Tuesday, September 25, 2018


Filed under: General — Tags: , — m759 @ 10:10 AM

See some posts related to three names
associated with Trinity College, Cambridge —

Atiyah + Shaw + Eddington .

Saturday, November 23, 2013

Frame Tale (continued)

Filed under: General — m759 @ 10:30 AM

See The X-Men Tree,  another tree,  and Trinity MOG.

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




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


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.

Saturday, September 1, 2018

Ron Shaw — D. 21 June 2016

The date of Ron Shaw's 2016 death appears to be June 21:


All other Internet sources I have seen omit the June 21 date.

This  journal on that date —


Thursday, June 21, 2018

Dirac and Geometry (continued)

Filed under: General,Geometry — Tags: , — m759 @ 10:04 AM

"Just fancy a scale model of Being 
made out of string and cardboard."

Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington

"… a treatise on Kummer's quartic surface."

The "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.

See Hudson + Kummer in this  journal.

See also posts tagged Dirac and Geometry.

Saturday, February 17, 2018

The Binary Revolution

Filed under: General,Geometry — Tags: , — m759 @ 5:00 PM

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • "The digital revolution based on the 2 symbols (0,1)"
  • "The algebra of George Boole"
  • "Binary codes"
  • "Dirac's spinors, with their up/down dichotomy"

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today's earlier post God's Dice and Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

God’s Dice

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

On a Trinity classmate of Ian Macdonald (see previous post)—

Atiyah's eulogy of Shaw in Trinity Annual Record 2017 
is on pages 137 through 146.  The conclusion —


Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 10:29 PM

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —


From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Monday, June 26, 2017

Upgrading to Six

This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.

Related material (click to enlarge) —

Detail —

From the work of Eddington cited in 1974 by von Franz —

See also Dirac and Geometry and Kummer in this journal.

Updates from the morning of June 27 —

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Friday, March 4, 2016

Cube Bricks 1984

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

An Approach to Symmetric Generation of the Simple Group of Order 168

Related aesthetics —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Monday, November 23, 2015

Dirac and Line Geometry

Filed under: General,Geometry — Tags: , , — m759 @ 2:29 AM

Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.

For a biography of Bruins translated by Google, click here.

For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 39-52 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992
 . (Springer, 2012,
originally published by Kluwer in 1993.)

For more-recent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post

Monday, November 24, 2008

Monday November 24, 2008

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

'Brick' octads in the Miracle Octad Generator (MOG) of R. T. Curtis

Click on image for details.

Tuesday, February 27, 2007

Tuesday February 27, 2007

Filed under: General,Geometry — m759 @ 7:59 AM
Continued from 2/06:

The Poetics of Space

Log24 yesterday:

+John Cardinal Farley,
Archbishop of New York”

Tom Hanks as Robert Langdon in The Da Vinci Code

Tom Hanks as Robert Langdon
in “The Da Vinci Code”

“… and by ‘+’ I mean
artistic vision.”

New York State Lottery
yesterday, Feb. 26, 2007:

Mid-day 206
Evening 888

For more on the artistic
significance of 206,
see 2/06.

For more on the artistic
significance of 888, see
St. Bonaventure on the
Trinity at math16.com.

A trinity:

Click on picture for further details.

Friday, May 6, 2005

Friday May 6, 2005

Filed under: General — Tags: , — m759 @ 2:56 PM

Trinity symbol
(See Sequel.)

The image “http://www.log24.com/theory/images/KleinDualInsideOut200.gif” cannot be displayed, because it contains errors.
Trinity symbol
by Greg Egan
(via John Baez)



"Difficult to understand because of intricacy: byzantine, complex, complicated, convoluted, daedal, Daedalian, elaborate, intricate, involute, knotty, labyrinthine, tangled."

— Roget's II: The New Thesaurus, Third Edition

See also the previous three entries,
as well as Symmetries.

Sunday, November 21, 2004

Sunday November 21, 2004

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

Trinity and Counterpoint

Today's Roman Catholic meditation is from Gerry Adams, leader of Sinn Fein, the political arm of the Irish Republican Army:

"I certainly regret what happened and I make no bones about that," Adams said on the 30th anniversary of pub bombings that killed 21 on Nov. 21, 1974, in Birmingham, England.

Those who care what Roman Catholics think of the Trinity may read the remarks of St. Bonaventure at math16.com.

That site also offers a less holy but more intelligible trinity based on the irrefutable fact that 3 x 8 = 24 and on a remarkable counterpoint between group actions on a 4×2 array and group actions on a 4×4 array.

For a Protestant view of this trinity, see a website at the University of Birmingham in England.

That site's home page links to Birmingham's City Evangelical Church.

Friday, April 30, 2004

Friday April 30, 2004

Filed under: General — Tags: — m759 @ 6:24 AM


These are the folios of April,
All the library of spring

Christopher Morley

The above quotation is dedicated to Quay A. McCune, M.D., whose copy of Bartlett's Familiar Quotations I purchased for two dollars at a Friends of the Library sale on July 2, 1999.  Dr. McCune's copy of Bartlett was the twelfth edition, of November 1948, in a February 1952 reprint.  It was edited by Christopher Morley.

Incidentally, Morley's father Frank, a professor of mathematics, is the discoverer of Morley's theorem, which says that the angle trisectors of any triangle, of whatever shape, determine an equilateral "Morley triangle" hidden within the original triangle.


Those familiar with Dorothy Sayers's explication of the Trinity, The Mind of the Maker, will recognize that this figure represents a triumph over the heresies she so skillfully describes in the chapter "Scalene Trinities."  From another chapter:

"… this is the Idea that is put forward for our response. There is nothing mythological about Christian Trinitarian doctrine: it is analogical. It offers itself freely for meditation and discussion; but it is desirable that we should avoid the bewildered frame of mind of the apocryphal Japanese gentleman who complained:

'Honourable Father, very good;
 Honourable Son, very good; but
 Honourable Bird
     I do not understand at all.'

'Honourable Bird,' however, has certain advantages as a pictorial symbol, since, besides reminding us of those realities which it does symbolise, it also reminds us that the whole picture is a symbol and no more."

In the Morley family trinity, if Frank is the Father and Christopher is the Son, we must conclude that the Holy Spirit is Christopher's mother — whose maiden name was, appropriately, Bird.

Sunday, June 15, 2003

Sunday June 15, 2003

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

Readings for Trinity Sunday

  1. Triune knot:
    Problems in Combinatorial Group Theory, 7 and 8, in light of the remark in Section 8.3 of Lattice Polygons and the Number 12 
  2. Cardinal Newman:
    Sermon 24
  3. Simon Nickerson:

For more on the structure
discussed by Nickerson, see

Raiders of the Lost Matrix:

For theology in general, see

Jews Telling Stories.

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