane, letter in the
American Mathematical Monthly 
1985 June-July issue

If the ancient Greeks had not been distracted by
investigations of triangular  (as opposed to square )
numbers, they might have done something with this fact.

A search for occurrences of the phrase

"n2 [i.e., n 2 ] congruent triangles

indicates only fairly recent (i.e., later than 1984) results.*

Some related material, updated this morning—

This suggests a problem
 

What mappings of a square  array of n 2 points to
a triangular  array of n 2 points are "natural"?

http://www.log24.com/log/pix12B/120708-SquareAndTriangle.jpg

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

 

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

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

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

Saturday, July 7, 2012

Quartet

Filed under: General,Geometry — m759 @ 1:23 PM

"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- Triangle cut into four congruent subtriangles
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

Étude

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

 IMAGE- Google Books ad for 'Geometric Etudes in Combinatorial Mathematics,' by Alexander Soifer

IMAGE- Triangle cut into four congruent subtriangles

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).

Saturday, March 24, 2012

The David Waltz…

Filed under: General,Geometry — m759 @ 9:00 AM

In Turing's Cathedral

"At the still point…" — T. S. Eliot

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

  1. The Log24 post of March 22 on the square-triangle theorem
  2. The March 18 post, Square-Triangle Diamond
  3. Remarks from the BBC on linguistic embedding
    that begin as follows—
         "If we draw a large triangle and embed smaller triangles in it,
          how does it look?"—
    and include discussion of a South American "tribe called Piranha" [sic ]
  4. The result of a Cartoon Bank search suggested by no. 3 above—
    (Click image for some related material.)
  5. A suggestion from the Cartoon Bank—
    IMAGE- 'Try our new grid view.'
  6. The following from the First of May, 2010

    The Nine Divisions of Heaven–

    Image-- Routledge Encyclopedia of Taoism, Vol. I, on the Nine Heavens, 'jiutian,' ed. by Fabrizio Pregadio

    Some context–

    IMAGE- The 3x3 ('ninefold') square as Chinese 'Holy Field'

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

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

  7. The phrase "embedding the stone" —

Monday, January 16, 2012

Mapping Problem

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

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

http://www.log24.com/log/pix12/120116-SquareAndTriangle.jpg

Here is a trial solution of the inverse problem—

http://www.log24.com/log/pix12/120116-trisquare-map-500w.jpg

(Click for larger version.)

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

Friday, February 25, 2011

Diamond Theorem Exposition

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

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

Saturday, February 5, 2011

Cover Art

Filed under: General,Geometry — m759 @ 3:17 AM

Click to enlarge

http://www.log24.com/log/pix11/110205-LatinSquaresOfTrianglesSm.jpg

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

Friday, October 8, 2010

Starting Out in the Evening

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

… 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.

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
Starting Out in the Evening

Right: Johnny Depp in
The Ninth Gate

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

“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)—

  1. Pale Rider by Alan Dean Foster
  2. Franny and Zooey by J. D. Salinger
  3. The Hobbit by J. R. R. Tolkien
  4. Le Petit Prince by Antoine de Saint Exupéry
  5. Literary Reflections by James A. Michener
  6. The Ninth Configuration by William Peter Blatty
  7. A Streetcar Named Desire by Tennessee Williams
  8. Nine Stories by J. D. Salinger
  9. A Midsummer Night’s Dream by William Shakespeare
  10. The Tempest by William Shakespeare
  11. Being There by Jerzy Kosinski
  12. What Dreams May Come by Richard Matheson
  13. Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig
  14. A Gathering of Spies by John Altman
  15. Selected Poems by Robinson Jeffers
  16. Hook— Tinkerbell’s Challenge by Tristar Pictures
  17. Rising Sun by Michael Crichton
  18. Changewar by Fritz Leiber
  19. The Painted Word by Tom Wolfe
  20. The Hustler by Walter Tevis
  21. The Natural by Bernard Malamud
  22. Truly Tasteless Jokes by Blanche Knott
  23. The Man Who Was Thursday by G. K. Chesterton
  24. 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.”

Friday, April 17, 2009

Friday April 17, 2009

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

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.”

Autobiography
of Goethe

(Vol. II, London, Bell & Daldy,
1868, at Google Books):

… 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:

Roth and Corleone in Havana

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:

  • Yesterday’s entry on Giordano Bruno and the Geometry of Language
  • James Joyce and Heraldry
  • “One might say that he [Joyce] invented a non-Euclidean geometry of language; and that he worked over it with doggedness and devotion….” —Unsigned notice in The New Republic, 20 January 1941
  • Joyce’s “collideorscape” (scroll down for a citation)
  • “A Hanukkah Tale” (Log24, Dec. 22, 2008)
  • Stevens’s phrase from “An Ordinary Evening in New Haven” (Canto XXV)

Some further context:

Roth’s entry of Nov. 3, 2006–
Why blog, sinners?“–
and Log24 on that date:
First to Illuminate.”

Monday, December 22, 2008

Monday December 22, 2008

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

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 —

The 16 Puzzle: transformations of a 4x4 square
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 accordion-pleated. 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.'”

Tuesday, December 16, 2008

Tuesday December 16, 2008

Filed under: General,Geometry — Tags: , — m759 @ 8:00 PM
The Square Wheel
(continued)

From The n-Category 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:

Wolfram's Theory of Everything
and the Gameplayers of Zan
.

Related pictures–

From Wolfram:

http://www.log24.com/log/pix08A/081216-WolframWalsh.gif

A Square

From me:

http://www.log24.com/log/pix08A/081216-IChingWheel.gif

A Wheel

Monday, November 10, 2008

Monday November 10, 2008

Filed under: General,Geometry — m759 @ 10:31 AM

Frame Tales

From June 30

("Will this be on the test?")

Frame Tale One:

Summer Reading

The King and the Corpse: Tales of the Soul's Conquest of Evil

Subtitle:
Tales of the Soul's
Conquest of Evil

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.

Plato's Diamond

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."

Twenty-four Variations on a Theme of Plato

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."

Monday, August 18, 2008

Monday August 18, 2008

Filed under: General,Geometry — m759 @ 9:00 AM
The Revelation Game
Revisited

(See also Jung’s birthday.)

Google logo, Aug. 18, 2008: Dragon playing Olympic ping pong

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

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).”

Friday, July 25, 2008

Friday July 25, 2008

Filed under: General,Geometry — Tags: , — m759 @ 6:01 PM

56 Triangles

Greg Egan's drawing of the 56 triangles on the Klein quartic 3-hole torus

John Baez on
Klein’s quartic:

“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 3-holed 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:The Eightfold Cube: The Beauty of Klein's Simple Group

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 eight-triangle 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 eight-triangle 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.

Wednesday, May 23, 2007

Wednesday May 23, 2007

Filed under: General,Geometry — m759 @ 7:00 AM
 
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:

PA Lottery May 22, 2007: Mid-day 515, Evening 062

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 image “http://www.log24.com/log/pix07/070523-Coxeter62.jpg” cannot be displayed, because it contains errors.

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:

The image “http://www.log24.com/log/pix07/070523-Hardy.jpg” cannot be displayed, because it contains errors.

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.)

Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 4:01 PM

Make a Différance

From Frida Saal's
Lacan The image “http://www.log24.com/log/pix05/050322-Diamond.gif” cannot be displayed, because it contains errors. 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."

 


From a Contemporary Literary Theory website:

"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:

The image “http://www.log24.com/log/pix05/050322-Fig39.gif” cannot be displayed, because it contains errors.22. Without using the Pythagorean Theorem prove that the hypotenuse of  an isosceles right triangle will have the length The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.  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 = The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors. .

 

— 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 image “http://www.log24.com/log/pix05/050322-Reba2.jpg” cannot be displayed, because it contains errors.

The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.

* See the remarks of Frida Saal above and of Barbara Johnson on mathematics (The Shining of May 29, cited in Readings for St. Patrick's Day).


† 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.

Tuesday, September 14, 2004

Tuesday September 14, 2004

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

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 square-wheel 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 finite-geometry 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 Error-Correcting Codes
Through Sphere Packings
To Simple Groups
,
by Thomas M. Thompson,
Mathematical Association of America, 1983

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General,Geometry — m759 @ 2:45 AM

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:

quilt

This block has a certain symmetry. The right half is a mirror image of the left, and the top half is a mirror of the bottom.”

© 1997-2003 Annenberg/CPB. All rights reserved.
Legal Policy

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

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

See also Geometry for Jews and related entries.

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

Pilate, Truth, and Friday the Thirteenth.

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

Thursday, September 19, 2002

Thursday September 19, 2002

Filed under: General,Geometry — m759 @ 2:16 PM

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 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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