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."
(A sequel to yesterday's post Publication )
Meanwhile, in this journal …
Les Miserables at the Academy Awards
"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
See also Steiner on Galois.
Les Miserables at the Academy Awards
"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.
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."
Illustrations—
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
Books 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).
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—
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
"Forget about your rainbow schemes,
Spin a little web of dreams."
Related material:
See "Mind of Winter" in this journal.
"And we may see the meadow in December…."
"Currently in post-production": The Zero Theorem.
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.
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.
Part I: Professor Marvin W. Meyer (this journal yesterday)
Part II: Producer/writer Richard Collins (Los Angeles Times today)
(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.”
But perhaps not a proper noun.
See the link to Good's Singularity
at the end of today's previous post.
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.
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—
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)…
See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model.
Related material:
|
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) |
Those who prefer story to structure may consult
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.
(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.
Related material:
See also remarks on Penrose linked to in Sacerdotal Jargon.
(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)
In memory of Rabbi David Hartman, who died yesterday.
The architecture is by Lou Gelehrter.
I do not know the logo designer's name.
Click image for some background.
Context:
and the following post from last October:
* Who is Kearney? See, for instance, this book.
Suggested by a recent review of a
book by Richard Kearney:
The World, the Flesh, and the Devil.
Talk amongst yourselves.
"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…
Detail:
An early version of quaternions.
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.
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…
|
THE SYMPLECTIZATION OF SCIENCE:
Mark J. Gotay
James A. Isenberg February 18, 1992 Acknowledgments:
We would like to thank Jerry Marsden and Alan Weinstein Published in: Gazette des Mathématiciens 54, 59-79 (1992). Opening:
"Physics is geometry . This dictum is one of the guiding |
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.
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 is Math Awareness Month.
This year's theme is "mathematics and art."
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)…
.
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:
… See also Bus 318 and 3/18 in 2012.
Act IV:
… Log24 post dated March 28, 2011
John Berryman in The New York Review of Books :
FEBRUARY 1, 1963 • VOLUME 1, NUMBER 1
"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."
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.
Voilà.
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.
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.
Related material: "universe of discourse"—
A Raven's Remark—
Related material:
Fish Story, Object Lesson, The Universe of Discourse,
Archimedes's Approximation of Pi, and…
From the 1984 New Orleans film Tightrope—
Related material: Walking the Tightrope and Transgressing.
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.
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 .
"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 —
A perceptive review of Missing Out: In Praise of the Unlived Life—
"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—
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
In memory of the late Ed Koch,
a poem and a link:
Poem — "The Shoebox," by Sheila Gogol
Link — Sheila in this journal
For Tony Kushner fans:
For logic fans:
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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