Tuesday, November 25, 2014

Euclidean-Galois Interplay

Filed under: Uncategorized — Tags: — m759 @ 11:00 AM

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

Oxley's 2004 drawing of the 13-point projective plane

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).


   The 3×3×3 Galois Cube 

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points  be naturally pictured as lines  within the 
5x5x5 Galois cube (vector 3-space over GF(5))?

Monday, November 24, 2014

“What Reality?”

Filed under: Uncategorized — m759 @ 10:00 PM

Creating new realities in DeLillo's 'Point Omega'

"We tried to create new realities overnight,
careful sets of words that resemble advertising slogans
in memorability and repeatability."

Our Most Important Product


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

— Burt Lancaster in Sam Peckinpah's last film

Homemade Aesthetics

Filed under: Uncategorized — m759 @ 12:48 PM

Continued from Tuesday, November 18, 2014

International poster for the 2010 film "Black Swan"—

"Death is a black swan." — Graciela Chichilnisky

Metaphysician in the Dark

Filed under: Uncategorized — m759 @ 1:00 AM

Continued from Friday, November 21:

Sunday, November 23, 2014

Remarks on Reality

Filed under: Uncategorized — m759 @ 5:09 PM

Wallace Stevens in "An Ordinary Evening in New Haven"
(1950) on "The Ruler of Reality" —

"Again, 'He has thought it out, he thinks it out,
As he has been and is and, with the Queen
Of Fact, lies at his ease beside the sea.'"

One such scene, from 1953 —

Another perspective, from "The Osterman Weekend" (1983) —


Filed under: Uncategorized — m759 @ 6:00 AM

Corrections to the NY Times  obituary of Alexander Grothendieck
are shown below. For the original Sunday, Nov. 16, NY Times  
print obituary (with its online date, Nov. 14), see a copy taken
from a weblog.

For another poetic remark in memory of Grothendieck,
see a Log24 post from November 13, the day of his death. 

Saturday, November 22, 2014

About Nothing

Filed under: Uncategorized — m759 @ 8:00 AM

See the easy grace link in this morning's previous post.
See also Michael Dubruiel and Log24 posts of Feb. 3
and Feb. 9, 2009.

The November 22 Candidate

Filed under: Uncategorized — m759 @ 5:24 AM

Welcome to the Garden Club, Pilgrim


"A journalist with a literary bent, Mr. Eder wrote with an easy grace
and a practiced eye for detail. In 1974, he assessed a cultural
malaise in England during an economic downturn.

'In the West Country town of Hereford,' he began, 'the president
of a women’s club told a year-end meeting that the January bingo
game would be canceled to save electricity. Then she proposed a
New Year’s resolution. "Let us all work to get England back on her
dear old feet," she said and bumped down pinkly into her chair,
overwhelmed by applause.'" — Bruce Weber, NY Times

See also Bingo  in this journal.

Friday, November 21, 2014


Filed under: Uncategorized — m759 @ 9:00 AM

"When Three Into One Equals MoreNew York Times  headline 

See also Trinity in this journal.  From that search:

                     … The actor is
A metaphysician in the dark….

— Wallace Stevens, 
   "Of Modern Poetry

Thursday, November 20, 2014

The Unit

Filed under: Uncategorized — m759 @ 12:12 PM

In memory of Mike Nichols

"Let me put you in this unit." — John Calley

Graduate Director

Filed under: Uncategorized — m759 @ 11:00 AM

The Manchurian Candidate

Filed under: Uncategorized — m759 @ 12:59 AM

Continued from Tuesday, Nov. 18

The above conclusion of a NY Times obituary
about a Monday death may also serve as the
missing conclusion of Monday's "A Search for
Missing Pieces
" —

Related material starring Einstein and
Thomas Mann: "A Riddle for Davos."

Wednesday, November 19, 2014

The Eye/Mind Conflict

Filed under: Uncategorized — m759 @ 10:25 AM

Harold Rosenberg, "Art and Words," 
The New Yorker , March 29, 1969. From page 110:

"An advanced painting of this century inevitably gives rise
in the spectator to a conflict between his eye and his mind; 
as Thomas Hess has pointed out, the fable of the emperor's 
new clothes is echoed at the birth of every modemist art 
movement. If work in a new mode is to be accepted, the 
eye/mind conflict must be resolved in favor of the mind; 
that is, of the language absorbed into the work. Of itself, 
the eye is incapable of breaking into the intellectual system 
that today distinguishes between objects that are art and 
those that are not. Given its primitive function of 
discriminating among things in shopping centers and on 
highways, the eye will recognize a Noland as a fabric
design, a Judd as a stack of metal bins— until the eye's 
outrageous philistinism has been subdued by the drone of 
formulas concerning breakthroughs in color, space, and 
even optical perception (this, too, unseen by the eye, of 
course). It is scarcely an exaggeration to say that paintings 
are today apprehended with the ears. Miss Barbara Rose, 
once a promoter of striped canvases and aluminum boxes, 
confesses that words are essential to the art she favored 
when she writes, 'Although the logic of minimal art gained 
critical respect, if not admiration, its reductiveness allowed
for a relatively limited art experience.' Recent art criticism 
has reversed earlier procedures: instead of deriving principles 
from what it sees, it teaches the eye to 'see' principles; the 
writings of one of America's influential critics often pivot on 
the drama of how he failed to respond to a painting or 
sculpture the first few times he saw it but, returning to the 
work, penetrated the concept that made it significant and
was then able to appreciate it. To qualify as a member of the 
art public, an individual must be tuned to the appropriate 
verbal reverberations of objects in art galleries, and his 
receptive mechanism must be constantly adjusted to oscillate 
to new vocabularies."

New vocabulary illustrated:

Graphic Design and a Symplectic Polarity —

Background: The diamond theorem
and a zero system .

Tuesday, November 18, 2014

For the Green Mountain Girls

Filed under: Uncategorized — Tags: — m759 @ 9:13 AM

Tom Wolfe in The Painted Word  (1975):

"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde  understood, they
were not only theories but … hot news,
straight from the studios, from the scene."

"Parable of American Painting," 1954 — From The Tradition of the New by Harold Rosenberg

"In this essay Rosenberg set out to explain what he believed to be definitively American about Abstract Expressionism. He did so by drawing on the American Revolutionary War for his metaphors, likening the new Americans to the coonskin trappers whose knowledge of their terrain enabled them to pick off the British soldiers (Redcoats), who followed the dictates of their military training. The professionally- trained soldiers were defeated because, as Rosenberg states, 'They were such extreme European professionals … they did not even see  the American trees.' 'Redcoatism' was, Rosenberg argued, a symptom of the old European world's stubborn rejection of the new. It did at one time also '[dominate] the history of American art,' he wrote, but with the advent of Abstract Expressionism, times had changed. And just as the Coonskins were victorious because they stood apart from the professional military, so the new American art was triumphant because, as Rosenberg saw it, it marked a profound break with the traditions of European art."

— TheArtStory.org

Lectures at Bennington, 1971

For example:

Art adapted today from the Google search screen. Discuss.

The Abacus Conundrum…

Filed under: Uncategorized — Tags: — m759 @ 1:44 AM



Prequel from 1961 (click image for context):

Detail that may be interpreted as the Chinese
3×3 "Holy Field" and a Chinese temple bell—

"Ting-a-ling." — Kurt Vonnegut.

Our Most Important Product

Filed under: Uncategorized — Tags: — m759 @ 12:45 AM

Hexagram 35:

"Then came a 'Robot Psychologist,' known as a Psychological Matrix Rotator,
developed for the Department of Defense. It is still used to literally 'see' that
the right man gets the right Army job."

Ronald Reagan, 1961 GE Sales Meeting

"Always with a little humor." — Yen Lo

In memory of Dr. Irving Peress,
who reportedly died on Thursday,
November 13, 2014.

Monday, November 17, 2014

A Search for Missing Pieces

Filed under: Uncategorized — m759 @ 9:00 AM

A detail from the image search below —

A Google image search today for
“portal del aguila de oro” “bella vista” —

Click for a larger (4.2 MB) version.

See also yesterday’s post.

Sunday, November 16, 2014

Along the Way

Filed under: Uncategorized — m759 @ 5:20 AM

“Bit by bit, putting it together.
Piece by piece, working out the vision night and day.
All it takes is time and perseverance
With a little luck along the way.”
— Stephen Sondheim

See also, in this journal,  528,  1963,  522, and  3273.

Saturday, November 15, 2014

A Dark Detective Story

Filed under: Uncategorized — m759 @ 7:20 PM

Alexander Grothendieck, Récoltes et Semailles , e,  p. 1181:

Pour mettre la joie à son comble, j’ajoute que le dénommé Saavedra
semble avoir disparu de la circulation sans plus laisser aucune trace….
Du coup, l’histoire prend des allures de sombre intrigue policière.

Google Translate version:

To the joy at its height, I would add that the so-called Saavedra
seems to have disappeared from circulation without leaving any trace….
Suddenly, the story looks like a dark detective story.

Or horror film

Enigma Variations

Filed under: Uncategorized — m759 @ 7:59 AM


A Math Enigma

Grothendieck reportedly died on Thursday, November 13, 2014.
From this  journal a year earlier:

A German Enigma 

After clicking on the link “a grave intellectual whole” from the Nov. 13, 2013, post
see also Benedict Cumberbatch’s less serious approach to Bletchley Park:

“I want to see this film; this film’s been up my ass for the last five years.”

Friday, November 14, 2014

Another Opening, Another Show

Filed under: Uncategorized — m759 @ 9:00 PM

“What happens when you mix the brilliant wit of Noel Coward
with the intricate plotting of Agatha Christie? Set during a
weekend in an English country manor in 1932, Death by Design
is a delightful and mysterious ‘mash-up’ of two of the greatest
English writers of all time. Edward Bennett, a playwright, and
his wife Sorel Bennett, an actress, flee London and head to
Cookham after a disastrous opening night. But various guests
arrive unexpectedly….”

Samuel French (theatrical publisher) on a play that
opened in Houston on September 9, 2011.

Related material:

Some Symplectic History

Filed under: Uncategorized — m759 @ 1:28 AM

A paper from 1976 on symplectic torsors  and finite geometry:

IMAGE- Saavedra-Rivano, 'Finite Geometries in the Theory of Theta Characteristics' (1976)

A pdf is now available:

Autor(en): Rivano, Neantro Saavedra
Objekttyp: Article
Zeitschrift: L’Enseignement Mathématique
Band (Jahr): 22 (1976)
PDF erstellt am: 14.11.2014
Persistenter Link: http://dx.doi.org/10.5169/seals-48185

(Received by the journal on February 20, 1976.)

Saavedra-Rivano was a student of Grothendieck, who reportedly died yesterday.

Thursday, November 13, 2014

Mort de Grothendieck

Filed under: Uncategorized — m759 @ 5:55 PM

Alexandre Grothendieck est mort jeudi matin
à l’hôpital de Saint-Girons (Ariège), à l’âge de 86 ans.”

Update of 6: 16 PM ET:  A memorial of sorts, from May 27 this year:

IMAGE- Massimo Vignelli, his wife Lella, and cube

Progressive Matrix

Filed under: Uncategorized — Tags: , — m759 @ 6:00 AM

Yesterday's post and recent Hollywood news suggest
a meditation on a Progressive Matrix —

Oct. 12-14, 2005:

'A Poem for Pinter,' conclusion: 'Tick Tick Hash.'

'The Interpreter'-- Sean Penn to Nicole Kidman-- 'My Card.'

Click to enlarge.

"My card."

Structurally related images —

A sample Raven's Progressive Matrices  test item
(such items share the 3×3 structure of the hash symbol above):

IMAGE- Raven's Progressive Matrices item with symbols from Cullinane's box-style I Ching

Structural background —

Wednesday, November 12, 2014

Harrowing Cuteness

Filed under: Uncategorized — m759 @ 10:22 AM

The title is a phrase from yesterday’s post.

An example of harrowing cuteness:

Charlize Theron in “Young Adult” (2011) —

Related material for older adults: Ravenna and Nietzsche.

Tuesday, November 11, 2014

Vonnegut’s Birthday

Filed under: Uncategorized — m759 @ 11:00 AM

Monday, November 10, 2014

Meanwhile, Back in 1962…

Filed under: Uncategorized — Tags: — m759 @ 11:59 PM

Remember him to Herald Square.

Narrative Line

Filed under: Uncategorized — Tags: — m759 @ 11:02 PM

We live entirely, especially if we are writers, by the imposition
of a narrative line upon 
disparate images….” — Joan Didion

Narrative Line:

IMAGE- R. D. Carmichael's 1931 construction of the Steiner system S(5, 8, 24)

IMAGE- Harvard senior Jeremy Booher in 2010 discusses Carmichael's 1931 construction of S(5, 8, 24) without mentioning Carmichael.

Disparate images:


Can the above narrative line be imposed in any sensible way
upon the above disparate images?

Sunday, November 9, 2014


Filed under: Uncategorized — Tags: , , , — m759 @ 11:00 AM

The Ideas

“We tell ourselves stories in order to live….
We interpret what we see, select the most workable
of multiple choices. We live entirely, especially if we
are writers, by the imposition of a narrative line upon
disparate images, by the ‘ideas’  with which we have
learned to freeze the shifting phantasmagoria
which is our actual experience.”
— Joan Didion

See Didion and the I Ching  and posts tagged Plato in China .


Filed under: Uncategorized — Tags: — m759 @ 1:00 AM

“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”

— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121

Cf. Raiders of the Lost Theorem, Oct. 13, 2014.

See also tonight’s previous post and
“Feng Shui” in this journal.

Saturday, November 8, 2014

At 11:59*

Filed under: Uncategorized — m759 @ 11:59 PM

Fantasy and the Buffered Self.”

*For the title, see Enormous Changes.

Friday, November 7, 2014

The Crosswicks Curse…

Filed under: Uncategorized — m759 @ 7:00 AM


There is  such a thing as an MBTI Tesseract.

See a thread at http://www.typologycentral.com/forums/
from August 17 and 18, 2010.

See also this journal on those dates: The Kermode Game.

Thursday, November 6, 2014

Working Backward

Filed under: Uncategorized — m759 @ 1:44 PM


This setting of the Ave Verum Corpus  text was composed
to celebrate the feast of Corpus Christi .” — Wikipedia

Ave Verum Corpus .”— Madison in the BBC America TV series
“Intruders,” Season 1, Episode 3: “Time Has Come Today.”

See also the Eucharistic meditation of Feb. 13, 2006, linked to in yesterday’s post
on Guy Fawkes Day. (That British holiday originally commemorated the Catholic
Gunpowder Plot of 1605.)

Wednesday, November 5, 2014

Dark Fields…

Filed under: Uncategorized — m759 @ 1:06 AM


From the first of previous Log24 posts tagged “Dark Fields”—

“A link in memory of Donald G. Higman,
dead on Feb. 13, 2006,
the day after Lincoln’s birthday:

On the Graphs of Hoffman-Singleton and Higman-Sims.

His truth is marching on.”

See also Foundation Square (October 25, 2014).

Monday, November 3, 2014

Plan 9 Continues…

Filed under: Uncategorized — m759 @ 10:00 PM

A link to “Nine Tailors” in this journal may serve as
a memorial to the late David M. Abshire, who
reportedly died at 88 on Halloween.

See also tonight’s previous post and a remark by
Mira Sorvino in her version of The Great Gatsby .

Wisconsin Death Trip*

Filed under: Uncategorized — m759 @ 8:59 PM

Courtesy of Mira Sorvino.

Enter Madison :

From “Intruders,” BBC America, Season 1, Episode 2, at 1:07 of 43:31.

You sure know how to show a girl a good time.

* The title is a reference to a Wisconsin-related Halloween post.

Gods and Giants

Filed under: Uncategorized — m759 @ 2:01 PM

A weblog reports Chris Rock’s remarks
on Saturday Night Live this past weekend:

“It’s America, we commercialize everything.
Look at what we did to Christmas.
Christmas.  Christmas is Jesus’ birthday.
It’s Jesus’ birthday.  Now, I don’t know Jesus
but from what I’ve read, Jesus is the least
materialistic person to ever roam the earth.
No bling on Jesus.
Jesus kept a low profile and we turned his
birthday into the most materialistic day of the
year.  Matter of fact, we have the Jesus birthday
season.  It’s a whole season of materialism.
Then, at the end of the Jesus birthday season
we have the nerve to have an economist come
on TV and tell you how horrible the Jesus birthday
season was this year.  Oh, we had a horrible Jesus’
birthday this year.  Hopefully, business will pick up
by his Crucifixion.”

Related music and image:

Show us the way to the next little girl …”

Natalie Wood in “Miracle on 34th Street” (1947)

Related non-materialistic meditations:
The Rhetoric of Abstract Concepts and Gods and Giants.

The Rhetoric of Abstract Concepts

Filed under: Uncategorized — m759 @ 12:48 PM

From a post of June 3, 2013:

New Yorker  editor David Remnick at Princeton today
(from a copy of his prepared remarks):

“Finally, speaking of fabric design….”

I prefer Tom and Harold:

Tom Wolfe in The Painted Word 

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

Harold Rosenberg in The New Yorker  (click to enlarge)

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

Symplectic :

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

— Steven H. Cullinane,
diamond theorem illustration

Sunday, November 2, 2014

A Singular Place

Filed under: Uncategorized — m759 @ 5:09 PM

“Macy’s Herald Square occupies a singular place
in American retailing.” — NY Times  today, in print
on page BU1 of the New York edition with the headline:

Makeover on 34th Street .

A Singular Time:

See Remember Me to Herald Square, at noon on
August 21, 2014, and related earlier Log24 posts.

Also on Aug. 21, 2014: from a blog post, ‘Tiles,’ by
Theo Wright, a British textile designer —

The 24 tile patterns displayed by Wright may be viewed
in their proper mathematical context at …


IMAGE - The Diamond Theorem

Right Through Hell

Filed under: Uncategorized — m759 @ 8:28 AM

… there is a path. — Malcolm Lowry

Saturday, November 1, 2014


Filed under: Uncategorized — m759 @ 10:00 PM

Related posts:
New Key and The Well-Tempered Monolith.

Hold the gin.

Not Saints

Filed under: Uncategorized — m759 @ 1:00 PM

“We are not saints.” — Alcoholics Anonymous , Chapter 5

The New York Times  on AA’s co-founder Bill Wilson in 1934:

Click for the rest of the story.

All Saints’ Day

Filed under: Uncategorized — m759 @ 12:06 PM

Friday, October 31, 2014

For the Late Hans Schneider

Filed under: Uncategorized — m759 @ 3:30 AM

See a University of Wisconsin obituary for Schneider,
a leading expert on linear algebra who reportedly died
at 87 on Tuesday, October 28, 2014.

Some background on linear algebra and “magic” squares:
tonight’s 3 AM (ET) post and a search in this
journal for Knight, Death, and the Devil.

Click image to enlarge.


Filed under: Uncategorized — m759 @ 3:00 AM

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

IMAGE- Introduction to 322,560 Affine Transformations of Dürer's 'Magic' Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

Thursday, October 30, 2014


Filed under: Uncategorized — m759 @ 5:09 PM

This journal Tuesday,  Oct. 28, 2014, at 5 PM ET:

“What is a tai chi master, and what is it that he unfolds?”

From an earlier post, Hamlet’s father’s ghost
on “the fretful porpentine”:

Hamlet , Act 1, Scene 5 —


“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combinèd locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood.”

Galway Kinnell:

“I roll
this way and that in the great bed, under
the quilt
that mimics this country of broken farms and woods”

— “The Porcupine”

For quilt-block designs that do not mimic farms or woods,
see the cover of Diamond Theory .  See also the quotations
from Wallace Stevens linked to in the last line of yesterday’s
post in memory of Kinnell.

“… a bee for the remembering of happiness” — Wallace Stevens

Wednesday, October 29, 2014

Dead Poet

Filed under: Uncategorized — m759 @ 12:00 PM

For poet Galway Kinnell, Princeton ’48:

Kinnell was named “Tiger of the Week” in a
Princeton Alumni Weekly  post of August 27, 2014.

See his obituary in today’s New York Times
as well as posts here  on August 27, 2014.

Symbology for Harvard

Filed under: Uncategorized — m759 @ 12:00 AM

See also “Satan’s School.”

Tuesday, October 28, 2014

Raiders of the Lost Symbol

Filed under: Uncategorized — m759 @ 5:00 PM

A print copy of next Sunday’s New York Times Book Review
arrived in today’s mail. From the front-page review:

Marcel Theroux on The Book of Strange New Things ,
a novel by Michel Faber —

“… taking a standard science fiction premise and
unfolding it with the patience and focus of a
tai chi master, until it reveals unexpected
connections, ironies and emotions.”

What is a tai chi master, and what is it that he unfolds?

Perhaps the taijitu  symbol and related material will help.

The Origin of Change

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

“Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come.”

Wallace Stevens,
“Notes Toward a Supreme Fiction,”
Canto IV of “It Must Change”

Go Figure

Filed under: Uncategorized — m759 @ 4:00 PM

For Karl Pribram and Katherine Neville,
a sequel to this morning’s Figural Processing —

See also Christmas 2013.

Figural Processing

Filed under: Uncategorized — m759 @ 4:22 AM

Part I:

Six-dimensional hypercube from 'Brain and Perception: Holonomy and Structure in Figural Processing,' by Karl H. Pribram

Part II:

Click images for some context.

Monday, October 27, 2014

Revolutions in Geometry

Filed under: Uncategorized — m759 @ 9:00 AM

A post in honor of Évariste Galois (25 October 1811 – 31 May 1832)

From a book by Richard J. Trudeau titled The Non-Euclidean Revolution

See also “non-Euclidean” in this journal.

One might argue that Galois geometry, a field ignored by Trudeau,
is also “non-Euclidean,” and  (for those who like rhetoric) revolutionary.

Sunday, October 26, 2014


Filed under: Uncategorized — m759 @ 12:00 PM

The “Chern” of today’s previous post is mathematician
Shiing-Shen Chern (b. Oct. 26, 1911, d. Dec. 3, 2004).

For an observance of the 2011 centennial of his birth,
see a website in China.

See also this journal on the centennial date —
Erlanger and Galois, a post of Oct. 26, 2011.

For Chern’s Birthday

Filed under: Uncategorized — m759 @ 11:00 AM

The Poem of Pure Reality

Saturday, October 25, 2014

Foundation Square

Filed under: Uncategorized — m759 @ 2:56 PM

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner’s abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Friday, October 24, 2014

New Key

Filed under: Uncategorized — m759 @ 9:00 PM

See Langer (Harvard U. Press, Third Edition, Jan. 31, 1957, pp. 3-4-5).

See also Old Key : Pythagoras, harmony, and the 3-4-5 triangle.

Thursday, October 23, 2014

Prime Cut

Filed under: Uncategorized — m759 @ 7:24 PM

See that phrase in this journal.

Wednesday, October 22, 2014

The Seventh Stage

Filed under: Uncategorized — m759 @ 7:01 PM

Robin Williams at Bunker Hill Community College

Robin Williams and the Stages of Math

i)   shock & denial
ii)  anger
iii) bargaining
iv) depression
v)  acceptance

And then…

vi)  checking
vii) Joan Rivers

See also

Claves Regni Caelorum

Filed under: Uncategorized — m759 @ 10:00 AM

Continued from Day at the Museum, last Sunday, October 19, 2014.

This post was suggested by…

  1.  A piece in the Bookends section of the New York Times
    Sunday Book Review
      (page BR31 last Sunday, Oct. 19):
    Daniel Mendelsohn on rereading The Catcher in the Rye .
  2. A detail in Day at the Museum— The New York Times ‘s
    appraisal of Joan Rivers: “A Comic Without a Shut-Off Switch.”
  3. A Sept. 7 Log24 post, Sunday School, in memory of Joan Rivers.

From The Catcher in the Rye , a passage just before the
museum passage quoted by Mendelsohn:

“She was having a helluva time tightening her skate.
She didn’t have any gloves on or anything and her hands
were all red and cold. I gave her a hand with it. Boy, I
hadn’t had a skate key in my hand for years. It didn’t feel
funny, though. You could put a skate key in my hand
fifty years from now, in pitch dark, and I’d still know
what it is. She thanked me and all when I had it tightened
for her. She was a very nice, polite little kid. God, I love it
when a kid’s nice and polite when you tighten their skate
for them or something. Most kids are. They really are.
I asked her if she’d care to have a hot chocolate or something
with me, but she said no, thank you. She said she had to meet
her friend. Kids always have to meet their friend. That kills me.

Even though it was Sunday and Phoebe wouldn’t be there
with her class or anything, and even though it was so damp
and lousy out, I walked all the way through the park over to
the Museum of Natural History. I knew that was the museum
the kid with the skate key meant.”

Tuesday, October 21, 2014

Eerie Twist

Filed under: Uncategorized — m759 @ 9:00 PM

(Continued from Nov. 15, 2011)

Ben Bradlee, legendary Washington Post editor, dies at 93

See also a post of Jan. 20, 2011, and an earlier post on Twelfth Night, 2010.



A star figure and the Galois quaternion.

The square root of the former is the latter.

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