Log24

Tuesday, June 20, 2017

All-Spark Notes

Filed under: Uncategorized — Tags: — m759 @ 1:55 PM

(Continued)

"For years, the AllSpark rested, sitting dormant
like a giant, useless art installation."

— Vinnie Mancuso at Collider.com yesterday

Related material —

Dormant cube

IMAGE- Britannica 11th edition on the symmetry axes and planes of the cube

Giant, useless art installation —

Sol LeWitt at MASS MoCA.  See also LeWitt in this journal.

Monday, April 3, 2017

Odd Core

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

 

3x3x3 Galois cube, gray and white

Saturday, September 17, 2016

Interior/Exterior

Filed under: Uncategorized — m759 @ 12:25 AM


3x3x3 Galois cube, gray and white

Saturday, October 10, 2015

Epiphany in Paris

Filed under: Uncategorized — m759 @ 10:00 PM

It's 10 PM …

    

Related material: Adam Gopnik, The King in the Window.

Wednesday, May 13, 2015

Space

Filed under: Uncategorized — Tags: — m759 @ 2:00 PM

Notes on space for day 13 of May, 2015 —

The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."

Related poetic material:

The ninefold square and Apollo, as well as 

http://www.log24.com/log/pix11A/110426-ApolloAndDionysus.jpg

Thursday, December 18, 2014

Platonic Analogy

Filed under: Uncategorized — 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.

Sunday, November 30, 2014

Two Physical Models of the Fano Plane

Filed under: Uncategorized — 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

Class Act

Filed under: Uncategorized — 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: Uncategorized — 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.

Monday, May 19, 2014

Un-Rubik Cube

Filed under: Uncategorized — m759 @ 10:48 AM

IMAGE- Britannica 11th edition on the symmetry axes and planes of the cube

See also Cube Symmetry Planes  in this journal.

Monday, January 13, 2014

A Prime for Marissa

Filed under: Uncategorized — m759 @ 10:00 PM

"I don't like odd numbers, and I really don't like primes."

Marissa Mayer

See Cube Symmetry Axes in this journal.

IMAGE- The 13 symmetry axes of the cube

Saturday, April 13, 2013

Narrative

Filed under: Uncategorized — m759 @ 6:16 PM

Two friends from Brooklyn —

"… both marveled at early Ingmar Bergman movies."

One of the friends' "humor was inspired by
surrealist painters and Franz Kafka."

Wikipedia

"Most of Marvel's fictional characters operate in
a single reality known as the Marvel Universe…."

This journal yesterday

    

Related material:  The Cosmic Cube.

Friday, April 12, 2013

Midnight in Paris

Filed under: Uncategorized — m759 @ 6:00 PM

Surreal requiem for the late Jonathan Winters:

"They 'burn, burn, burn like fabulous yellow roman candles
exploding like spiders across the stars,'
as Jack Kerouac once wrote. It was such a powerful
image that Wal-Mart sells it as a jigsaw puzzle."

— "When the Village Was the Vanguard,"
       by Henry Allen, in today's Wall Street Journal

See also Damnation Morning and the picture in
yesterday evening's remarks on art:

    

Thursday, April 11, 2013

Naked Art

Filed under: Uncategorized — m759 @ 9:48 PM

The New Yorker  on Cubism:

"The style wasn’t new, exactly— or even really a style,
in its purest instances— though it would spawn no end
of novelties in art and design. Rather, it stripped naked
certain characteristics of all pictures. Looking at a Cubist
work, you are forced to see how you see. This may be
gruelling, a gymnasium workout for eye and mind.
It pays off in sophistication."

Online "Culture Desk" weblog, posted today by Peter Schjeldahl

Non-style from 1911:

IMAGE- Britannica 11th edition on the symmetry axes and planes of the cube

See also Cube Symmetry Planes  in this  journal.

A comment at The New Yorker  related to Schjeldahl's phrase "stripped naked"—

"Conceptualism is the least seductive modern-art movement."

POSTED 4/11/2013, 3:54:37 PM BY CHRISKELLEY

(The "conceptualism" link was added to the quoted comment.)

Tuesday, February 19, 2013

Configurations

Filed under: Uncategorized — 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).

Saturday, January 14, 2012

Defining Form (continued)

Filed under: Uncategorized — Tags: , — m759 @ 12:00 PM

Detail of Sylvie Donmoyer picture discussed
here on January 10

http://www.log24.com/log/pix12/120114-Donmoyer-Still-Life-CubeDetail.jpg

The "13" tile may refer to the 13 symmetry axes
in the 3x3x3 Galois cube, or the corresponding
13 planes through the center in that cube. (See
this morning's post and Cubist Geometries.)

Damnation Morning*

Filed under: Uncategorized — Tags: , — m759 @ 5:24 AM

(Continued)

The following is adapted from a 2011 post

IMAGE- Galois vs. Rubik

* The title, that of a Fritz Leiber story, is suggested by
   the above picture of the symmetry axes of the square.
   Click "Continued" above for further details. See also
   last Wednesday's Cuber.

Friday, May 13, 2011

Apollo’s 13

Filed under: Uncategorized — Tags: — m759 @ 6:36 AM

Continued … See related previous posts.

IMAGE- The 13 symmetry axes of the cube

Those who prefer narrative to mathematics
may consult Wikipedia on The Cosmic Cube.

Monday, December 13, 2010

Mathematics and Narrative continued…

Filed under: Uncategorized — m759 @ 7:20 AM

Apollo's 13: A Group Theory Narrative —

I. At Wikipedia —

http://www.log24.com/log/pix10B/101213-GroupTheory.jpg

II. Here —

See Cube Spaces and Cubist Geometries.

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A note from 1985 describing group actions on a 3×3 plane array—

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985<br />
note by Cullinane

Pegg gives no reference to the 1985 work on group actions.

Monday, June 28, 2010

Shall I Compare Thee

Filed under: Uncategorized — Tags: — m759 @ 11:02 AM

Margaret Soltan on a summer's-day poem by D.A. Powell

first, a congregated light, the brilliance of a meadowland in bloom
and then the image must fail, as we must fail, as we

graceless creatures that we are, unmake and befoul our beds
don’t tell me deluge.     don’t tell me heat, too damned much heat

"Specifically, your trope is the trope of every life:
 the organizing of the disparate parts of a personality
 into a self (a congregated light), blazing youth
 (a meadowland in bloom), and then the failure
 of that image, the failure of that self to sustain itself."

Alternate title for Soltan's commentary, suggested by yesterday's Portrait:

Smart Jewish Girl Fwows Up.

Midrash on Soltan—

Congregated Light

The 13 symmetry axes 
of the cube

Meadowland

Appalachian meadow

Failure

Wert thou my enemy, O thou my friend,
How wouldst thou worse, I wonder, than thou dost
Defeat, thwart me?

Coda

"…meadow-down is not distressed
For a rainbow footing…."

Sunday, June 27, 2010

Sunday at the Apollo

Filed under: Uncategorized — m759 @ 9:00 PM
 

27

 

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the 27-part (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

Saturday, March 13, 2010

Space Cowboy

Filed under: Uncategorized — m759 @ 9:00 AM

From yesterday's Seattle Times

According to police, employees of a Second Avenue mission said the suspect, clad in black and covered in duct tape, had come into the mission "and threatened to blow the place up." He then told staffers "that he was a vampire and wanted to eat people."

The man… also called himself "a space cowboy"….

This suggests two film titles…

Plan 9 from Outer Space

Rebecca Goldstein and a Cullinane quaternion

and Apollo's 13

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

Saturday, February 27, 2010

Cubist Geometries

Filed under: Uncategorized — Tags: , — m759 @ 2:01 PM

"The cube has…13 axes of symmetry:
  6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 3x3x3 geometer's cube, with coordinates

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

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

A later version of the 13-point plane
by Ed Pegg Jr.–

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

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