Log24

Saturday, February 16, 2019

Melancholy for Dürer

Filed under: General — Tags: — m759 @ 11:53 AM

The title refers to a 1514 engraving.

See also Angel Particle in this  journal.

Tuesday, February 12, 2019

A Long Time

Filed under: General — Tags: , — m759 @ 2:01 PM

This  journal on the above date, October 17, 2008

“Every musician wants to do something of lasting quality,
something which will hold up for a long time, and
I guess we did it with ‘Stairway.'”

— Jimmy Page on “Stairway to Heaven

Scholium —

"Kummer " in German means "sorrow."

Related material —

Other posts now tagged Dolmen.

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: , — m759 @ 12:00 AM

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Saturday, January 26, 2019

Installasjon

Filed under: General — Tags: , — m759 @ 5:13 AM

'Josefine Lyche,' 'Interlock, Interlac, Interweave'

The above cryptic search result indicates that there may
soon be a new Norwegian art installation based on this page
of Eddington (via Log24) —

See also other posts tagged Kummerhenge.

Friday, December 14, 2018

Small Space Odyssey

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:00 AM

References in recent posts to physical space and 
to mathematical space suggest a comparison.

Physical space is well known, at least in the world
of mass entertainment.

Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.

A figure from each space —

The source of the Conway-Sloane brick —

Quote from a mathematics writer —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.

Wednesday, December 12, 2018

Kummerhenge Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 7:24 PM

Those pleased by what Ross Douthat today called
"The Return of Paganism" are free to devise rituals
involving what might be called "the sacred geometry
of the Kummer 166  configuration."

As noted previously in this journal, 

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also earlier posts also tagged "Kummerhenge" and 
another property of the remarkable Kummer 166 

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

For some related literary remarks, see "Transposed" in  this journal.

Some background from 2001 —

An Inscape for Douthat

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:41 AM

Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

In finite geometry and combinatorics,
an inscape  is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:


 

Related material — the phrase
"Quantum Tesseract Theorem" and  

A.  An image from the recent
      film "A Wrinkle in Time" — 

B.  A quote from the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

Friday, December 7, 2018

The Angel Particle

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 7:15 PM

(Continued from this morning)

Majorana spinors and fermions at ncatlab

The Gibbons paper on the geometry of Majorana spinors and the Kummer configuration

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also other Log24 posts tagged Kummerhenge.

Thursday, November 22, 2018

Rosenhain and Göpel Meet Kummer in Projective 3-Space

Filed under: General,Geometry — Tags: — m759 @ 2:07 PM

For further details, see finitegeometry.org/sc/35/hudson.html.

Geometric Incarnation

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Note also the four 4×4 arrays surrounding the central diamond
in the chi  of the chi-rho  page of the Book of Kells

From a Log24 post
of March 17, 2012

"Interlocking, interlacing, interweaving"

— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science

Sunday, November 18, 2018

Space Music

Filed under: General,Geometry — Tags: , — m759 @ 9:27 AM

'The Eddington Song,' based on 'The Philosophy of Physical Science,' p. 141 (1939)

Update of Nov. 19 —

"Design is how it works." — Steve Jobs

See also www.cullinane.design.

Monday, July 16, 2018

Greatly Exaggerated Report

Filed under: General,Geometry — Tags: — m759 @ 1:21 PM

"The novel has a parallel narrative that eventually
converges with the main story."

— Wikipedia on a book by Foer's novelist brother
 

Public Squares

An image from the online New York Times 
on the date, July 6,
of  the above Atlantic  article —

An image from "Blackboard Jungle," 1955 —

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

"Through the unknown, remembered gate . . . ."

— T. S. Eliot, Four Quartets

Thursday, July 12, 2018

Kummerhenge Illustrated

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

      

“… the utterly real thing in writing is the only thing that counts…."

— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935

"Omega is as real  as we need it to be."

— Burt Lancaster in "The Osterman Weekend"

Friday, July 6, 2018

Something

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

"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."

— T. S. Eliot, "Burnt Norton," 1936

"Read something that means something."

Advertising slogan for The New Yorker

The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —

"… we have sixteen elements with which to form a group-structure" —

See as well posts tagged Dirac and Geometry.

Wednesday, June 27, 2018

Taken In

Filed under: General,Geometry — Tags: — m759 @ 9:36 AM

A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :

From Mere Christianity , by C. S. Lewis (1952) —

"Book IV – Beyond Personality:
or First Steps in the Doctrine of the Trinity"
. . . .

I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully.

You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares.

Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels.

Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine.

In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.

You may ask, "If we cannot imagine a three-personal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like.

. . . .

But beware of being drawn into the personal life of the Happy Family .

https://www.jstor.org/stable/24966339

"The colorful story of this undertaking begins with a bang."

And ends with

Martin Gardner on Galois

"Galois was a thoroughly obnoxious nerd,
 suffering from what today would be called
 a 'personality disorder.'  His anger was
 paranoid and unremitting."

Sunday, June 24, 2018

For 6/24

Filed under: General,Geometry — Tags: — m759 @ 10:12 AM

A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M24

Related material —

See too the diamond-theorem correlation.

Saturday, June 23, 2018

Meanwhile …

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

Backstory for fiction fans, from Log24 on June 11 —

Related non -fiction —

See as well the structure discussed in today's previous post.

Plan 9 from Inner Space

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

From Nanavira Thera, "Early Letters," in Seeking the Path —

"nine  possibilities arising quite naturally" —

Compare and contrast with Hudson's parametrization of the
4×4 square by means of 0 and the 15  2-subsets of a 6-set —

Friday, June 22, 2018

For the Late Charles Krauthammer

Filed under: General — Tags: — m759 @ 2:00 AM

"… lo lidchok et haketz …."

Acceptance speech, Guardian of Zion award, 2002

Eric Voegelin and 'immanentizing the eschaton'

Also on February 20, 2012 —

Thursday, June 21, 2018

Models of Being

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

A Buddhist view —

"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 Christian view —

A formal view —

From a Log24 search for High Concept:

See also Galois Tesseract.

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.

Cavell’s Matrix

Filed under: General — Tags: , — m759 @ 3:00 AM

From an obituary for Stanley Cavell, Harvard philosopher
who reportedly died at 91 on Tuesday,  June 19:

The London Review of Books  weblog yesterday —

"Michael Wood reviewed [Cavell’s] 
Philosophy the Day after Tomorrow  in 2005:

'The ordinary slips away from us. If we ignore it, we lose it.
If we look at it closely, it becomes extraordinary, the way
words or names become strange if we keep staring at them.
The very notion turns into a baffling riddle.' "

See also, in this  journal, Tuesday morning's Ici vient M. Jordan  and
this  morning's previous post.

Update of 3:24 AM from my RSS feed —

Kummerhenge

Filed under: General,Geometry — Tags: , — m759 @ 2:19 AM

See also the Omega Matrix in this  journal.

Tuesday, June 19, 2018

Ici vient M. Jordan

Filed under: General,Geometry — Tags: , — m759 @ 2:13 AM

NY Times correction, online June 16, about 'Here Comes Mr. Jordan' and 'Heaven Can Wait'

See also this  journal on Saturday morning, June 16.

Saturday, June 16, 2018

Kummer’s (16, 6) (on 6/16)

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

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

See too "The Ruler of Reality" in this journal.

Related material —

A more esoteric artifact: The Kummer 166 Configuration . . .

An array of Göpel tetrads appears in the background below.

"As you can see, we've had our eye on you
for some time now, Mr. Anderson."

For June 16

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

"But perhaps the desire for story
is what gets us into trouble to begin with."

Sarah Marshall on June 5, 2018

"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence  is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."

— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018

Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce

Monday, June 11, 2018

Arty Fact

Filed under: General,Geometry — Tags: , — m759 @ 10:35 PM

The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.

The Eye of ARTI —

See also a post of May 19, "Uh-Oh" —

— and a post of June 6, "Geometry for Goyim" — 

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8 

An arty fact I prefer, suggested by the triangular computer-eye forms above —

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

This is from the July 29, 2012, post The Galois Tesseract.

See as well . . .

Monday, September 12, 2016

The Kummer Lattice

Filed under: General,Geometry — Tags: , , , — m759 @ 2:00 PM

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Wednesday, May 25, 2016

Kummer and Dirac

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 AM

From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng  and H. van Dam, 
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239

(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)

" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "

1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is 
related to that of Kummer’s 166 configuration . . . ."

[4]

O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef

E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135

F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished


A remark of my own on the structure of Kummer’s 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

Wednesday, June 10, 2015

Epistemic* Tetrads

Filed under: General,Geometry — Tags: — m759 @ 12:45 PM

"Those that can be obtained…." —

Related music video: Waterloo.

* "In defense of the epistemic view of quantum states:
a toy theory," by Robert W. Spekkens, Perimeter Institute
for Theoretical Physics, Waterloo, Canada 

Friday, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 AM

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

From Klein's 1893 Lectures on Mathematics —

"The varieties introduced by Wirtinger may be called Kummer varieties…."
E. Spanier, 1956

From this journal on March 10, 2013 —

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Update of 10 PM ET March 7, 2014 —

The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Saturday, September 21, 2013

Geometric Incarnation

The  Kummer 166  configuration  is the configuration of sixteen
6-sets within a 4×4 square array of points in which each 6-set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.

See Configurations and Squares.

The Wikipedia article Kummer surface  uses a rather poetic
phrase* to describe the relationship of the 166 to a number
of other mathematical concepts — "geometric incarnation."

Geometric Incarnation in the Galois Tesseract

Related material from finitegeometry.org —

IMAGE- 4x4 Geometry: Rosenhain and Göpel Tetrads and the Kummer Configuration

* Apparently from David Lehavi on March 18, 2007, at Citizendium .

Sunday, June 5, 2011

Edifice Complex

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM

"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."

— Wallace Stevens, "To an Old Philosopher in Rome"

The following edifice may be lacking in grandeur,
and its properties as a configuration  were known long
before I stumbled across a description of it… still…

"What we do may be small, but it has
 a certain character of permanence…."
 — G.H. Hardy, A Mathematician's Apology

The Kummer 166 Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)

IMAGE-- 16_6 configuration from '2-Transitive Symmetric Designs,' by William M. Kantor (AMS Transactions, 1969)

For some background, see Configurations and Squares.

For some quite different geometry of the 4×4 square that  is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do  claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)

Related material— The Schwartz Notes of June 1.

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