Log24

Saturday, July 29, 2017

MSRI Program

Filed under: General,Geometry — Tags: — m759 @ 8:29 PM

"The field of geometric group theory emerged from Gromov’s insight
that even mathematical objects such as groups, which are defined
completely in algebraic terms, can be profitably viewed as geometric
objects and studied with geometric techniques."

— Mathematical Sciences Research Institute, 2016:

Geometric Group theory at MSRI (pronounced 'Misery')

See also some writings of Gromov from 2015-16:

For a simpler example than those discussed at MSRI
of both algebraic and geometric techniques applied to
the same group, see a post of May 19, 2017,
"From Algebra to Geometry." That post reviews
an earlier illustration —

For greater depth, see "Eightfold Cube" in this journal.

Monday, April 8, 2019

Misère Play

Filed under: General — Tags: , — m759 @ 5:21 PM

Facebook on Bloomsday 2017 —

Also on that Bloomsday —

Chalkroom Jungle Revisited —

Sunday, April 7, 2019

Chess King

Filed under: General — Tags: — m759 @ 10:10 PM

Meanwhile . . .

Front page top center, online NY Times: Bobby Fischer Dead at 64

Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:43 AM

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018

http://www.math.sci.hiroshima-u.ac.jp/
branched/files/2018/abstract/Aitchison.txt

Iain AITCHISON

Title:

Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II

Abstract:

Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'.

Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve.

See also yesterday morning's post, "Character."

Update: For a followup, see the next  Log24 post.

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 4:28 PM

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Friday, August 4, 2017

Clay

Filed under: General — Tags: — m759 @ 4:08 PM

Landon T. Clay, founder of the Clay Mathematics Institute,
reportedly died on Saturday, July 29, 2017.

See related Log24 posts, now tagged Prize Problem,
from the date of Clay's death and the day before.
 

Update of 9 PM ET on August 4, 2017 —

Other mathematics discussed here on the date of Clay's death —

MSRI Program. Here MSRI is pronounced "Misery."
 

Update of 9:45 PM ET on August 4, 2017 —

Sunday, July 30, 2017

Sermon: MS R I

Filed under: General,Geometry — Tags: — m759 @ 9:57 AM

From Solomon's Cube

"Here MSRI, an acronym for Mathematical Sciences Research Institute,
is pronounced 'Misery.' See Stephen King [and] K.C. Cole . . . ."

From a manuscript by Mikhail Gromov cited yesterday in MSRI Program —

Quotes from a founder of geometric group theory

Tuesday, February 10, 2015

In Memoriam…

Filed under: General,Geometry — m759 @ 12:25 PM

industrial designer Kenji Ekuan —

Eightfold Design.

The adjective "eightfold," intrinsic to Buddhist
thought, was hijacked by Gell-Mann and later 
by the Mathematical Sciences Research Institute
(MSRI, pronounced "misery").  The adjective's
application to a 2x2x2 cube consisting of eight
subcubes, "the eightfold cube," is not intended to
have either Buddhist or Semitic overtones.  
It is pure mathematics.

Wednesday, October 1, 2014

Misery

Filed under: General — m759 @ 6:01 PM

The title is the usual pronunciation of MSRI,
the Mathematical Sciences Research Institute
at 17 Gauss Way, Berkeley, California.

The late Scandinavian novelist Stieg Larsson
might prefer to call this street Gardner Way.

I do not.

Tuesday, June 26, 2012

Looking Deeply

Filed under: General,Geometry — Tags: , — m759 @ 3:48 PM

Last night's post on The Trinity of Max Black  and the use of
the term "eightfold" by the Mathematical Sciences Research Institute
at Berkeley suggest a review of an image from Sept. 22, 2011

IMAGE- Eightfold cube with detail of triskelion structure

The triskele  detail above echoes a Buddhist symbol found,
for instance, on the Internet in an ad for meditation supplies—

Related remarks

http://www.spencerart.ku.edu/about/dialogue/fdpt.shtml

Mary Dusenbury (Radcliffe '64)—

"… I think a textile, like any work of art, holds a tremendous amount of information— technical, material, historical, social, philosophical— but beyond that, many works of art are very beautiful and they speak to us on many layers— our intellect, our heart, our emotions. I've been going to museums since I was a very small child, thinking about what I saw, and going back to discover new things, to see pieces that spoke very deeply to me, to look at them again, and to find more and more meaning relevant to me in different ways and at different times of my life. …

… I think I would suggest to people that first of all they just look. Linger by pieces they find intriguing and beautiful, and look deeply. Then, if something interests them, we have tried to put a little information around the galleries to give a bit of history, a bit of context, for each piece. But the most important is just to look very deeply."

http://en.wikipedia.org/wiki/Nikaya_Buddhism

According to Robert Thurman, the term "Nikāya Buddhism" was coined by Professor Masatoshi Nagatomi of Harvard University, as a way to avoid the usage of the term Hinayana.[12] "Nikaya Buddhism" is thus an attempt to find a more neutral way of referring to Buddhists who follow one of the early Buddhist schools, and their practice.

12. The Emptiness That is Compassion:
An Essay on Buddhist Ethics, Robert A. F. Thurman, 1980
[Religious Traditions , Vol. 4 No. 2, Oct.-Nov. 1981, pp. 11-34]

http://dsal.uchicago.edu/cgi-bin/philologic/getobject.pl?c.2:1:6.pali

Nikāya [Sk. nikāya, ni+kāya]
collection ("body") assemblage, class, group

http://en.wiktionary.org/wiki/नि

Sanskrit etymology for नि (ni)

From Proto-Indo-European *ni …

Prefix

नि (ni)

  • down
  • back
  • in, into

http://www.rigpawiki.org/index.php?title=Kaya

Kaya (Skt. kāya སྐུ་, Tib. ku Wyl. sku ) —
the Sanskrit word kaya literally means ‘body’
but can also signify dimension, field or basis.

སྐུ། (Wyl. sku ) n. Pron.: ku

structure, existentiality, founding stratum ▷HVG KBEU

gestalt ▷HVG LD

Note that The Trinity of Max Black  is a picture of  a set
i.e., of an "assemblage, class, group."

Note also the reference above to the word "gestalt."

"Was ist Raum, wie können wir ihn
erfassen und gestalten?"

Walter Gropius

Wednesday, December 21, 2011

The Purloined Diamond

Filed under: General — Tags: , — m759 @ 9:48 AM

Stephen Rachman on "The Purloined Letter"

"Poe’s tale established the modern paradigm (which, as it happens, Dashiell Hammett and John Huston followed) of the hermetically sealed fiction of cross and double-cross in which spirited antagonists pursue a prized artifact of dubious or uncertain value."

For one such artifact, the diamond rhombus formed by two equilateral triangles, see Osserman in this journal.

Some background on the artifact is given by John T. Irwin's essay "Mysteries We Reread…" reprinted in Detecting Texts: The Metaphysical Detective Story from Poe to Postmodernism .

Related material—

Mathematics vulgarizer Robert Osserman died on St. Andrew's Day, 2011.

A Rhetorical Question

Osserman in 2004

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales— regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer

http://www.log24.com/log/pix11C/111130-SunshineCleaning.jpg

Above: Amy Adams in "Sunshine Cleaning"

Wednesday, March 3, 2010

Plato’s Ghost

Filed under: General,Geometry — Tags: , — m759 @ 11:07 AM

Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton, 2008–

"Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious— rather than a naïve relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve. This brisk definition…."

Brisk? Consider Caesar's "The die is cast," Gray in "Solomon's Cube," and yesterday's post

Group of 8 cube-face permutations generated by reflections in midplanes parallel to faces

This is the group of "8 rigid motions
generated by reflections in midplanes"
of Solomon's Cube.

Related material:

"… the action of G168 in its alternative guise as SL(3; Z/2Z) is also now apparent. This version of G168 was presented by Weber in [1896, p. 539],* where he attributed it to Kronecker."

— Jeremy Gray, "From the History of a Simple Group," in The Eightfold Way, MSRI Publications, 1998

Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber.

*H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig, 1896. Reprinted by Chelsea, New York, 1961.

Wednesday, February 24, 2010

Transvections

Filed under: General,Geometry — m759 @ 4:24 PM

A topic related to A Simple Reflection Group of Order 168

Transvection groups over GF(2). See, for instance…

  1. Binary Coordinate Systems, by Steven H. Cullinane, 1984
     
  2. Classification of the Finite N-Generator Transvection Groups Over Z2, by Jizhu Nan and Jing Zhao, 2009, Advances in Applied Mathematics Vol. 44 Issue 3 (March 2010), 185–202
     
  3. Anne Shepler, video of a talk on Nov. 4, 2004, "Reflection Groups and Modular Invariant Theory"

Monday, September 14, 2009

Monday September 14, 2009

Filed under: General,Geometry — m759 @ 3:09 PM
Figure

Generating permutations for the Klein simple group of order 168 acting on the eightfold cube

The Sept. 8 entry on non-Euclidean* blocks ended with the phrase “Go figure.” This suggested a MAGMA calculation that demonstrates how Klein’s simple group of order 168 (cf. Jeremy Gray in The Eightfold Way) can be visualized as generated by reflections in a finite geometry.

* i.e., other than Euclidean. The phrase “non-Euclidean” is usually applied to only some of the geometries that are not Euclidean. The geometry illustrated by the blocks in question is not Euclidean, but is also, in the jargon used by most mathematicians, not “non-Euclidean.”

Monday, July 21, 2008

Monday July 21, 2008

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

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.

Wednesday, April 23, 2008

Wednesday April 23, 2008

Filed under: General — Tags: — m759 @ 9:00 AM

Upscale Realism

or, "Have some more
wine and cheese, Barack."

(See April 15, 5:01 AM)

  Allyn Jackson on Rebecca Goldstein
in the April 2006 AMS Notices (pdf)

"Rebecca Goldstein’s 1983 novel The Mind-Body Problem has been widely admired among mathematicians for its authentic depiction of academic life, as well as for its exploration of how philosophical issues impinge on everyday life. Her new book, Incompleteness: The Proof and Paradox of Kurt Gödel, is a volume in the 'Great Discoveries' series published by W. W. Norton….

In March 2005 the Mathematical Sciences Research Institute (MSRI) in Berkeley held a public event in which its special projects director, Robert Osserman, talked with Goldstein about her work. The conversation, which took place before an audience of about fifty people at the Commonwealth Club in San Francisco, was taped….

A member of the audience posed a question that has been on the minds of many of Goldstein’s readers: Is The Mind-Body Problem based on her own life? She did indeed study philosophy at Princeton, finishing her Ph.D. in 1976 with a thesis titled 'Reduction, Realism, and the Mind.' She said that while there are correlations between her life and the novel, the book is not autobiographical….

She… talked about the relationship between Gödel and his colleague at the Institute for Advanced Study, Albert Einstein. The two were very different: As Goldstein put it, 'Einstein was a real mensch, and Gödel was very neurotic.' Nevertheless, a friendship sprang up between the two. It was based in part, Goldstein speculated, on their both being exiles– exiles from Europe and intellectual exiles. Gödel's work was sometimes taken to mean that even mathematical truth is uncertain, she noted, while Einstein's theories of relativity were seen as implying the sweeping view that 'everything is relative.' These misinterpretations irked both men, said Goldstein. 'Einstein and Gödel were realists and did not like it when their work was put to the opposite purpose.'"


Related material:

From Log24 on
March 22 (Tuesday of
Passion Week), 2005:

 
"'What is this Stone?' Chloe asked…. 'It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

Many Dimensions,
by Charles Williams, 1931

For more on this theme
appropriate to Passion Week
Jews playing God — see

The image “http://www.log24.com/log/pix05/050322-Trio.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein
in conversation with
Bob Osserman
of the
Mathematical Sciences
Research Institute
at the
Commonwealth Club,
San Francisco,
Tuesday, March 22.

Wine and cheese
reception at 5:15 PM
(San Francisco time).

From
UPSCALE,
a website of the
physics department at
the University of Toronto:

Mirror Symmetry

Robert Fludd: Universe as mirror image of God

"The image [above]
is a depiction of
the universe as a
mirror image of God,
drawn by Robert Fludd
in the early 17th century.

The caption of the
upper triangle reads:

'That most divine and beautiful
counterpart visible below in the
flowing image of the universe.'

The caption of the
lower triangle is:

'A shadow, likeness, or
reflection of the insubstantial*
triangle visible in the image
of the universe.'"

* Sic. The original is incomprehensibilis, a technical theological term. See Dorothy Sayers on the Athanasian Creed and John 1:5.

For further iconology of the
above equilateral triangles,
see Star Wars (May 25, 2003),
Mani Padme (March 10, 2008),
Rite of Sping (March 14, 2008),
and
Art History: The Pope of Hope
(In honor of John Paul II
three days after his death
in April 2005).

Happy Shakespeare's Birthday.

Friday, November 3, 2006

Friday November 3, 2006

Filed under: General,Geometry — m759 @ 9:00 AM

First to Illuminate

From the History of a Simple Group” (pdf), by Jeremy Gray:

“The American mathematician A. B. Coble [1908; 1913]* seems to have been the first to illuminate the 27 lines and 28 bitangents with the elementary theory of geometries over finite fields.

The combinatorial aspects of all this are pleasant, but the mathematics is certainly not easy.”

* [Coble 1908] A. Coble, “A configuration in finite geometry isomorphic with that of the 27 lines on a cubic  surface,” Johns Hopkins University Circular 7:80-88 (1908), 736-744.

   [Coble 1913] A. Coble, “An application of finite geometry to the characteristic theory of the odd and even theta functions,” Trans. Amer. Math. Soc. 14 (1913), 241-276.

Related material:

Geometry of the 4x4x4 Cube,

Christmas 2005.

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM
The Fano Plane
Revisualized:

 

 The Eightfold Cube

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
 
The image “http://www.log24.com/theory/images/Fano.gif” cannot be displayed, because it contains errors.
 

Every permutation of the plane's points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

 

Fano plane with cubes as points
 
The cubes' partitioning planes are added in binary (1+1=0) fashion.  Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.

 

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

 

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

 

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

 

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon's Cube.

 

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General — Tags: , — m759 @ 7:59 PM

The God Factor

Reba McEntire on
Make a Difference Day:

"Kids who may never get out of their town will be able to see the world through books. But I'm talking about my passion. What's yours?"

"There is the God factor…."

— NickyJett, Xanga comment

"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"

Many Dimensions,
by Charles Williams, 1931

For more on this theme
appropriate to Passion Week
Jews playing God — see

The image “http://www.log24.com/log/pix05/050322-Trio.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein
in conversation with
Bob Osserman
of the
Mathematical Sciences Research Institute
at the Commonwealth Club, San Francisco,
Tuesday, March 22.  Wine and cheese
reception at 5:15 PM (San Francisco time).
 
For the meaning of the diamond,
see the previous entry.

Tuesday, April 20, 2004

Tuesday April 20, 2004

Filed under: General — Tags: — m759 @ 3:00 PM

Rhetorical Question

Yesterday's Cartesian theatre continues….

Robert Osserman, a professor emeritus of mathematics at Stanford University, is special-projects director at the Mathematical Sciences Research Institute, in Berkeley, Calif.

Osserman at aldaily.com today:

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales — regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all — into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

Answer:

William Randolph Hearst III.

"Trained as a mathematician at Harvard, he now likes to hang out with Ken Ribet and the other gurus at the University of California, Berkeley's prestigious Mathematical Sciences Research Institute. Two years ago, he moderated a panel of math professors discussing Princeton professor Andrew Wiles's historic proof of Fermat's Last Theorem."

—   Wired magazine, June 1995

See also

Hearst Gift Spurs Math Center Expansion and

Review of Rational Points on Elliptic Curves by Joseph H. Silverman and John T. Tate (pdf), Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 248–252,

by William Randolph Hearst III
and Kenneth A. Ribet.

 

Chet Atkins summarizes:

"And that's the secret of frog kissin', and you can do it too if you'll just listen.

Just slow down, turn around, bend down and kiss you a frog! Ribet! Ribet!"
 

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