## Sunday, May 19, 2019

### Rural Relativity

## Friday, July 28, 2017

### Prize Problem

The last page of a novel published on Sept. 2, 2014 —

**Related material —**

The 2017 film *Gifted* presents a different approach to the Navier-Stokes

problem.

The figure below perhaps represents the above *novel *'s Millennium Prize

winner reacting, in the afterlife, to the *film* 's approach in *Gifted *.

*Bustle* online magazine last April —

*Gifted *’s Millennium Prize Problems

Are Real & They Will Hurt Your Brain

**By JOHNNY BRAYSON Apr 11 2017**

See also other news from the above *Bustle* date — April 11, 2017.

## Tuesday, December 15, 2015

### Square Triangles

Click image for some background.

Exercise: Note that, modulo color-interchange, the set of 15 two-color

patterns above is invariant under the group of six symmetries of the

equilateral triangle. **Are there any other such sets** of 15 two-color triangular

patterns that are closed *as sets *, modulo color-interchange, under the six

triangle symmetries *and* under the 322,560 permutations of the 16

subtriangles induced by actions of the affine group AGL(4,2)

on the 16 subtriangles' *centers , *given a suitable coordinatization?

## Tuesday, June 17, 2014

### Finite Relativity

Anyone tackling the *Raumproblem* described here

on Feb. 21, 2014 should know the history of coordinatizations

of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis

and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis

in a 1989 paper. Curtis seemed in that paper to be saying, falsely, that

his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly

definitive and natural canonical coordinatization of the 4×6 MOG

array by the symbols for the 24 points of the projective line over GF(23)—

{**∞, 0, 1, 2, 3… , 21, 22}:**

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the

*octads* , and not the projective-line *coordinates *, in his earlier papers were

mirror images of the *octads *that resulted later from the Conway coordinates,

as in the images below.

## Friday, February 21, 2014

### Raumproblem*

Despite the blocking of Doodles on my Google Search

screen, some messages get through.

Today, for instance —

"Your idea just might change the world.

Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

Clearly there is a problem here analogous to

the square-triangle coordinatization problem,

but with the 4×6 rectangle of the R. T. Curtis

Miracle Octad Generator playing the role of

the square.

I once studied this 24-triangle-hexagon

coordinatization problem, but was unable to

obtain any results of interest. Perhaps

someone else will have better luck.

* For a rather different use of this word,

see Hermann Weyl in the Stanford

Encyclopedia of Philosophy.

## Saturday, January 18, 2014

### The Triangle Relativity Problem

A sequel to last night's post The 4×4 Relativity Problem —

In other words, how should the triangle corresponding to

the above square be *coordinatized* ?

See also a post of July 8, 2012 — "Not Quite Obvious."

Context — "Triangles Are Square," a webpage stemming

from an *American Mathematical Monthly * item published

in 1984.

## Friday, January 17, 2014

### The 4×4 Relativity Problem

The sixteen-dot square array in yesterday’s noon post suggests

the following remarks.

“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

The *Galois tesseract* appeared in an early form in the journal

*Computer Graphics and Art* , Vol. 2, No. 1, February 1977—

The 1977 matrix Q is echoed in the following from 2002—

A different representation of Cullinane’s 1977 square model of the

16-point affine geometry over the two-element Galois field GF(2)

is supplied by Conway and Sloane in *Sphere Packings, Lattices and Groups *

(first published in 1988) :

Here **a, b, c, d** are basis vectors in the vector 4-space over GF(2).

(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

## Thursday, November 22, 2012

### Finite Relativity

**(Continued from 1986)**

S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them. — H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8-sets. A representative coordinatization:
0000 0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them. — H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |

For some background on the triangular version,

see the Square-Triangle Theorem,

noting particularly the linked-to coordinatization picture.

## Tuesday, September 20, 2011

### Relativity Problem Revisited

A footnote was added to Finite Relativity—

**Background:**

Weyl on what he calls *the relativity problem*—

"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, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"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, 1946, *The Classical Groups *, Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, *M*_{ 24} (containing the original group), acts on the larger array. There is no obvious solution to Weyl's relativity problem for *M*_{ 24}. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or *symbol-strings* ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is *M*_{ 24}. ….

**Footnote of Sept. 20, 2011:**

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction "The generators of *M*_{ 24} are defined… as permutations of twenty-four 7-cycles in the action of PSL_{2}(7) on seven letters…."

See "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," by R.T. Curtis, *Mathematical Proceedings of the Cambridge Philosophical Society* (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, "Natural Constructions of the Mathieu groups," *Math. Proc. Cambridge Philos. Soc. * (1989), Vol. 106, pp. 423-429

R.T. Curtis. "Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups *M*_{ 12} and *M*_{ 24}" In *Proceedings of 1990 LMS Durham Conference 'Groups, Combinatorics and Geometry'* (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, "A Survey of Symmetric Generation of Sporadic Simple Groups," in *The Atlas of Finite Groups: Ten Years On* , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

## Saturday, May 18, 2019

## Thursday, May 9, 2019

### The Birdseye Requiem

From *The Boston Globe* yesterday evening —

" Ms. Adams 'had this quiet intelligence that made you feel like

she understood you and she loved you. She was a true friend —

a true generous, generous friend. This is the kind of person

you keep in your life,' Birdseye added.

'And she had such a great sense of humor,' Birdseye said.

“She would always have the last laugh. She wasn’t always

the loudest, but she was always the funniest, and in the

smartest way.' "

"Ms. Adams, who lived in Waltham, was 55 when she died April 9 . . . ."

See as well April 9 in the post Math Death and a post from April 8,

also now tagged "Berlekamp's Game" — Horses of a Dream.

"When logic and proportion have fallen sloppy dead

And the white knight is talking backwards . . . ."

— Grace Slick in a song from yesterday's post "When the Men"

## Friday, August 4, 2017

### Clay

Landon T. Clay, founder of the Clay Mathematics Institute,

reportedly died on Saturday, July 29, 2017.

See related Log24 posts, now tagged Prize Problem,

from the date of Clay's death and the day before.

**Update of 9 PM ET on August 4, 2017 —**

Other mathematics discussed here on the date of Clay's death —

MSRI Program. Here MSRI is pronounced "Misery."

**Update of 9:45 PM ET on August 4, 2017 —**

## Saturday, July 29, 2017

### Science News

## Friday, July 28, 2017

### Creeds

From a novel featuring the Navier-Stokes problem —

A search for "Creed" in this journal yields

a different sort of Shiva —

For further reviews, click on the Penguin below.

### Aesthetic Distance

In memory of a Disney "imagineer" who reportedly died yesterday.

**From the opening scene of a 2017 film, "Gifted"****:**

Frank calls his niece Mary to breakfast on the morning she is

to enter first grade. She is dressed, for the first time, for school —

- Hey! Come on. Let's move! - No! - Let me see. - No. - Come on, I made you special breakfast. - You can't cook. - Hey, Mary, open up. (She opens her door and walks out.) - You look beautiful. - |

## Monday, September 28, 2015

### Cracker Jack Prize

From a post of July 24, 2011 —

A review —

“The story, involving the Knights Templar, the Vatican, sunken treasure,

the fate of Christianity and a decoding device that looks as if it came out of

a really big box of medieval Cracker Jack, is the latest attempt to combine

Indiana Jones derring-do with ‘Da Vinci Code’ mysticism.”

A feeble attempt at a purely mathematical "decoding device"

from this journal earlier this month —

For some background, see a question by John Baez at Math Overflow

on Aug. 20, 2015.

The nonexistence of a 24-cycle in the large Mathieu group

might discourage anyone hoping for deep new insights from

the above figure.

See Marston Conder's "Symmetric Genus of the Mathieu Groups" —

## Saturday, September 19, 2015

### Geometry of the 24-Point Circle

The latest *Visual Insight* post at the American Mathematical

Society website discusses group actions on the McGee graph,

pictured as 24 points arranged in a circle that are connected

by 36 symmetrically arranged edges.

Wikipedia remarks that …

"The automorphism group of the McGee graph

is of order 32 and doesn't act transitively upon

its vertices: there are two vertex orbits of lengths

8 and 16."

The partition into 8 and 16 points suggests, for those familiar

with the Miracle Octad Generator and the Mathieu group M_{24},

the following exercise:

Arrange the 24 points of the projective line

over GF(23) in a circle in the natural cyclic order

( ∞, 1, 2, 3, … , 22, 0 ). Can the McGee graph be

modeled by constructing edges in any natural way?

In other words, if the above set of edges has no

"natural" connection with the 24 points of the

projective line over GF(23), does some *other*

set of edges in an isomorphic McGee graph

have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez

at Math Overflow on August 20.

## Thursday, June 9, 2011

### Upshot

**Suggested by this afternoon’s NY Lottery number, 541—**

**Related material: Finite Relativity and The Schwartz Notes.**

## Wednesday, March 2, 2011

### Labyrinth of the Line

“Yo sé de un laberinto griego que es una línea única, recta.”

—Borges, “La Muerte y la Brújula”

“I know of one Greek labyrinth which is a single straight line.”

—Borges, “Death and the Compass”

Another single-line labyrinth—

Robert A. Wilson on the projective line with 24 points

and its image in the Miracle Octad Generator (MOG)—

**Related material —**

The remarks of Scott Carnahan at Math Overflow on October 25th, 2010

and the remarks at Log24 on that same date.

A search in the latter for **miracle octad** is updated below.

This search (here in a customized version) provides some context for the

Benedictine University discussion described here on February 25th and for

the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”

Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.

## Sunday, February 20, 2011

### Sunday School

## Wednesday, April 23, 2008

### Wednesday April 23, 2008

**Upscale Realism**

or, "Have some more

wine and cheese, Barack."

** ** Allyn Jackson on Rebecca Goldstein

in the April 2006 *AMS Notices *(pdf)

"Rebecca Goldstein’s 1983 novel The Mind-Body Problem has been widely admired among mathematicians for its authentic depiction of academic life, as well as for its exploration of how philosophical issues impinge on everyday life. Her new book, Incompleteness: The Proof and Paradox of Kurt Gödel, is a volume in the 'Great Discoveries' series published by W. W. Norton….
In March 2005 the Mathematical Sciences Research Institute (MSRI) in Berkeley held a public event in which its special projects director, Robert Osserman, talked with Goldstein about her work. The conversation, which took place before an audience of about fifty people at the Commonwealth Club in San Francisco, was taped…. A member of the audience posed a question that has been on the minds of many of Goldstein’s readers: Is She… talked about the relationship between Gödel and his colleague at the Institute for Advanced Study, Albert Einstein. The two were very different: As Goldstein put it, 'Einstein was a real mensch, and Gödel was very neurotic.' Nevertheless, a friendship sprang up between the two. It was based in part, Goldstein speculated, on their both being exiles– exiles from Europe and intellectual exiles. Gödel's work was sometimes taken to mean that even mathematical truth is uncertain, she noted, while Einstein's theories of relativity were seen as implying the sweeping view that 'everything is relative.' These misinterpretations irked both men, said Goldstein. 'Einstein and Gödel were realists and did not like it when their work was put to the opposite purpose.'" |

Related material:

Related material:

**From Log24 on**

March 22 (Tuesday of

Passion Week), 2005:

"'What is this Stone?' Chloe asked…. 'It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
— For more on this theme
appropriate to Passion Week — Jews playing God — see
Rebecca Goldstein Wine and cheese |

From

**UPSCALE**,

a website of the

physics department at

the University of Toronto:

Mirror Symmetry"The image [above] The caption of the 'That most divine and beautiful The caption of the 'A shadow, likeness, or |

For further iconology of the

above equilateral triangles,

see Star Wars (May 25, 2003),

*Mani Padme* (March 10, 2008),

Rite of Sping (March 14, 2008),

and

Art History: The Pope of Hope

(In honor of John Paul II

three days after his death

in April 2005).

Happy Shakespeare's Birthday.

## Friday, January 18, 2008

### Friday January 18, 2008

… Todo lo sé por el lucero puro

que brilla en la diadema de la Muerte.

— Rubén Darío,

born January 18, 1867

## Thursday, January 17, 2008

### Thursday January 17, 2008

**Well, she was**

**just seventeen…**

(continued)

(continued)

"Mazur introduced the topic of prime numbers with a story from *Don Quixote* in which Quixote asked a poet to write a poem with 17 lines. Because 17 is prime, the poet couldn't find a length for the poem's stanzas and was thus stymied."

— Undated American Mathematical Society news item about a Nov. 1, 2007, event

**Related material:**

*Desconvencida,
Jueves, Enero 17, 2008*

Horses of a Dream

(Log24, Sept. 12, 2003)

Knight Moves

(Log24 yesterday–

anniversary of the

Jan. 16 publication

of *Don Quixote)*

Windmill and Diamond

(St. Cecilia's Day 2006)

## Wednesday, January 16, 2008

### Wednesday January 16, 2008

**Knight Moves:**

Geometry of the

Eightfold Cube

Geometry of the

Eightfold Cube

Click on the image for a larger version

and an expansion of some remarks

quoted here on Christmas 2005.

## Friday, February 20, 2004

### Friday February 20, 2004

**Finite Relativity**

Today is the 18th birthday of my note

“The Relativity Problem in Finite Geometry.”

That note begins with a quotation from Weyl:

“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

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, *Symmetry*, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.