Log24

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.

Friday, January 18, 2019

The Woke Grids …

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

… as opposed to The Dreaming Jewels .

A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —

"The properties of G24 and M24 are visualized by
four geometric objects:  the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."

Some "geometric objects"  — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.

A related image from a post of Dec. 1, 2018 —

Tuesday, April 14, 2015

Sacramental Geometry:

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

The Dreaming Jewels  continued

"… the icosahedron and dodecahedron have the same properties
of symmetry. For the centres of the twenty faces of an icosahedron
may be joined to form a regular dodecahedron, and conversely, the
twelve vertices of an icosahedron can be placed at the centres
of the faces of a suitable dodecahedron. Thus the icosahedral and
dodecahedral groups are identical , and either solid may be used to
examine the nature of the group elements."

— Walter Ledermann, Introduction to the Theory
of Finite Groups  (Oliver and Boyd, 1949, p. 93)

Salvador Dali, The Sacrament of the Last Supper

Omar Sharif and Gregory Peck in Behold a Pale Horse

Above: soccer-ball geometry.
              See also …

             See as well
"In Sunlight and in Shadow."

Sunday, April 12, 2015

Symbol of Heaven

Filed under: General — Tags: — m759 @ 7:00 pm

Today is Easter Sunday in the Orthodox Church.

Two readings:

"Ancient Symbol of Heaven"

From "Misunderstood Masterpiece," an essay
in the Jesuit weekly America  on Salvador Dali's
"The Sacrament of the Last Supper" —

"The setting is distinctive: a dodecahedron,
or 12-sided space, that we perceive in the
pentagon-shaped windowpanes behind the
table. The architecture is also transparent.
The dodecahedron is an ancient symbol of
heaven, where this event is taking place.
This is the realm of the Father…."

— Michael Anthony Novak, Nov. 5, 2012

Scholarship, Not Rhetoric

A PDF of the Kotrc paper is available online.

The Greek Fifth Element:

Filed under: General — Tags: — m759 @ 11:00 am

The Dodecahedron .

This Platonic solid appears, for instance, on the cover 
of a colorful text titled The Heart of Mathematics 
(Wiley, third edition, 2009) —

For serious  students, here is a better book, more in
keeping with the above authors' later interpretation  
of the fifth element as change :

Sunday, December 28, 2014

A Christmas Carol

Filed under: General — Tags: , , , — m759 @ 7:00 pm

See also Sagan Dodecahedron, which includes 
an image posted at 12 AM ET December 25, 2014:

The image stands for the
phrase "five by five,"
meaning "loud and clear."

Friday, December 19, 2014

Colorful Tale

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

Wikipedia on a tale about one Hippasus of Metapontum,
who supposedly was drowned by Pythagoreans for his
discovery of irrational numbers and/or of the dodecahedron —

"In the hands of modern writers this combination of vague
ancient reports and modern guesswork has sometimes
evolved into a much more emphatic and colourful tale."

See, for instance, a tale told by the late Carl Sagan,
who was bitterly anti-Pythagorean (and anti-Platonic):

IMAGE- Sagan in 'Cosmos' on the Pythagoreans

For a related colorful tale, see "Patrick Blackburn" in this journal.

Thursday, December 18, 2014

Platonic Analogy

Filed under: General,Geometry — Tags: , , — m759 @ 2:23 pm

(Five by Five continued)

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

See posts tagged Galois-Plane Models.

Wednesday, November 26, 2014

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 corner points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of corners, totalling 13 axes (the octahedron simply interchanges the roles of faces and corners); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of corners, totalling 31 axes (the icosahedron again interchanging roles of faces and corners). 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.

Thursday, February 28, 2013

Paperweights

Filed under: General,Geometry — Tags: , — m759 @ 1:06 pm

A different dodecahedral space (Log24 on Oct. 3, 2011)—

R. T. Curtis, symmetric generation of M12 in a dodecahedron

Monday, October 3, 2011

Mathieu Symmetry

Filed under: General,Geometry — Tags: , — m759 @ 7:08 am

The following may help show why R.T. Curtis calls his approach
to sporadic groups symmetric  generation—

(Click to enlarge.)

http://www.log24.com/log/pix11C/111003-Curtis10YrsOn-Dodecahedron-320w.jpg

Related material— Yesterday's Symmetric Generation Illustrated.

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