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Web Journal of
Steven H.
Cullinane
Music for Pegasus

m759
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Name: Steven
Country: United States
State: Pennsylvania
Gender: Male


Interests: Mathematics, literature.
Occupation: Retired
Industry: Computers (Software)


Message: message me
Website: visit my website


Member Since: 7/20/2002
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Archived Entries:
See log24.com.

Selected Past Entries:

Three Days
of the
Saint, 2002

12/6:
Santa vs.
the Volcano


12/7:
Satori at
Pearl Harbor


12/8:
Architecture
of Eternity


Some may feel that the Saint in question is Philip Berrigan, who joined Saburo Ienaga and Ivan Illich on Dec. 6, 2002.

Others may feel that the Saint is Don Ameche, who died on Dec. 6, 1993.

"Things change."

— SHC 12/9/02

Sequel

Stan Rice died on Dec. 9, 2002. A poem of his tells what happened next.

Eight is a Gate

Hollywood producer dies Dec. 14, meets Bach at Heaven's Gate. Realistic comedy.

The Diamond Project

Notes on dance, mortality, and "the still point" on the date of Irene Diamond's death.

Immortal Diamond,
or
NASA Meets Jesus

Thoughts on John O'Hara and G. M. Hopkins for James Joyce's birthday.

Blackbird Singing

The Fred Rogers memorial koan.

Art Wars

LeWitt vs. Witt

Stone, not Wood

best describes St. Peter

The Word

in the Desert

Art Wars:

Fahne Hoch

and

Thorny Crown


O'Hara's Crucifixion


Unity and Reciprocity

in mathematics

The Quality of Diamond


Da Vinci Code ,

Crimson Passion,

Cubist Crucifixion.

Truth and Style


The Line


Bush Mutiny


Symmetry and Change


A Shot at Redemption


Mathematics and Narrative


The Judas Seat


Countdown


My math sites:

Finitegeometry.org

Finitegeometry.org/sc

The Diamond 16 Puzzle

Notes on Finite Geometry

The Diamond Theorem

The Geometry of Qubits

Diamond Theory

Diamond Theory
in 1937


Galois Geometry

A Four-Color Theorem

Latin-Square Geometry

Walsh Functions

The Fano Plane Revisualized

Knight Moves

The MOG

Inscapes

The Diamond Theory of Truth

Logos and Logic

Literary-Philosophical
Puzzle Notes


A Mathematician's Aesthetics

Reflection Groups in Finite Geometry

A Reflection Group of Order 168

Reflection Groups: The Missing Link

Geometry of
the I Ching


The Diamond Archetype

Modal Theology

The Eightfold Way and Solomon's Seal

Crystal and Dragon in Diamond Theory

Poetry's Bones

Time Fold

War of Ideas

The Proof
and the Lie


Lemniscate
to Langlands


Symmetry Groups

Block Designs

Finite Relativity

Cognitive Blending

Geometry of the 4x4 Square

Visualizing GL(2,p)

Pattern Groups

Ideas and Art

Jung's Imago

Theme and Variations

The Geometry of Logic

Space-Time and a Finite Model

Quilt Geometry

Duality and Symmetry

Polster on Pictures

Kaleidoscope

The Dharwadker Files

Certified Crank

Dharwadker at Wikipedia

Coset Representatives

Archived Journal


Radio I Like

Plano TX KHYI

WAMU 88.5FM

WHRB Harvard

BBC 3

Live365.com


Favorite Books

The Practical Cogitator

Style

The Reader Over Your Shoulder

The Oxford Book of English Prose

Fancies and Goodnights


Other Online Commonplace Books

David Lavery

Peter J. Cameron

A. M. Kuchling

Constant Reader

Identity Theory

J. Jacobs

M. Magnus

ChrisNet

Anonymous

Sites I Read:

Bloglines list

Ping form

SubscriptionsSites I Read

Blogrings
: : : HARVARD : : :
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Monday, July 21, 2008

Mathematics and Narrative, continued:

Knight Moves:
The Relativity Theory
of Kindergarten Blocks


(Continued from
January 16, 2008)

"Hmm, next paper... maybe
 'An Unusually Complicated
Theory of Something.'"

-- Garrett Lisi at
Physics Forums, July 16

Something:

From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift: A cube made up of eight subcubes

Click on image for details.


An Unusually
Complicated Theory:


From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for details.


For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168
."

For an even more
complicated theory of
Klein's simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.


Saturday, July 19, 2008

Annals of Mathematics:


Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievably intricate internal structure."

Hermann Weyl on the hard core of objectivity:

"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind-- as Eddington puts it-- the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)

Steven H. Cullinane on the symmetries of a 4x4 array of points:

A Structure-Endowed Entity

"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.  You can expect to gain a deep insight into the constitution of S in this way."

-- Hermann Weyl in Symmetry

Let us apply Weyl's lesson to the following "structure-endowed entity."

4x4 array of dots

What is the order of the resulting group of automorphisms?


The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
  called a "colorful tale"--

The Diamond 16 Puzzle

The Diamond 16 Puzzle


This puzzle shows
that the 4x4 array can
also be viewed in
thousands of ways.

"You can make 322,560
pairs of patterns. Each
 pair pictures a different
symmetry of the underlying
16-point space."

-- Steven H. Cullinane,
July 17, 2008

For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings
.


Friday, July 18, 2008

Mathematics and Narrative, continued:

Hard Core

David Corfield quotes Weyl in a weblog entry, "Hierarchy and Emergence," at the n-Category Cafe this morning:

"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind-- as Eddington puts it-- the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science [Princeton, 1949], p. 237)

For the same quotation in a combinatorial context, see the foreword by A. W. Tucker, "Combinatorial Problems," to a special issue of the IBM Journal of Research and Development, November 1960 (1-page pdf).

See also yesterday's Log24 entry.


Thursday, July 17, 2008

Annals of Philosophy:

CHANGE
 FEW CAN BELIEVE IN

Continued from June 18.

Jungian Symbols
of the Self --


User icons (identicons) from Secret Blogging Seminar
Compare and contrast:

Jung's four-diamond figure from
Aion -- a symbol of the self --

Jung's four-diamond figure showing transformations of the self as Imago Dei

Jung's Map of the Soul,
by Murray Stein:

"... Jung thinks of the self as undergoing continual transformation during the course of a lifetime.... At the end of his late work Aion, Jung presents a diagram to illustrate the dynamic movements of the self...."

For related dynamic movements,
see the Diamond 16 Puzzle
and the diamond theorem.


Tuesday, July 15, 2008

Elliptic comment:

My comment on a discussion of elliptic curves and modular forms at Secret Blogging Seminar, about 10 PM tonight:

How does this affect popularized discussions of the Taniyama-Shimura conjecture-- for instance, Ivars Peterson's, in "Curving Beyond Fermat," November 1999-- which claim, for instance, that "Elliptic curves and modular forms are mathematically so different that mathematicians initially [in the 1950's, the early days of the conjecture] couldn't believe that the two are related."?

Update of about 10:45 PM tonight:

A reply by the author of the discussion, Scott Carnahan:

I don’t think anyone doubted that there is a connection between elliptic curves and modular forms on the level I described above. However, the Taniyama-Shimura conjecture refers to a more advanced idea about a deeper connection.

Carnahan then gives a one-paragraph summary, definitely not popularized, of the deeper connection.



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