(Adapted from Encyclopaedia Britannica,

Eleventh Edition (1911), *Crystallography .*)

## Friday, December 13, 2019

### Apollo’s 13 Revisited

## Friday, October 13, 2017

## Friday, July 7, 2017

### A Prime Radiant for Krugman

**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

(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

"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

"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

**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 —

**The 3×3×3 Galois Cube**

See also Rubik in this journal.

## Monday, June 27, 2016

### Interplay

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

**The 3×3×3 Galois Cube**

**A tune suggested by the first image above —**

## Tuesday, November 25, 2014

### Euclidean-Galois Interplay

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.

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)).

**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

## Friday, May 13, 2011

### Apollo’s 13

**Continued … See related previous posts.**

Those who prefer narrative to mathematics

may consult Wikipedia on The Cosmic Cube.

## Monday, December 13, 2010

### Mathematics and Narrative continued…

**Apollo's 13: A Group Theory Narrative —**

I. At Wikipedia —

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–**

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

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—

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

## Monday, June 28, 2010

### Shall I Compare Thee

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—**

Wert thou my enemy, O thou my friend,

How wouldst thou worse, I wonder, than thou dost

Defeat, thwart me?

"…meadow-down is not distressed

For a rainbow footing…."

## Sunday, June 27, 2010

### Sunday at the Apollo

**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–**

## Saturday, March 13, 2010

### Space Cowboy

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…

**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–

## Saturday, February 27, 2010

### Cubist Geometries

"The cube has…13 axes of symmetry:

6 *C*_{2} (axes joining midpoints of opposite edges),

4 *C*_{3} (space diagonals), and

3*C*_{4} (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 13 symmetry axes of the (Euclidean) cube–

exactly one axis for each pair of opposite

subcubes in the (Galois) 3×3×3 cube–

A closely related structure–

the finite projective plane

with 13 points and 13 lines–

A later version of the 13-point plane

by Ed Pegg Jr.–

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)–

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.