Log24

Wednesday, March 6, 2019

The Relativity Problem and Burkard Polster

Filed under: General,Geometry — Tags: — m759 @ 11:28 AM
 

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

For some context, see Relativity Problem  in this journal.

In the case of PG(3,2), there is a choice of geometric models 
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.

The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.

For comparison of the two models, see a figure posted here on
May 21, 2014 —

Labeling the Tetrahedral Model  (Click to enlarge) —

"Citation needed" —

The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)
," a "reliable source."

Wednesday, December 30, 2020

The Sixteenth Subset

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

A four-set has sixteen subsets.  Fifteen of these symbolize the points
of “the smallest perfect universe,”* PG(3,2).  The sixteenth is empty.

In memory of . . .

Polish this — “The Nothing That Is.”

* Phrase by Burkard Polster.

Thursday, February 27, 2020

Occult Writings

Filed under: General — Tags: — m759 @ 12:44 PM

From the author who in 2001 described "God's fingerprint"
(see the previous post) —

From the same publisher —

From other posts tagged Triskele in this journal —

IMAGE- Eightfold cube with detail of triskelion structure

Other geometry for enthusiasts of the esoteric —

Monday, November 4, 2019

As Above, So Below*

Filed under: General —
Tags:  —
m759 @ 5:43 AM 

Braucht´s noch Text?

       — Deutsche Schule Montevideo

* An "established rule of law
across occult writings.
"

Wednesday, February 26, 2020

“Perfect”

Filed under: General — m759 @ 9:59 PM

Usage example —

(Click to enlarge.)

See also the previous post as well as PG(3,2),
Schoolgirl Space, and Tetrahedron vs. Square.

Friday, December 20, 2019

Identity Theory*

Filed under: General — Tags: — m759 @ 1:56 AM

The van Dam cited by Polster should not be confused
with the fictional   Vandamm of "North by Northwest."

See Pursued by a Biplane (Log24, May 23, 2017).

* For the title, see posts tagged March 8, 2018.

Triangles, Spreads, Mathieu…

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

Continued.

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

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 —

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.

Thursday, May 11, 2017

In Memoriam

Filed under: General — m759 @ 9:00 PM

See also Chandrasekharan in a Log24 search for Weyl+Schema.

Update of 6:16 AM Friday, May 12, 2017 —

The phrase "smallest perfect universe" is from Burkard Polster (2001).

Monday, December 19, 2016

Tetrahedral Cayley-Salmon Model

Filed under: General,Geometry — Tags: , — m759 @ 9:38 AM

The figure below is one approach to the exercise
posted here on December 10, 2016.

Tetrahedral model (minus six lines) of the large Desargues configuration

Some background from earlier posts —


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

Click the image below to enlarge it.

Polster's tetrahedral model of the small Desargues configuration

Sunday, December 18, 2016

Two Models of the Small Desargues Configuration

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

Click image to enlarge.

Polster's tetrahedral model of the small Desargues configuration

See also the large  Desargues configuration in this journal.

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.

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .

Update, morning of the following day (7:07 ET) — related material:

See also “56 spreads” in this  journal.

Thursday, September 15, 2016

Metaphysics at Scientific American

Filed under: General,Geometry — Tags: — m759 @ 9:36 PM

In 2011 Scientific American  magazine ran
the following promotional piece for one of their articles —

"Why 5, 8 and 24 Are the Strangest Numbers 
in the Universe
," by Michael Moyer, "the editor
in charge of physics and space coverage."

This is notably bad metaphysics. Numbers are, of course,
not  "in  the universe" — the universe, that is, of physics.

A passage from G. H. Hardy's Mathematician's Apology 
is relevant:

The contrast between pure and applied mathematics
stands out most clearly, perhaps, in geometry.
There is the science of pure geometry, in which there
are many geometries, projective geometry, Euclidean
geometry, non-Euclidean geometry, and so forth. Each
of these geometries is a model , a pattern of ideas, and
is to be judged by the interest and beauty of its particular
pattern. It is a map  or picture , the joint product of many
hands, a partial and imperfect copy (yet exact so far as
it extends) of a section of mathematical reality. But the
point which is important to us now is this, that there is
one thing at any rate of which pure geometries are not
pictures, and that is the spatio-temporal reality of the
physical world. It is obvious, surely, that they cannot be,
since earthquakes and eclipses are not mathematical
concepts.

By an abuse of language such as Burkard Polster's
quoted in the previous post, numbers may be said to be
in  the various "universes" of pure mathematics.

The Scientific American  article above is dated May 4, 2011.
See also Thomas Mann on metaphysics in this  journal
on that date.

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.

Monday, May 30, 2016

Perfect Universe

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

(A sequel to the previous post, Perfect Number)

Since antiquity,  six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its 
proper divisors — in the case of six: 1, 2, and 3.

The properties of a six-element set (a "6-set") 
divided into three 2-sets and divided into two 3-sets
are those of what Burkard Polster, using the same 
adjective in a different sense, has called 
"the smallest perfect universe" — PG(3,2), the projective
3-dimensional space over the 2-element Galois field.

A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance, 
that might please Yahoo CEO Marissa Mayer on her birthday —

The semantic  turnaround here in the meaning  of "perfect"
is accompanied by a model  turnaround in the picture  of PG(3,2) as
Polster's tetrahedral  model is replaced by Cullinane's square  model.

Further background from the previous post —

See also Kirkman's Schoolgirl Problem.

Monday, December 29, 2014

Dodecahedron Model of PG(2,5)

Filed under: General,Geometry — Tags: , , — m759 @ 2:28 PM

Recent posts tagged Sagan Dodecahedron 
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.  

For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:

For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.

Wednesday, December 3, 2014

Pyramid Dance

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

Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).

My response —

Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid"

and remarks from a Log24 post of August 14, 2013 :

Norway dance (as interpreted by an American)

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

I prefer a different, Norwegian, interpretation of "the dance of four."

Related material:
The clash between square and tetrahedral versions of PG(3,2).

See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —

(Click image below to enlarge.)

Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :

From On Art and Magic (May 5, 2011) —

http://www.log24.com/log/pix11A/110505-ThemeAndVariations-Hofstadter.jpg

http://www.log24.com/log/pix11A/110505-BlockDesignTheory.jpg

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows  symbol—
Two blocks short of  a design.

 

(Updated at about 7 PM ET on Dec. 3.)

Tuesday, December 2, 2014

Models

Filed under: General,Geometry — m759 @ 6:45 PM

Continued from November 30, 2014

"Number right Everything right." — Burkard Polster

See also the six  posts of November 30, St. Andrew's Day.

Related material —

Peter J. Cameron today discussing Julia Kristeva on poetry

"This seems to be saying that the Kolmogorov
complexity of poetry is very low: the entire poem
can be generated from a small amount of information."

… and this  journal on St. Andrew's day :

From "A Piece of the Storm,"
by the late poet Mark Strand —

A snowflake, a blizzard of one….

Sunday, November 30, 2014

Two Physical Models of the Fano Plane

Filed under: General,Geometry — Tags: , — m759 @ 1:23 AM

The Regular Tetrahedron

The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains 
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three 
edge-to-edge axes.

(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book pp. 16-17.)

The Cube

There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the 
other two members.

(This is the eightfold cube  discussed at finitegeometry.org.)

Wednesday, November 26, 2014

A Tetrahedral Fano-Plane Model

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

Update of Nov. 30, 2014 —

It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by 
Burkard Polster (Springer, 1998).

"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"

—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya

For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."

The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay" 
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.

Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.

Class Act

Filed under: General,Geometry — Tags: — m759 @ 7:18 AM

Update of Nov. 30, 2014 —

For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie  I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge, 
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science 
, 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…


… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled.  So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge.  It’s been a rich life.  I’m grateful. 
 
Steve
 

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.

Tuesday, November 25, 2014

Euclidean-Galois Interplay

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

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

Oxley's 2004 drawing of the 13-point projective plane

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube 

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points  be naturally pictured as lines  within the 
5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.

Tuesday, October 14, 2014

The Judas Seat

Filed under: General,Geometry — m759 @ 6:30 PM

My own contribution to an event of the Mathematical Association of America:

Rick’s Tricky Six  and  The Judas Seat.

The Polster tetrahedral model of a finite geometry appears, notably,
in a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Wednesday, May 21, 2014

The Tetrahedral Model of PG(3,2)

Filed under: General,Geometry — Tags: , — m759 @ 10:15 PM

The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).

The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.

Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square  model,
based on the 4×4 array I call the Galois tesseract  (at left below).

(Click to enlarge.)

Wednesday, January 1, 2014

The 56 Spreads in PG(3,2)

Filed under: General,Geometry — m759 @ 11:07 PM

IMAGE- The 56 spreads in PG(3,2)

Click for a larger image

For a different pictorial approach, see Polster's
1998 Geometrical Picture Book , pp. 77-80.

Update:  Added to finitegeometry.org on Jan. 2, 2014.
(The source of the images of the 35 lines was the image
"Geometry of the Six-Element Set," with, in the final two
of the three projective-line parts, the bottom two rows
and the rightmost two columns interchanged.)

Tuesday, February 19, 2013

Configurations

Filed under: General,Geometry — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Monday, November 5, 2012

Sitting Specially

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

Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.

Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.

A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.

Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Click for a pdf of the article.

Related material:

The Religion of Cubism” (May 9, 2003) and “Art and Lies
(Nov. 16, 2008).

This  post was suggested by following the link in yesterday’s
Sunday School post  to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date

Cryptology

IMAGE- The ninefold square

— The Delphic Corporation

The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.

Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).

Friday, March 18, 2011

Defining Configurations*

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

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.

http://www.log24.com/log/pix11/110320-MoebiusKantorConfig500w.jpg

* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.

_____________________________________________________________

Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—

http://www.log24.com/log/pix11/110322-DolgachevIntro.gif

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Thursday, February 17, 2011

Paradigms

Filed under: General,Geometry — Tags: — m759 @ 4:16 PM

"These passages suggest that the Form is a character or set of characters
common to a number of things, i.e. the feature in reality which corresponds
to a general word. But Plato also uses language which suggests not only
that the forms exist separately (χωριστά ) from all the particulars, but also
that each form is a peculiarly accurate or good particular of its own kind,
i.e. the standard particular of the kind in question or the model (παράδειγμα )
[i.e. paradigm ] to which other particulars approximate….

… Both in the Republic  and in the Sophist  there is a strong suggestion
that correct thinking is following out the connexions between Forms.
The model is mathematical thinking, e.g. the proof given in the Meno
that the square on the diagonal is double the original square in area."

— William and Martha Kneale, The Development of Logic,
Oxford University Press paperback, 1985

Plato's paradigm in the Meno

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

Changed paradigm in the diamond theorem (2×2 case) —

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Aspects of the paradigm change* —

Monochrome figures to
colored figures

Areas to
transformations

Continuous transformations to
non-continuous transformations

Euclidean geometry to
finite geometry

Euclidean quantities to
finite fields

Some pedagogues may find handling all of these
conceptual changes simultaneously somewhat difficult.

* "Paradigm shift " is a phrase that, as John Baez has rightly pointed out,
should be used with caution. The related phrase here was suggested by Plato's
term παράδειγμα  above, along with the commentators' specific reference to
the Meno  figure that serves as a model. (For "model" in a different sense,
see Burkard Polster.) But note that Baez's own beloved category theory
has been called a paradigm shift.

Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: , , , — m759 @ 11:07 AM

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).

Notes:

  Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Compare Jung’s image in Aion  of the Self as a four-diamond figure:

http://www.log24.com/log/pix10A/100615-JungImago.gif

and Cullinane’s purely geometric four-diamond figure:

http://www.log24.com/log/pix10A/100615-FourD.gif

For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system  of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009.  Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Image-- Josefine Lyche's combination of Polster's phrase with<br /> Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest<br /> Perfect Universe -- Points and Hyperplanes'

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

Sunday, May 23, 2010

For Your Consideration —

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

Cannes Festival Readies for Awards Night

Uncertified Copy

Image-- Uncertified copy of 1986 figures by Cullinane in a 2009 art exhibit in Oslo

The pictures in the detail are copies of
figures created by S. H. Cullinane in 1986.
They illustrate his model of hyperplanes
and points in the finite projective space
known as PG(3,2) that underlies
Cullinane's diamond theorem.

The title of the pictures in the detail
is that of a film by Burkard Polster
that portrays a rival model of PG(3,2).

The artist credits neither Cullinane nor Polster.

Friday, May 21, 2010

The Oslo Version

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

From an art exhibition in Oslo last year–

Image-- Josefine Lyche's combination of Polster's phrase with Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest Perfect Universe -- Points and Hyperplanes'

The artist's description above is not in correct left-to-right order.
Actually the hyperplanes above are at left, the points at right.

Compare to "Picturing the Smallest Projective 3-Space,"
a note of mine from April 26, 1986—

Image-- Points and hyperplanes in the finite 3-space PG(3,2), April 1986, by Cullinane

Click for the original full version.

Compare also to Burkard Polster's original use of
the phrase "the smallest perfect universe."

Polster's tetrahedral model of points and hyperplanes
is quite different from my own square version above.

See also Cullinane on Polster.

Here are links to the gallery press release
and the artist's own photos.

Wednesday, February 28, 2007

Wednesday February 28, 2007

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

Elements
of Geometry

The title of Euclid’s Elements is, in Greek, Stoicheia.

From Lectures on the Science of Language,
by Max Muller, fellow of All Souls College, Oxford.
New York: Charles Scribner’s Sons, 1890, pp. 88-90 –

Stoicheia

“The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.

Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row….

Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:–

  1. stix, gen. stichos, a row, a line of soldiers
  2. stichos, a row, a line; distich, a couplet
  3. steichoestichon, to march in order, step by step; to mount
  4. stoichos, a row, a file; stoichein, to march in a line

In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount….

Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”

Monday, September 4, 2006

Monday September 4, 2006

Filed under: General,Geometry — Tags: , — m759 @ 7:20 PM
In a Nutshell:
 
The Seed

"The symmetric group S6 of permutations of 6 objects is the only symmetric group with an outer automorphism….

This outer automorphism can be regarded as the seed from which grow about half of the sporadic simple groups…."

Noam Elkies, February 2006

This "seed" may be pictured as

The outer automorphism of a six-set in action

group actions on a linear complex

within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3-space over the 2-element field.

Related material: yesterday's entry for Sylvester's birthday.

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM
The Fano Plane
Revisualized:

 

 The Eightfold Cube

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
 
The image “http://www.log24.com/theory/images/Fano.gif” cannot be displayed, because it contains errors.
 

Every permutation of the plane's points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

 

Fano plane with cubes as points
 
The cubes' partitioning planes are added in binary (1+1=0) fashion.  Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.

 

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

 

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

 

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

 

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon's Cube.

 

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

Thursday, December 2, 2004

Thursday December 2, 2004

Filed under: General,Geometry — m759 @ 8:23 PM

The Poem of Pure Reality

                                       “We seek
The poem of pure reality, untouched
By trope or deviation,
    straight to the word,
Straight to the transfixing object,
    to the object
At the exactest point at which it is itself,
Transfixing by being purely what it is….”

Wallace Stevens (1879-1955)
“An Ordinary Evening in New Haven” IX,
from The Auroras of Autumn (1950)
(Collected Poems, pp. 465-489)

I have added new material to Geometry of the 4×4 Square, including links to a new commentary on a paper by Burkard Polster.

“It is a good light, then, for those
That know the ultimate Plato,
Tranquillizing with this jewel
The torments of confusion.”

— Wallace Stevens,
Collected Poetry and Prose, page 21,
The Library of America, 1997

Sunday, April 25, 2004

Sunday April 25, 2004

Filed under: General,Geometry — m759 @ 3:31 PM

Small World

Added a note to 4×4 Geometry:

The 4×4 square model  lets us visualize the projective space PG(3,2) as well as the affine space AG(4,2).  For tetrahedral and circular models of PG(3,2), see the work of Burkard Polster.  The following is from an advertisement of a talk by Polster on PG(3,2).

The Smallest Perfect Universe

“After a short introduction to finite geometries, I’ll take you on a… guided tour of the smallest perfect universe — a complex universe of breathtaking abstract beauty, consisting of only 15 points, 35 lines and 15 planes — a space whose overall design incorporates and improves many of the standard features of the three-dimensional Euclidean space we live in….

Among mathematicians our perfect universe is known as PG(3,2) — the smallest three-dimensional projective space. It plays an important role in many core mathematical disciplines such as combinatorics, group theory, and geometry.”

— Burkard Polster, May 2001

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




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