Log24

Monday, February 6, 2012

Savage Logic

Filed under: General — m759 @ 1:00 PM

(Continued)

Saturday, October 16, 2010

Quaternion Day

m759 @ 12:00 PM

Brightness at Noon continued

i2 = j2 = k2 = ijk = -1

– Sir William Rowan Hamilton,
   
Oct. 16, 1843

See also this journal on 1/09, 2010.

This post was suggested by the date
of a user comment in Wikipedia.

Wednesday, November 10, 2010

Scavenger Hunt

Filed under: General — m759 @ 3:09 AM

A description in Pynchon's Against the Day  of William Rowan Hamilton's October 16th, 1843, discovery of quaterions—

"The moment, of course, is timeless. No beginning, no end, no duration, the light in eternal descent, not the result of conscious thought but fallen onto Hamilton, if not from some Divine source then at least when the watchdogs of Victorian pessimism were sleeping too soundly to sense, much less frighten off, the watchful scavengers of Epiphany."

New York Lottery yesterday, on Hermann Weyl's birthday— Midday 106, Evening 865.

http://www.log24.com/log/pix10B/101110-WeylAndDiamond.jpg

Here 106 suggests 1/06, the date of Epiphany, and 865 turns out to be the title number of Weyl's Symmetry  at Princeton University Press—

http://press.princeton.edu/titles/865.html.

Symmetry and quaternions are, of course, closely related.

Friday, November 5, 2010

Seize the Day*

Filed under: General — m759 @ 3:24 AM

From last night's post

"… right now…. winning the day…."

— President Obama on the 16th of October (between 5:19 and 5:37 PM EDT)

This journal on the 16th of October—

Saturday, October 16, 2010

Quaternion Day

m759 @ 12:00 PM

Brightness at Noon continued

i2 = j2 = k2 = ijk = -1

– Sir William Rowan Hamilton, Oct. 16, 1843

See also this journal on 1/09, 2010.

Related religious material from Thomas Pynchon—

http://www.log24.com/log/pix10B/101105-QuaternionDayByPynchon.jpg

* Material related to this post's title, "Seize the Day"—

Indirectly related — an ad for the new film Black Swan
accompanying a Halloween story in yesterday's online New York Times

http://www.log24.com/log/pix10B/101104-ObamaNYTSm.jpg

More directly related —

Black Swan Theory at Wikipedia —

http://www.log24.com/log/pix10B/101105-BlackSwanTheory.jpg

Saturday, October 16, 2010

Quaternion Day

Filed under: General — m759 @ 12:00 PM

Brightness at Noon continued

i2 = j2 = k2 = ijk = -1

— Sir William Rowan Hamilton, Oct. 16, 1843

See also this journal on 1/09, 2010.

Friday, August 4, 2006

Friday August 4, 2006

Filed under: General — m759 @ 2:00 PM
The Double Cross

The following symbol
has been associated
with the date
December 1:

The image “http://www.log24.com/log/pix06A/060804-DWA2.gif” cannot be displayed, because it contains errors.

Click on the symbol
for details.

That date is connected
to today’s date since
Dec. 1 is the feast
i.e., the deathday– of
a saint of mathematics:
G. H. Hardy, author of
the classic
A Mathematician’s Apology
(online, pdf, 52 pp. ),
while today is the birthday
of three less saintly
mathematical figures:
Sir William Rowan Hamilton,

For these birthdays, here is
a more cheerful version of
the above symbol:

The image “http://www.log24.com/theory/images/PeirceBox.bmp” cannot be displayed, because it contains errors.

For the significance of
this version, see
Chinese Jar Revisited
(Log24, June 27, 2006),
a memorial to mathematician
Irving Kaplansky
(student of Mac Lane).

This version may be regarded
as a box containing the
cross of St. Andrew.
If we add a Greek cross
(equal-armed) to the box,
we obtain the “spider,”
or “double cross,” figure

The image “http://www.log24.com/theory/images/PeirceSpider.bmp” cannot be displayed, because it contains errors.

of my favorite mythology:
Fritz Leiber’s Changewar.

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

In a 1940 letter to his sister Simone,  André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.  

John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:

“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”

For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,

The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3).  For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).

The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3).  It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Thursday, August 4, 2005

Thursday August 4, 2005

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

Today’s mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.

It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.

The figures below illustrate this fact.

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Related material: Visualizing GL(2,p)

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

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