The title refers to a 1514 engraving.

See also Angel Particle in *this* journal.

*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.

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 M_{24 }with the aid of the MOG.

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.

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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.”

The former practice yields reflections like those of Conway and Sloane.

The latter practice is not recommended.

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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 16_{6 }configuration."

As noted previously in this journal,

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

— T. S. Eliot in *Four Quartets*

See also earlier posts also tagged "Kummerhenge" and

another property of the remarkable Kummer 16_{6} —

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

Some background from 2001 —

Comments Off on Kummerhenge Continues.

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" —

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."

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(Continued from this morning)

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

— T. S. Eliot in *Four Quartets*

See also other Log24 posts tagged Kummerhenge.

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

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"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*

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"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 —

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

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“… 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"

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"… 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.

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A passage that may or may not have influenced Madeleine L'Engle's

writings about the *tesseract* :

From
"Book IV – Beyond Personality: 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 …

"Galois was a thoroughly obnoxious nerd,

suffering from what today would be called

a 'personality disorder.' His anger was

paranoid and unremitting."

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A clue to the relationship between the Kummer (16, 6)

configuration and the large Mathieu group M_{24} —

Related material —

See too the diamond-theorem correlation.

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**Backstory for fiction fans, from Log24 on June 11 —**

**Related non -fiction —**

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

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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 —

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"… lo lidchok et haketz …."

— Acceptance speech, Guardian of Zion award, 2002

Also on February 20, 2012 —

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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.

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"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.

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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 —

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"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 16 _{6} 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."**

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"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 —

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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 —

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

See as well . . .

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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 M_{24 }"

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.

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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 16

**[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 16_{6} configuration . . . .

*this* journal, for instance —

See as well yesterday morning's post.

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"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

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The Dream of the Expanded Field continues**…**

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 —

Two such considerations —

**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 E_{7}, *W *(E_{7}):

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 *.

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The *Kummer 16 _{6} 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 16_{6} to a number

of other mathematical concepts — "geometric incarnation."

**Related material from finitegeometry.org —**

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

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"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 16 _{6} Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)**

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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