Log24

Monday, December 16, 2013

The Seventh Square

Filed under: General,Geometry — m759 @ 10:00 PM

The above image is from Geometry of the 4×4 Square.

(The link "Visible Mathematics" in today's previous post, Quartet,
led to a post linked to that page, among others.)

Note that the seventh square above, at top right of the array of 35,
is the same as the image in Quartet.

Related reading

Quartet

Filed under: General,Geometry — m759 @ 12:00 PM

IMAGE- Four quadrants of a Galois tesseract, and a figure from 'Lawrence of Arabia'

Happy Beethoven's Birthday.

Related material:  Abel 2005 and, more generally, Abel.

See also Visible Mathematics.

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
In honor of Paul Newman’s age today, 81:

On Beauty

Elaine Scarry, On Beauty (pdf), page 21:

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 92 + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

The image “http://www.log24.com/theory/images/060126-planes2.jpg” cannot be displayed, because it contains errors.

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

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

 

Saturday, December 24, 2005

Saturday December 24, 2005

Filed under: General — m759 @ 9:00 PM
Nine is a Vine
(continued)

The image “http://www.log24.com/log/pix05B/051224-Stars.jpg” cannot be displayed, because it contains errors.

The figures are:
 
A symbol of Apollo from
Balanchine’s Birthday and
A Minature Rosetta Stone,

a symbol of pure reason from
Visible Mathematics and
Analogical Train of Thought,

a symbol of Venus from
Why Me?
and
To Graves at the Winter Solstice,

and, finally, a more
down-to-earth symbol,
adapted from a snowflake in

The image “http://www.log24.com/log/pix05B/051224-RebaCard2.jpg” cannot be displayed, because it contains errors.

an online Christmas card.

Those who prefer their
theological art on the scary side
may enjoy the
Christian Snowflake
link in the comments on
the “Logos” entry of
Orthodox Easter (May 1), 2005.

Thursday, August 25, 2005

Thursday August 25, 2005

Filed under: General,Geometry — m759 @ 3:09 PM
Analogical
Train of Thought

Part I: The 24-Cell

From S. H. Cullinane,
 Visualizing GL(2,p),
 March 26, 1985–

Visualizing the
binary tetrahedral group
(the 24-cell):

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

Another representation of
the 24-cell
:

The image “http://www.log24.com/theory/images/24-cell.jpg” cannot be displayed, because it contains errors.

 From John Baez,
This Week’s Finds in
Mathematical Physics (Week 198)
,”
September 6, 2003: 

Noam Elkies writes to John Baez:

Hello again,

You write:

[…]

“I’d like to wrap up with a few small comments about last Week.  There I said a bit about 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! And I’ve learned a bit more about this thing from here:”

[…]

Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL2(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.

Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.

NDE


The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota

Like footprints erased in the sand….

Part II: Discrete Space

The James Joyce School
 of Theoretical Physics
:


Log24, May 27, 2004

  “Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.” 

  “A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?”

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

   — Peter Szekeres, abstract of Discrete Space-Time

   “A theory…. predicts that space and time are indeed made of discrete pieces.”

   — Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

   “… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

   — Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

   “Theories of discrete space-time structure are being studied from a variety of perspectives.”

   — Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London

Disclaimer:

The above speculations by physicists
are offered as curiosities.
I have no idea whether
 any of them are correct.

Related material:

Stephen Wolfram offers a brief
History of Discrete Space.

For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.

Part III: Quaternions
in a Discrete Space

Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
 (Aug. 4, 2005):

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

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — m759 @ 1:25 PM

The Fugue

   "True joy is a profound remembering, and true grief is the same.
    Thus it was, when the dust storm that had snatched Cal up finally died, and he opened his eyes to see the Fugue spread out before him, he felt as though the few fragile moments of epiphany he'd tasted in his twenty-six years– tasted but always lost– were here redeemed and wed. He'd grasped fragments of this delight before. Heard rumour of it in the womb-dream and the dream of love; known it in lullabies. But never, until now, the whole, the thing entire.
    It would be, he idly thought, a fine time to die.
    And a finer time still to live, with so much laid out before him."

— Clive Barker,
Weaveworld,
 Book Two:
The Fugue

From Monday:

Weaveworld,
Book Three:
Out of the
Empty Quarter

"The wheels of its body rolled,
the visible mathematics
    of its essence turning on itself…."

From Friday:

The image “http://www.log24.com/log/pix05B/050806-Square.bmp” cannot be displayed, because it contains errors.

  For the meaning
of this picture, see
Geometry of the
4×4 Square.

For graphic designs
based on this geometry,
see Theme and Variations
and Diamond Theory.

For these designs in the
context of a Bach fugue,
see Timothy A. Smith's
essay (pdf) on

Fugue No. 21 in B-Flat Major
from Book II of
The Well-Tempered Clavier
by Johann Sebastian Bach.

Smith also offers a
Shockwave movie
that uses diamond theory
to illustrate this fugue.

Friday, August 5, 2005

Friday August 5, 2005

Filed under: General,Geometry — Tags: , — m759 @ 4:23 PM

For Sir Alec

From Elegance:

"Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday…."

— Bernard Holland, page C12,
    The New York Times,
    Monday, May 20, 1996.

Holland was pondering the identity of the Juilliard String Quartet, which had just given a series of concerts celebrating its fiftieth anniversary.

"Elegant"

— Page one,
    The New York Times,
    Monday, August 7, 2000.
 
The Times was describing the work of Sir Alec Guinness, who died on 8/5/00.

An example of the Holland name problem:

Monday, August 1, 2005 — Visible Mathematics:

    "Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
   '
"Empty" is a misnomer…  the Rub' al-Khali contains many hidden riches.'"

Friday, August 5, 2005 —  

The image “http://www.log24.com/log/pix05B/050805-Rag.jpg” cannot be displayed, because it contains errors.

Related material:

Geometry for Prince Harry

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

Tuesday, August 2, 2005

Tuesday August 2, 2005

Filed under: General,Geometry — Tags: — m759 @ 7:00 AM
Today's birthday:
Peter O'Toole

"What is it, Major Lawrence,
 that attracts you personally
 to the desert?"

"It's clean."

Visible Mathematics,
continued —

From May 18:

Lindbergh's Eden

"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."

— Anne Morrow Lindbergh,
Earth Shine, p. xii
 

 
On Beauty
 
"Beauty is the proper conformity
of the parts to one another
and to the whole."

— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974

Related material:

The Eightfold Cube

The Eightfold Cube

(in Arabic, ka'b)

and

The image “http://www.log24.com/log/pix05B/050802-Geom.jpg” cannot be displayed, because it contains errors.
 

Tuesday August 2, 2005

Filed under: General — Tags: — m759 @ 5:24 AM

Final Arrangements, continued
 

Kismet

From yesterday's Log24
Clive Barker's Weaveworld:

Another of the angel's attributes rose from memory now, and with it a sudden shock of comprehension.  Uriel had been the angel left to stand guard at the gates of Eden.
    Eden.
    At the word, the creature blazed.  Though the ages had driven it to grief and forgetfulness, it was still an angel: its fires unquenchable.  The wheels of its body rolled, the visible mathematics of its essence turning on itself and preparing for new terrors.
    There were others here, the Seraph said, that called this place Eden.  But I never knew it by that name.
    "What, then?" Shadwell asked.
    Paradise, said the Angel, and at the word a new picture appeared in Shadwell's mind.  It was the garden, in another age….
    This was a place of making, the Angel said.  Forever and ever.  Where things came to be.
    "To be?"
    To find a form, and enter the world.

If I stand starry-eyed
That's a danger in paradise
For mortals who stand beside
  An angel like you.

Robert Wright and George Forrest

The image “http://www.log24.com/log/pix05B/050802-NYTobits2.jpg” cannot be displayed, because it contains errors.
 

Monday, August 1, 2005

Monday August 1, 2005

Filed under: General — Tags: — m759 @ 12:00 PM
Visible Mathematics

    "Earlier, there had been mapping projects in Saudi Arabia's Rub' al-Khali, the Empty Quarter in the south and west of the country….
     '
"Empty" is a misnomer…  the Rub' al-Khali contains many hidden riches.'"

Maps from the Sky,
   Saudi Aramco World, March/April 1995

From Weaveworld

Book Three:
Out of the Empty Quarter,
 by Clive Barker, 1987:


… As a child he'd learned the names of all the angels and archangels by heart: and among the mighty, Uriel was of the mightiest.  The archangel of salvation: called by some the flame of God…. What had he done, stepping into the presence of such power?  This was Uriel, of the principalities….
    Another of the angel's attributes rose from memory now, and with it a sudden shock of comprehension.  Uriel had been the angel left to stand guard at the gates of Eden.
    Eden.
    At the word, the creature blazed.  Though the ages had driven it to grief and forgetfulness, it was still an angel: its fires unquenchable.  The wheels of its body rolled, the visible mathematics of its essence turning on itself and preparing for new terrors.
    There were others here, the Seraph said, that called this place Eden.  But I never knew it by that name.
    "What, then?" Shadwell asked.
    Paradise, said the Angel, and at the word a new picture appeared in Shadwell's mind.  It was the garden, in another age….
    This was a place of making, the Angel said.  Forever and ever.  Where things came to be.
    "To be?"
    To find a form, and enter the world.

 

"The serpent's eyes shine
As he wraps around the vine
In the Garden of Allah."

Don Henley, 1995  
 

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