Thursday, February 28, 2013

Parts of a World

Filed under: General — m759 @ 10:31 PM

Bruce Reynolds, chief architect of
the Great Train Robbery, who reportedly
died today:

"We all have our benchmarks….
it’s the same madness, I suppose,
that drives people to bivouac on
the north face of the Eiger."

For the Eiger in this journal, see "Parts of a World."

Two-Part Invention

Filed under: General,Geometry — m759 @ 10:01 PM




Filed under: General,Geometry — Tags: , — m759 @ 1:06 PM

A different dodecahedral space (Log24 on Oct. 3, 2011)—

R. T. Curtis, symmetric generation of M12 in a dodecahedron

Wednesday, February 27, 2013


Filed under: General — m759 @ 11:02 PM

(A sequel to yesterday's post Publication )

IMAGE- A Death in Paris on February 26

Meanwhile, in this journal 

Les Miserables  at the Academy Awards


Filed under: General — m759 @ 9:00 PM

News summary in today's previous post:

Van Cliburn Dies. Pope Bids Farewell.


Friday, July 11, 2008, 7:11 PM ET and Plato's Ghost.


Filed under: General — m759 @ 1:06 PM

The News Today:

Van Cliburn Dies

Pope Bids Farewell

Image search from 2011:

Claves  (2 MB)

Tuesday, February 26, 2013


Filed under: General,Geometry — Tags: — m759 @ 4:00 PM

"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and 
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'

'Didn’t publish!'

That cold tenure- joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."

— George Steiner, 2002 talk at York University

Related material

See also Steiner on Galois.

Les Miserables  at the Academy Awards


Filed under: General — m759 @ 2:01 AM

"Hans Castorp is a searcher after the Holy Grail.
You would never have thought it when you read
his story—if I did myself, it was both more and
less than thinking. Perhaps you will read the
book again from this point of view. And perhaps
you will find out what the Grail is: the knowledge
and the wisdom, the consecration, the highest
reward, for which not only the foolish hero but
the book itself is seeking. You will find it in the
chapter called 'Snow'…."

— Thomas Mann, "The Making of
     The Magic Mountain "

In related entertainment news…

Click image for some backstory.

Mann's tale is set in Davos, Switzerland.
See also Mayer  at Davos.

Monday, February 25, 2013

Off the Road

Filed under: General — Tags: — m759 @ 6:00 PM

Starring Snow White! 

IMAGE- Winter storm; stay off roads, authorities say.

IMAGE- Off the road in 'New in Town' (2009 romantic comedy).

See also Thomas Pynchon’s remark in the previous post.

Thursday, February 21, 2013

Galois Space

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


The previous post suggests two sayings:

"There is  such a thing as a Galois space."

— Adapted from Madeleine L'Engle

"For every kind of vampire, there is a kind of cross."

Thomas Pynchon


(Click to enlarge.)

Tuesday, February 19, 2013


Filed under: General,Geometry — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Monday, February 18, 2013


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

Inscribed hexagon (1984)

The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry

Inscribed hexagon (2013)

A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

Related material

"Here is another way to present the deep question 1984  raises…."

— "The Quest for Permanent Novelty," by Michael W. Clune,
     The Chronicle of Higher Education , Feb. 11, 2013

“What we do may be small, but it has a certain character of permanence.”

— G. H. Hardy, A Mathematician’s Apology

Sunday, February 17, 2013

FROM an Entertainer

Filed under: General — m759 @ 9:00 PM

IMAGE- 'Michael Chabon can write like a magical spider....'

IMAGE- Book cover with 'WONDER BOYS' typewriter key

See also Back Space and Shift Lock .

For an Entertainer

Filed under: General — m759 @ 1:06 PM

"Forget about your rainbow schemes,
Spin a little web of dreams."

Song lyric

Related material: 

Big Time and The Lost Tesseract.

Zero Theorem

Filed under: General — m759 @ 12:25 AM

See "Mind of Winter" in this journal.

"And we may see the meadow in December…."

FROM Christoph Waltz

Filed under: General — m759 @ 12:12 AM

"Currently in post-production": The Zero Theorem.

For Christoph Waltz

Filed under: General — m759 @ 12:00 AM

Raiders of the Lost Tesseract  continues…

SOCRATES: Is he not better off in knowing his ignorance?
MENO: I think that he is.
SOCRATES: If we have made him doubt, and given him the 'torpedo's shock,' have we done him any harm?
MENO: I think not.

Torpedo… LOS!

Saturday, February 16, 2013

Second Life

Filed under: General — m759 @ 11:19 PM

Part I: The Midnight Ending Lincoln's Birthday, 2013

Part II: The Death of Barnaby Conrad on that day

Part III: The Second Life of John Wilkes Booth

Whether Conrad now enjoys a second life, I do not know.

Friday, February 15, 2013

Hollywood Valentine

Filed under: General — m759 @ 12:31 PM

Part I: Professor Marvin W. Meyer (this journal yesterday)

Part II: Producer/writer Richard Collins (Los Angeles Times  today)

Thursday, February 14, 2013

Chapman’s Homer

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

(Continued from August 23, 2012)

“Good is a noun. That was it.
That was what Phaedrus had been looking for.
That was the homer over the fence
that ended the ballgame.”

Robert M. Pirsig

But perhaps not a proper  noun.
See the link to Good's Singularity
at the end of today's previous post.

Annals of Quantum Hype

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

Yesterday's link to Aaronson and Turing suggests a review
of events on August 16, 2012 in the light of Log24 on that date.

Exhibit A: New Institute for Quantum Studies at Chapman University—

Exhibit B: The Aug. 16, 2012, death of Chapman University's Indiana Jones—

Whether this Indiana Jones successfully transgressed
the boundaries of space and time, I do not know.

Exhibit C: Related quantum hype— 

Chapman Professor Lands Discover  Cover Story,
Chapman University Happenings, March 18, 2010

See also Good's Singularity.

Wednesday, February 13, 2013


Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the 
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties 
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Transgressing the Boundary

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

The title refers not to the 1996 Sokal hoax (which has
Boundaries , plural, in the title), but to the boundary
discussed in Monday's Penrose diamond post

"Science is a differential equation.
Religion is a boundary condition."

Alan Turing in the epigraph to the
first chapter of a book by Terence Tao

From the Tao book, page 170—

"Typically the transformed solution extends to the
boundary of the Penrose diamond and beyond…."

Transgressing the boundary between science
and religion is the topic of a 1991 paper available
at JSTOR for $29.

For the Pope on Ash Wednesday:

"Think you might have access 
to this content via your library?" —JSTOR

See also Durkheim at Harvard.

Midnight in the Garden

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

(Continued, to mark Tuesday's birthdays of Lincoln and Darwin.)

A British reporter who died at 97 on Tuesday is said to have
"covered the space race in its entirety." In his honor, here
in review are posts containing the phrase Space Race
and, more generally, the two words Galois + Space.

Monday, February 11, 2013

The Penrose Diamond

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

IMAGE- The Penrose Diamond

Related material:

(Click to enlarge.)

See also remarks on Penrose linked to in Sacerdotal Jargon.

(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)

Pope To Resign

Filed under: General — m759 @ 12:00 PM


Filed under: General — m759 @ 3:09 AM

In memory of Rabbi David Hartman, who died yesterday.

The architecture is by Lou Gelehrter.
I do not know the logo designer's name.

Please Mister Please

Filed under: General — m759 @ 12:25 AM

IMAGE- Mumford & Sons wins Album of Year for 'Babel'

Related material:  Group Actions and B17.

Sunday, February 10, 2013

Semiotics for Kearney*

Filed under: General — Tags: — m759 @ 4:16 PM

Click image for some background.


and the following post from last October:

* Who is Kearney? See, for instance, this book.


Filed under: General — m759 @ 12:25 AM

Suggested by a recent review of a
book by Richard Kearney:

The World,  the Flesh, and the Devil.

Talk amongst yourselves.

The Cleaning

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

Arthur Jaffe CV:

"In 2005 Arthur Jaffe succeeded Sir Michael Atiyah as
Chair of the Board of the Dublin Institute for Advanced Study,
School of Theoretical Physics."

Related material:

Biddies in this journal and


An early version of quaternions.

Saturday, February 9, 2013

Snow Dance

Filed under: General,Geometry — m759 @ 2:09 PM

The Snow White dance from last Nov. 14
features an ad that was originally embedded
in an American Mathematical Society Notices
review describing three books of vulgarized
mathematics. These books all use "great
equations" as a framing device.

This literary strategy leads to a more abstract
snow dance. See the ballet blanc  in this journal
on Balanchine's birthday (old style) in 2003.
That dance involves equation (C) below.

Recall that in a unit ring ,
"0" denotes the additive identity,
"1" the multiplicative identity, and "-1" the
additive inverse of the multiplicative identity.

Three classic equations:

(A)  1 + 1 = 2    (Characteristic 0, ordinary arithmetic)

(B)  1 + 1 = 0   (Characteristic 2 arithmetic, in which 2 = 0)

(C)  1 + 1 = -1 (Characteristic 3 arithmetic, in which 2 = -1)

Cases (B) and (C), in which the characteristic is prime,
occur in Galois geometry.

For a more elaborate snow dance, see Master Class.

Friday, February 8, 2013


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

A review of the life of physicist Arthur Wightman,
who died at 90 on January 13th, 2013. yields 
the following.

Wightman at Wikipedia:
"His graduate students include
 Arthur Jaffe,  Jerrold Marsden, and Alan Sokal."

"I think of Arthur as the spiritual leader
of mathematical physics and his death
really marks the end of an era."

— Arthur Jaffe in News at Princeton , Jan. 30

Marsden at Wikipedia
"He [Marsden] has laid much of the foundation for
symplectic topology." (Link redirects to symplectic geometry.)

A Wikipedia reference in the symplectic geometry article leads to

Symplectic Geometry Lies at the Very
Foundations of Physics and Mathematics

Mark J. Gotay
Department of Mathematics
University of Hawai‘i

James A. Isenberg
Institute of Theoretical Science and Department of Mathematics
University of Oregon

February 18, 1992


We would like to thank Jerry Marsden and Alan Weinstein
for their comments on previous drafts.

Published in: Gazette des Mathématiciens  54, 59-79 (1992).


"Physics is geometry .  This dictum is one of the guiding
principles of modern physics. It largely originated with
Albert Einstein…."

A different account of the dictum:

The strange term Geometrodynamics 
is apparently due to Wheeler.

Physics may or may not be geometry, but
geometry is definitely not physics.

For some pure geometry that has no apparent 
connection to physics, see this journal
on the date of Wightman's death.

Thursday, February 7, 2013

Mathematics and Narrative, continued…

Filed under: General — m759 @ 2:01 PM

Primes Are Forever

"If diamonds are a girl's best friend,
prime numbers are a mathematician's….

A Mersenne prime is of the form 2P-1,
where the variable P is itself a prime—
making the Mersenne an elite sort of prime,
a James Bond among spies."

— Anonymous author at
    Fox News, Feb. 5, 2013

The author notes that the smaller
Mersenne primes include 7.

Related Material

April 7, 2003:

April is Math Awareness Month.

This year's theme is "mathematics and art."

Mathematics and Art: Totentanz from Seventh Seal

Update of 2:56 PM Feb. 7:

See also Paul Bateman and, in this journal, the date of Bateman's death.

For mathematics rather than narrative, see (for instance)


Wednesday, February 6, 2013

A Bus Named Desire

Filed under: General — m759 @ 12:00 PM

For Kiefer Sutherland, Hasty Pudding Man of the Year, 2013

Act I:

Current Validity for Erlangen…?

… MathOverflow question dated March 28, 2011

Act II:


… Starring Elke Sommer, former Erlangen student

Act III:

The Sweet Smell of Avon

… See also Bus 318 and 3/18 in 2012.

Act IV:


… Log24 post dated March 28, 2011

Tuesday, February 5, 2013

Serious Error

Filed under: General — m759 @ 6:00 PM

IMAGE- News item: 'Prime numbers... have little mathematical importance.'

Related material:  G. H. Hardy on seriousness.


Filed under: General — m759 @ 5:10 PM

John Berryman in The New York Review of Books :


"Now he has become, abrupt, an industry.
Professional-Friends-Of-Robert-Frost all over
open their mouths
while the quirky medium of so many truths
is quiet. Let’s all be quiet. Let’s listen:
while he begins to talk with Horace."


Filed under: General — Tags: — m759 @ 3:33 PM

Today's online Telegraph  has an obituary of The Troggs' 
lead singer Reg Presley, who died yesterday at 71.

The unusually brilliant style  of of the unsigned obituary
suggests a review of the life of a fellow Briton— 
F. L. Lucas (1894-1967), author of Style .

According to Wikipedia, Virginia Woolf described Lucas as
"pure Cambridge: clean as a breadknife, and as sharp."

Lucas's acerbic 1923 review of The Waste Land  suggests,
in the context of Woolf's remark and of the Blade and Chalice
link at the end of today's previous post, a search for a grail.



The previous post discussed some fundamentals of logic.

The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not  the algebra
needed for Galois geometry . See below.

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.

See also yesterday’s Universe of Discourse.


Filed under: General — Tags: — m759 @ 12:24 PM

From January 26, 2013

IMAGE- Yahoo CEO Marissa Mayer at Davos and the ontology of entities

Related material: "universe of discourse"

Monday, February 4, 2013

The Universe of Discourse

Filed under: General — Tags: — m759 @ 10:00 PM

A Raven's Remark—

Related material:

Fish Story, Object Lesson, The Universe of Discourse,
Archimedes's Approximation of Pi, and

Sunday, February 3, 2013


Filed under: General — Tags: , — m759 @ 11:30 PM

From the 1984 New Orleans film Tightrope


Related material: Walking the Tightrope and Transgressing.


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

Blake on the Bus

Note the contemptible adolescent misinformation
about Jim Morrison, The Doors, and Blake.

The Doors may have been named after neither
Blake's original version of the phrase "the doors
of perception" nor  Aldous Huxley's 1954
drug-related book by that title.

See also The Perception of Doors in this journal.

The Gospel According to Cartier

Filed under: General — Tags: , , , — m759 @ 10:30 AM

Yesterday's 11 AM post Mad Day concluded
with a link to a 2001 American Mathematical Society
article by Pierre Cartier that sums up the religion and
politics of many mathematicians

"Here ends the infancy narrative of the gospel…."

"… while Simone Weil's Catholicism was violently
anti-Semitic (in 1942!), Grothendieck's Buddhism
bears a strong resemblance to the practices of
his Hasidic ancestors."

See also Simone Weil in this journal.

Note esp. a post of April 6, 2004 that provides
a different way of viewing Derrida's notion of
inscription .

Saturday, February 2, 2013

Catholic Schools Week

Filed under: General — Tags: , — m759 @ 6:00 PM

"The theme for the National Catholic Schools Week 2013
is 'Catholic Schools Raise the Standards.' The annual
observance starts the last Sunday in January and runs
all week, which in 2013 is January 27 to February 2." 

"After all, tomorrow is another day." —Scarlett O'Hara,
quoted here in a post of May 9, 2005.

"Dr.  Tomorrow is another guy ." —A comment on that post.

The Dr. Tomorrow link leads to a page promoting something
called the Institute of Noetic Sciences. This in turn leads to
the 2009 Dan Brown novel The Lost Symbol .

For related material in this journal, see
Raiders of the Lost Dingbat.

As for raising the standards, see the conclusion of
Adolf Holl's The Left Hand of God 

Mad Day

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

A perceptive review of Missing Out: In Praise of the Unlived Life

IMAGE- The perception of doors

"Page 185: 'Whatever else we are, we are also mad.' "

Related material— last night's Outside the Box and, from Oct. 22 last year

"Some designs work subtly.
Others are successful through sheer force."

Par exemple—

IMAGE- The Cartier diamond ring from 'Inside Man'

See also Cartier in this journal.

The Cartier link leads to, among other things

A Mad Day’s Work: From Grothendieck to Connes and Kontsevich.
The Evolution of Concepts of Space and Symmetry
by Pierre Cartier, Bulletin of the American Mathematical Society ,
Vol. 38 (2001) No. 4, pages 389-408

Outside the Box

Filed under: General — m759 @ 1:00 AM

In memory of the late Ed Koch
a poem and a link:

Poem — "The Shoebox," by Sheila Gogol

Link  Sheila in this journal

Friday, February 1, 2013

Get Quotes

Filed under: General — Tags: , — m759 @ 4:01 PM

For Tony Kushner fans:

For logic fans:

IMAGE- NY Times market quotes, American Express Gold Card ad, Kevin Spacey in 'House of Cards' ad

John Searle on Derrida:

On necessity, possibility, and 'necessary possibility'

In the box-diamond notation, the axiom Searle quotes is


"The euclidean property guarantees the truth of this." — Wikipedia

Linking to Euclid

Clicking on "euclidean" above yields another Wikipedia article

"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."

Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.

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