Log24

Friday, October 13, 2017

Speak, Memra

Filed under: General,Geometry — Tags: — m759 @ 11:59 AM

The above was suggested by a Log24 review of October 13, 2002,
which in turn suggested a Log24 search for Carousel that yielded
(from Bloomsday Lottery) —

See as well Asimov's "prime radiant," and an illustration
of the number 13 as a radiant prime

"The Prime Radiant can be adjusted to your mind,
and all corrections and additions can be made
through mental rapport. There will be nothing to
indicate that the correction or addition is yours.
In all the history of the Plan there has been no
personalization. It is rather a creation of all of us 
together. Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, 
    Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."
— Optimus Prime

See also Transformers in this journal.

Friday, July 7, 2017

A Prime Radiant for Krugman

Filed under: General — Tags: — m759 @ 6:00 AM

Paul Krugman:
Asimov's Foundation  novels grounded my economics

In the Foundation  novels of Isaac Asimov

"The Prime Radiant can be adjusted to your mind, and all
corrections and additions can be made through mental rapport.
There will be nothing to indicate that the correction or addition
is yours. In all the history of the Plan there has been no
personalization. It is rather a creation of all of us together.
Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."

See also Transformers in this journal.

Sunday, January 1, 2017

Like the Horizon

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

(Continued from a remark by art critic Peter Schjeldahl quoted here
last  year on New Year's Day in the post "Art as Religion.")

"The unhurried curve got me. 
It was like the horizon of a world
that made a non-world of
all of the space outside it."

— Peter Schjeldahl, "Postscript: Ellsworth Kelly,"
The New Yorker , December 30, 2015

This suggests some further material from the paper 
that was quoted here yesterday on New Year's Eve —

"In teaching a course on combinatorics I have found
students doubting the existence of a finite projective
plane geometry with thirteen points on the grounds
that they could not draw it (with 'straight' lines)
on paper although they had tried to do so. Such a
lack of appreciation of the spirit of the subject is but
a consequence of the elements of formal geometry
no longer being taught in undergraduate courses.
Yet these students were demanding the best proof of
existence, namely, production of the object described."

— Derrick Breach (See his obituary from 1996.)

A related illustration of the 13-point projective plane 
from the University of Western Australia:

Projective plane of order 3

(The four points on the curve
at the right of the image are
the points on the line at infinity .)

The above image is from a post of August 7, 2012,
"The Space of Horizons."  A related image — 

Click on the above image for further remarks.

Friday, July 1, 2016

Transparent Core

Filed under: General — m759 @ 2:28 PM

"At the point of convergence the play of similarities and differences
cancels itself out in order that identity alone may shine forth.
The illusion of motionlessness, the play of mirrors of the one:
identity is completely empty; it is a crystallization and
in its transparent core the movement of analogy begins all over
once again." — The Monkey Grammarian  by Octavio Paz,
translated by Helen Lane 

A more specific "transparent core" —

See all references to this figure
in this journal.

For a more specific "monkey grammarian," 
see W. Tecumseh Fitch in this journal.

Thursday, June 30, 2016

Design Luminosity

Filed under: General — Tags: — m759 @ 3:13 PM

Peter Woit today

"At CERN the LHC has reached design luminosity,* and is
breaking records with a fast pace of new collisions. This may
have something to do with the report that the LHC is also 
about to tear open a portal to another dimension." 

See also the following figure from the Log24 Bion posts

— and Greg Egan's short story "Luminous":

"The theory was, we’d located part of the boundary
between two incompatible systems of mathematics –
both of which were physically true, in their respective
domains. Any sequence of deductions which stayed
entirely on one side of the defect – whether it was the
'near side', where conventional arithmetic applied, or
the 'far side', where the alternative took over – would
be free from contradictions. But any sequence which
crossed the border would give rise to absurdities –
hence S could lead to not-S."

Greg Egan, Luminous
   (Kindle Locations 1284-1288). 

* See a definition.

Wednesday, June 29, 2016

Space Jews

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

For the Feast of SS. Peter and Paul

In memory of Alvin Toffler and Simon Ramo,
a review of figures from the midnight that began
the date of their deaths, June 27, 2016 —

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

   The 3×3×3 Galois Cube

See also Rubik in this journal.

Monday, June 27, 2016

Interplay

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

From a search in this journal for Euclid + Galois + Interplay

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

   The 3×3×3 Galois Cube
 

A tune suggested by the first image above —

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.

Monday, January 13, 2014

A Prime for Marissa

Filed under: General,Geometry — 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

Friday, May 13, 2011

Apollo’s 13

Filed under: General,Geometry — 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: General,Geometry — 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: General,Geometry — 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: General,Geometry — 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: General,Geometry — 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: General,Geometry — 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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