Translation Plane
for Rosh Hashanah

Figure A

From the website of Priv.-Doz. Dr. H. Klein, Arbeitsgruppe Geometrie, Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel -- The Translation Plane of Order Nine There are exactly four projective planes of order nine, and one of these planes is a non-Desarguesian translation plane. Theorem. Up to isomorphism, there exists exactly one non-Desarguesian translation plane of order 9. This translation plane is defined by a spreadset in a 2-dimensional vector space over the field GF(3), consisting of the following matrices.   As it turns out, the coordinatizing quasifield is a nearfield. Moreover the non-Desarguesian translation plane of order 9 has Lenz-Barlotti type IVa.3.

Two versions of the defining spreadset for this plane are shown in Figure A.  In the left part of Fig. A, the matrices of Dr. Klein are altered by the use of "2" instead of "-1" (since these are the same, modulo 3).  In the right part of Fig. A, the corresponding figures from my 1985 note Visualizing GL(2, p) are shown.

The Square Wheel

Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions.  George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel.  Harmonic analysis based on square functions-- the "square wheel," as it were-- is also not without its importance.

For some observations of Stephen Wolfram on square-wheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002).  Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finite-geometry hyperplanes that I wrote in 1986.  A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.

That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh functions-- which have, in fact, properties very like those of the trig functions.  For details, see Symmetry of Walsh Functions, updated today.

"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake.  Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."

-- From Error-Correcting Codes
Through Sphere Packings
To Simple Groups,
by Thomas M. Thompson,
Mathematical Association of America, 1983

The Turning

A way a lone
a last a loved
a long the

PARIS,
1922-1939

Click on pictures
for further details.

"For the essence and the end
Of his labor is beauty... 
one beauty, the rhythm of that Wheel"

Robinson Jeffers,
"Point Pinos and Point Lobos"

Dark Lady

"Each time we come closer to Shakespeare's life, we escape from the aridity of formal criticism or the cheap generalities of social history into a recognizable world of real experience.  When A. L. Rowse insists that Emilia Bassano Lanier, the tempestuous, adulterous, musical, poetic wife of a court musician, was the original 'Dark Lady' of the Sonnets, we can buy it or not, as we please. But the very existence of a woman like Emilia demonstrates that the clichéd images of Elizabethan women, as subservient wives or unruly whores, are too grossly tuned to capture the reality of Shakespeare's world. Whether she is the Dark Lady or not, Emilia is a dark lady. Good biographical criticism dissolves determinisms, and replaces them not with gossipy puzzle-solution certainties but with glimpses of life as it is lived, and art as it is made."

-- Adam Gopnik, The New Yorker, issue dated Sept. 13, 2004

"En librairie depuis le 12 septembre 2003, ce livre correspond au désir du mouvement 'ni putes ni soumises' de briser l'omerta et de poursuivre les débats engagés depuis la marche des femmes. À travers ce récit, ce sont les voix de milliers de jeunes femmes qui se font entendre, exprimant leurs interrogations et leur révolte."

-- niputesnisoumises.com

On Samira Bellil, who died on Sept. 3:

"Bellil was considered the 'godmother' of the womens' rights group 'Ni Putes Ni Soumises' (Neither Whores Nor Submissive.)"

-- Hendersonville (NC) Times-News

September Morn

"Now is the time for turning. The leaves are beginning to turn from green to red to orange. The birds are beginning to turn and are heading once more toward the south. The animals are beginning to turn to storing their food for the winter. For leaves, birds and animals, turning comes instinctively. But for us, turning does not come so easily."

-- President Clinton,
Prayer Breakfast,
September 11, 1998

Philosophy

For Samira Bellil,

who died in Paris on
Friday, Sept. 3, 2004... 

From the link at
Symmetry and Change
in the Dreamtime,
Part 8, Friday,
Sept. 3, 2004,
Noon...

Under heaven 
thunder rolls...

Log24 on Sept. 10, 2002--

Three songs from Sept. 10
in various preceding years--

"Good morning little schoolgirl
Good morning little schoolgirl
Can I come home with
Can I come home with you"

-- Rod Stewart, Sept. 10, 1964

"Tell your mamma, girl, I can't stay long
We got things we gotta catch up on
Mmmm, you know
You know what I'm sayin' "

-- Neil Diamond, Sept. 10, 1966

"A time of war, a time of peace
A time of love, a time of hate
A time you may embrace
A time to refrain from embracing"

-- The Byrds, Sept. 10, 1965

Further verses from the Byrds
seem appropriate on this, the day
of Samira Bellil's funeral:

To everything, turn, turn, turn,
there is a season, turn, turn, turn...

Tournante

"It's not even called rape. They call it
a tournante, or pass-round.
The banality is deliberate:
a joint, a girl - same difference."

... and a time to every purpose
under heaven.

"... The kind of school where teacher
Fabrice Genestal kept hearing
the word "tournante" and didn't click
what it meant, till he and Sillam
sat the kids down in after-school
workshops, and got talking."

-- Metropolitan Police Service, London

An Invariant Feast

In memory of philosopher Robert D. Cumming, who took part in the liberation of Paris on the Feast of St. Louis, 1944, and who died on that same feast day, August 25, in 2004:

"We'll always have Paris."

A Story of Sorts

Sometimes one's journal entries seem to be telling a story...

This was the case for Log24 entries of Tuesday through Friday last week.  Unfortunately, the story they told is about as coherent as Finnegans Wake.

Anyone interested can find the story, put into chronological order and prefaced with a summary, at

Symmetry and Change
in the Dreamtime.

Symmetry and Change, conclusion...

Ite, Missa Est

3:17:13 PM ET

Hexagram 13
Fellowship With Men:

The Image

Heaven together with fire.

"A pretty girl --
is like a melody ---- !"

 For details, see
A Mass for Lucero.

Symmetry and Change, Part 9...

De Nada

Helen Lane

1:13:49 PM ET

Hexagram 49
Revolution:

The Image

"I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night.... But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? ...And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart."

-- Malcolm Lowry, Under the Volcano