Log24

Saturday, August 23, 2014

Explicatio

Filed under: General — m759 @ 2:20 AM

(The title is from a post of July 8, 2010.)

“What is important is the ability to tell stories through character.”

Diana Gabaldon, author of the Outlander  series of novels

An image from the bottom line of images in the previous post:

In memory of Scottish folk singer Jean Redpath,
who reportedly died on Thursday, August 21:

See also this journal on August 21.

Saturday, May 24, 2014

Lyric Stupidity

Filed under: General — m759 @ 7:25 PM

From a song discussed in yesterday’s online NY Times :

“Blue, blue, my love is blue.”*

Trigger warning from SNL’s Weekend Update on April 12, 2014:

“It was announced this week that in an upcoming issue of
Life With Archie , the main character Archie Andrews
will die, following a lifelong struggle with blue balls.”

* Misheard version of Bryan Blackburn‘s “blue, blue, my world is blue”
translation of the Pierre Cour lyric “bleu, bleu, l’amour est bleu 

Sunday, January 26, 2014

Blazing Bride’s Chair

Filed under: General,Geometry — m759 @ 10:30 PM

A sequel to last night's link Shear —

Some dead poet's words —

The "bride's chair" is the figure illustrating Euclid's proof
of the Pythagorean theorem (click image to enlarge) —

See also

Not since Madeline Kahn in Blazing Saddles 

A Dead Poet’s Word

Filed under: General — m759 @ 1:06 AM

For the Tin Men —

The word:  Shear.

Sunday, March 18, 2012

Square-Triangle Diamond

Filed under: General,Geometry — Tags: — m759 @ 5:01 AM

The diamond shape of yesterday's noon post
is not wholly without mathematical interest …

The square-triangle theorem

"Every triangle is an n -replica" is true
if and only if n  is a square.

IMAGE- Square-to-diamond (rhombus) shear in proof of square-triangle theorem

The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.

(See the diamond lattice  in Weyl's Symmetry .)

Similarly for any square n , not just 16.

There is a group of 322,560 natural transformations
that permute the centers  of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers  of the 16 subtriangles
in a 16-part triangular array.

(Updated March 29, 2012, to correct wording and add Weyl link.)

Wednesday, February 8, 2012

Lichtung!

Filed under: General — m759 @ 12:00 PM

(Continued from Dec. 5, 2002)

IMAGE- NY Times: 'For Romney, Night Goes from Bad to Worse,' by Ashley Parker and Michael D. Shear

From Bad…

Shear

Braucht´s noch Text?

To Verse—

lichtung

manche meinen
lechts und rinks
kann man nicht
velwechsern.
werch ein illtum!

by Ernst Jandl

Tuesday, February 15, 2011

Road House

Filed under: General — Tags: — m759 @ 2:02 AM

A 1948 classic

Again, this couldn't happen again.
This is that "once in a lifetime,"
this is the thrill divine.

The great 1949 days (according to Jack Kerouac)—

IMAGE-- Scene from 'Blackboard Jungle,' 1955

On the Road

Shearing began to play his chords; they rolled out of the piano in great rich showers, you'd think the man wouldn't have time to line them up. They rolled and rolled like the sea. Folks yelled for him to "Go!" Dean was sweating; the sweat poured down his collar. "There he is! That's him! Old God! Old God Shearing! Yes! Yes! Yes!" And Shearing was conscious of the madman behind him, he could hear every one of Dean's gasps and imprecations, he could sense it though he couldn't see. "That's right!" Dean said. "Yes!" Shearing smiled; he rocked. Shearing rose from the piano, dripping with sweat; these were his great 1949 days before he became cool and commercial. When he was gone Dean pointed to the empty piano seat. "God's empty chair," he said.

Sunday, February 21, 2010

Reflections

Filed under: General,Geometry — m759 @ 12:06 PM

From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005as revised on Nov. 25, 2009

Historically, (Coxeter 1934) proved that every reflection group [Euclidean, by the current Wikipedia definition] is a Coxeter group (i.e., has a presentation where all relations are of the form ri2 or (rirj)k), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group [again, Euclidean], and classified finite Coxeter groups.

Finite fields

This section requires expansion.

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as −1=1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).

Related material:

"A Simple Reflection Group of Order 168," by Steven H. Cullinane, and

"Determination of the Finite Primitive Reflection Groups over an Arbitrary Field of Characteristic Not 2,"

by Ascher Wagner, U. of Birmingham, received 27 July 1977

Journal   Geometriae Dedicata
Publisher   Springer Netherlands
Issue   Volume 9, Number 2 / June, 1980

Ascher Wagner's 1977 dismissal of reflection groups over fields of characteristic 2

[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]

Clearly the eightfold cube is a counterexample.

Friday, October 6, 2006

Friday October 6, 2006

Filed under: General — m759 @ 12:00 PM
A Visual Proof

The great mathematician
Robert P. Langlands
is 70 today.

In honor of his expository work–
notably, lectures at
The Institute for Advanced Study
on “The Practice of Mathematics
and a very acerbic review (pdf) of
a book called Euclid’s Window
here is a “Behold!” proof of
the Pythagorean theorem:

The image “http://www.log24.com/log/pix06A/Pythagorean_Theorem.jpg” cannot be displayed, because it contains errors.

The picture above is adapted from
 a sketch by Eves of a “dynamical”
proof suitable for animation.

The proof has been
 described by Alexander Bogomolny
as “a variation on” Euclid I.47.
Bogomolny says it is a proof
by “shearing and translation.”

It has, in fact, been animated.
The following version is
by Robert Foote:
The image “http://www.log24.com/log/pix06A/RobertFooteAnimation.gif” cannot be displayed, because it contains errors.

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