Today is Kelli O'Hara's last Saturday matinee in "The King and I."
A show that some may prefer —
Related to the plot of Dante's film —
|
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , Chapter 5, "The Tesseract" |
and versions of "Both Sides Now"
See a New York Times version of "Both Sides Now."
I prefer a version by Umberto Eco.
Related material for storytellers and the Church of Synchronology —
This journal on the date of the above shooting script, 03/19/15.
(Continued from Dec. 9, 2013)
|
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , |
From a media weblog yesterday, a quote from the video below —
"At 12:03 PM Eastern Standard Time, January 12th, 2016…."
This weblog on the previous day (January 11th, 2016) —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a young-adult novel
* For some backstory, see a Caltech page.
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
“Galois Switching Functions and Their Applications,”
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978
“Switching functions constructed by Galois extension fields,”
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
“There is such a thing as harmonic analysis of switching functions.”
— Saying adapted from a young-adult novel
"And the Führer digs for trinkets in the desert."
|
“By groping toward the light we are made to realize how deep the darkness is around us.” — Arthur Koestler, The Call Girls: A Tragi-Comedy, Random House, 1973, page 118 |
"The Tesseract is where it belongs: out of our reach."
— Samuel L. Jackson as Nick Fury,
quoted here on Epiphany 2013
Earlier … (See Jan. 27, 2012) …
"And the Führer digs for trinkets in the desert."
Click to enlarge:
For the hypercube as a vector space over the two-element field GF(2),
see a search in this journal for Hypercube + Vector + Space .
For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 1978-1986.
For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 5-7.
There is such a thing as geometry.*
* Proposition adapted from A Wrinkle in Time , by Madeleine L'Engle.
For geeks* —
" Domain, Domain on the Range , "
where Domain = the Galois tesseract and
Range = the four-element Galois field.
This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.
* A term from the 1947 film "Nightmare Alley."
On the artist Hilma af Klint (1862-1944):
"She belonged to a group called 'The Five'…."
Related material — Real Life (Log24, May 20, 2015).
From that post:
From the Milwaukee Journal Sentinel Tuesday afternoon —
A 46-year-old Jesuit priest who was a Marquette University
assistant professor of theology collapsed on campus
Tuesday morning and died, President Michael Lovell
announced to the campus community in an email….
"Rev. Lúcás (Yiu Sing Luke) Chan, S.J., died after
collapsing this morning in Marquette Hall. Just last Sunday,
Father Chan offered the invocation at the Klingler College
of Arts and Sciences graduation ceremony…."
Synchronicity check…
From this journal on the above publication date of
Chan's book — Sept. 20, 2012 —
From a Log24 post on the preceding day, Sept. 19, 2012 —
“The Game in the Ship cannot be approached as a job,
a vocation, a career, or a recreation. To the contrary,
it is Life and Death itself at work there. In the Inner Game,
we call the Game Dhum Welur , the Mind of God."
— The Gameplayers of Zan
The incidences of points and planes in the
Möbius 84 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.† In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "Self-Dual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413-455
† The subscripts' usual 1-2-3-4 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16-element Galois field GF(24). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
(Continued from July 16, 2014.)
Some background from Wikipedia:
"Friedrich Ernst Peter Hirzebruch ForMemRS[2]
(17 October 1927 – 27 May 2012)
was a German mathematician, working in the fields of topology,
complex manifolds and algebraic geometry, and a leading figure
in his generation. He has been described as 'the most important
mathematician in Germany of the postwar period.'
[3][4][5][6][7][8][9][10][11]"
A search for citations of the A. E. Brouwer paper in
the previous post yields a quotation from the preface
to the third ("2013") edition of Wolfgang Ebeling's
Lattices and Codes: A Course Partially Based
on Lectures by Friedrich Hirzebruch , a book
reportedly published on September 19, 2012 —
|
"Sadly, on May 27 this year, Friedrich Hirzebruch, Hannover, July 2012 Wolfgang Ebeling "
(Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry, |
Also sadly …
From H. S. M. Coxeter's 1950 paper
"Self-Dual Configurations and Regular Graphs,"
a 4×4 array and a more perspicuous rearrangement—
(Click image to enlarge.)
The above rearrangement brings Coxeter's remarks into accord
with the webpage The Galois Tesseract.
Update of Thursday, March 26, 2015 —
For an explanation of Coxeter's Fig. 24, see Thursday's later
post titled "The Möbius Hypercube."
Continued from June 17, 2013
(John Baez as a savior for atheists):
As an atheists-savior, I prefer Galois…
The geometry underlying a figure that John Baez
posted four days ago, "A Hypercube of Bits," is
Galois geometry —
See The Galois Tesseract and an earlier
figure from Log24 on May 21, 2007:
For the genesis of the figure,
see The Geometry of Logic.
From Zettel (repunctuated for clarity):
249. « Nichts leichter, als sich einen 4-dimensionalen Würfel
vorstellen! Er schaut so aus… »
"Nothing easier than to imagine a 4-dimensional cube!
It looks like this…
[Here the editor supplied a picture of a 4-dimensional cube
that was omitted by Wittgenstein in the original.]
« Aber das meine ich nicht, ich meine etwas wie…
"But I don't mean that, I mean something like…
…nur mit 4 Ausdehnungen! »
but with four dimensions!
« Aber das ist nicht, was ich dir gezeigt habe,
eben etwas wie…
"But isn't what I showed you like…
…nur mit 4 Ausdehnungen? »
…only with four dimensions?"
« Nein; das meine ich nicht! »
"No, I don't mean that!"
« Was aber meine ich? Was ist mein Bild?
Nun der 4-dimensionale Würfel, wie du ihn gezeichnet hast,
ist es nicht ! Ich habe jetzt als Bild nur die Worte und
die Ablehnung alles dessen, was du mir zeigen kanst. »
"But what do I mean? What is my picture?
Well, it is not the four-dimensional cube
as you drew it. I have now for a picture only
the words and my rejection of anything
you can show me."
"Here's your damn Bild , Ludwig —"
Context: The Galois Tesseract.
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.
Introducing a group of 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.)
Version from "The Avengers" (2012) —
Version from Josefine Lyche (2009) —
See also this journal on the date that the above Avengers video was uploaded.
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
* For related remarks, see posts of May 26-28, 2012.
Nick Fury takes the Tesseract…
… which travels back to 1955
(see The Call Girls, Nov. 3, 2013)…
Above: A 1955 cover design by Robert Flynn.
Images from December 1955…
… and a fictional image imagined in an earlier year:
The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).
|
The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.
Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square model,
based on the 4×4 array I call the Galois tesseract (at left below).
(Click to enlarge.)
Rivka Galchen, in a piece mentioned here in June 2010—
On Borges: Imagining the Unwritten Book
"Think of it this way: there is a vast unwritten book that the heart reacts to, that it races and skips in response to, that it believes in. But it’s the heart’s belief in that vast unwritten book that brought the book into existence; what appears to be exclusively a response (the heart responding to the book) is, in fact, also a conjuring (the heart inventing the book to which it so desperately wishes to respond)."
Related fictions
Galchen's "The Region of Unlikeness" (New Yorker , March 24, 2008)
Ted Chiang's "Story of Your Life." A film adaptation is to star Amy Adams.
… and non-fiction
"There is such a thing as a 4-set." — January 31, 2012
The search in the previous post for the source of a quotation from Poincaré yielded, as a serendipitous benefit, information on an interesting psychoanalyst named Wilfred Bion (see the Poincaré quotation at a webpage on Bion). This in turn suggested a search for the source of the name of author Madeleine L'Engle's son Bion, who may have partly inspired L'Engle's fictional character Charles Wallace. Cynthia Zarin wrote about Bion in The New Yorker of April 12, 2004 that
"According to the family, he is the person for whom L’Engle’s insistence on blurring fiction and reality had the most disastrous consequences."
Also from that article, material related to the name Bion and to what this journal has called "the Crosswicks Curse"*—
"Madeleine L’Engle Camp was born in 1918 in New York City, the only child of Madeleine Hall Barnett, of Jacksonville, Florida, and Charles Wadsworth Camp, a Princeton man and First World War veteran, whose family had a big country place in New Jersey, called Crosswicks. In Jacksonville society, the Barnett family was legendary: Madeleine’s grandfather, Bion Barnett, the chairman of the board of Jacksonville’s Barnett Bank, had run off with a woman to the South of France, leaving behind a note on the mantel. Her grandmother, Caroline Hallows L’Engle, never recovered from the blow. ….
… The summer after Hugh and Madeleine were married, they bought a dilapidated farmhouse in Goshen, in northwest Connecticut. Josephine, born in 1947, was three years old when they moved permanently to the house, which they called Crosswicks. Bion was born just over a year later."
* "There is such a thing as a tesseract."
"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."
— G. H. Hardy, A Mathematician's Apology
Part I: An Inch Deep
Part II: An Inch Wide
See a search for "square inch space" in this journal.
See also recent posts with the tag depth.
Raiders of the Lost (Continued)
"Socrates: They say that the soul of man is immortal…."
From August 16, 2012—
In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.
An 1892 figure by Jowett illustrating Plato's Meno—
A more correct version, from hermes-press.com —
|
Socrates: He only guesses that because the square is double, the line is double.Meno: True.
Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this-that is to say of eight feet; and I want to know whether you still say that a double square comes from double line? [Boy] Yes. Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)? [Boy] Certainly.
Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet? [Boy] Yes. Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet? [Boy] Yes. Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN) [Boy] True. Socrates: And is not that four times four? [Boy] Certainly. Socrates: And four times is not double? [Boy] No, indeed. Socrates: But how much? [Boy] Four times as much. Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much. [Boy] True. Socrates: Four times four are sixteen— are they not? [Boy] Yes. |
As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:
A 4×4 square version, by Coxeter in 1950, of a tesseract—
This square version we may call the Galois tesseract.
"There is such a thing as a tesseract."
— Saying from Crosswicks
See also March 5, 2011.
Adapted from the above passage —
"So did L'Engle understand four-dimensional geometry?"
The archived Java rotatable hypercube of
Harry J. Smith is no longer working.
For an excellent JavaScript replacement,
see Pete Michaud's
http://petemichaud.github.io/4dhypercube/.
This JavaScript version can easily be saved.
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
(On His Dies Natalis )…
This is asserted in an excerpt from…
"The smallest non-rank 3 strongly regular graphs
which satisfy the 4-vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152-212—
(Click for clearer image)
Note that Theorem 46 of Klin et al. describes the role
of the Galois tesseract in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric part of the above
exceptional geometric-combinatorial isomorphism.
"How about another hand for the band?
They work real hard for it.
The Cherokee Cowboys, ladies and gentlemen."
— Ray Price, video, "Danny Boy Mid 80's Live"
Other deathly hallows suggested by today's NY Times—
Click the above image for posts from December 14.
That image mentions a death on August 5, 2005, in
"entertainment Mecca" Branson, Missouri.
Another note from August 5, 2005, reposted here
on Monday—
Happy birthday, Keith Richards.
Happy Beethoven's Birthday.
Related material: Abel 2005 and, more generally, Abel.
See also Visible Mathematics.
Odin's Jewel
Jim Holt, the author of remarks in yesterday's
Saturday evening post—
"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."
— Vanity Fair Daily , July 16, 2012
Backstory: Odin + Jewel in this journal.
See also Odin on the Kyoto school —
For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —
The title refers to a post of April 26, 2009.
Or: The Naked Blackboard Jungle
|
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , |
Related material: Machete Math and…
Starring the late Eleanor Parker as Swiftly Mrs. Who.
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."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.
The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—
As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
|
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
Here is the link to an MIT Scratch project from the above comment.
See also a comment by a Random Norwegian Dude:
For related art, see
"4D AMBASSADOR (HYPERCUBE)" for Steven H. Cullinane
by the Norwegian artist Josefine Lyche.
The Daily Princetonian today:
A different cover act, discussed here Saturday:
See also, in this journal, the Galois tesseract and the Crosswicks Curse.
"There is such a thing as a tesseract." — Crosswicks saying
The hypercube model of the 4-space over the 2-element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4-space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).
The thirty-five 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book co-authored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the Conway-Sloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345-353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M 24 ,"
arXiv.org > hep-th > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
|
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array |
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
"Why history?
Well, the essence of history is story ,
and a good story is an end in itself."
— Barry Mazur, "History of Mathematics as a tool,"
February 17, 2013
This journal on February 17, 2013:FROM Christoph Waltz"Currently in post-production": The Zero Theorem. For Christoph WaltzRaiders of the Lost Tesseract continues… SOCRATES: Is he not better off in knowing his ignorance? |
See also today's previous post.
"There is such a thing as a tesseract." —A novel from Crosswicks
Related material from a 1905 graduate of Princeton,
"The 3-Space PG(3,2) and Its Group," is now available
at Internet Archive (1 download thus far).
The 3-space paper is relevant because of the
connection of the group it describes to the
"super, overarching" group of the tesseract.
The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013. The talk's subtitle was
"Oswald Veblen and the Six-hundred-acre Woods."
Meanwhile…
Thursday, March 21, 2013
|
Related material for those who prefer narrative
to mathematics:
|
Log24 on June 6, 2006:
The Omen :
|
Related material for those who prefer mathematics
to narrative:
What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry. As the British
geometer H. F. Baker pointed out, 6 is a rather important number
in that discipline. For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.
See also last night's Veblen and Young in Light of Galois.
"There is such a thing as a tesseract." — Madeleine L'Engle
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
The previous post discussed some tesseract–
related mathematics from 1905.
Returning to the present, here is some arXiv activity
in the same area from March 11, 12, and 13, 2013.
From the prologue to the new Joyce Carol Oates
novel Accursed—
"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.
1905!—the very year of the Curse."
Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract of Madeleine L'Engle.
The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —
"There is such a thing as a tesseract."
A tesseract is a 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also
be viewed as a 4×4 array (with opposite edges
identified).
Meanwhile, back in 1905…
For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15-point projective
Galois space PG(3,2).
See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.
For readers of The Daily Princetonian :
(From a site advertised in the
Princetonian on March 11, 2013)
For readers of The Harvard Crimson :
For some background, see Crimson Easter Egg and the Diamond 16 Puzzle.
For some (very loosely) related narrative, see Crosswicks in this journal
and the Crosswicks Curse in a new novel by Joyce Carol Oates.
"There is such a thing as a tesseract."
— Crosswicks author Madeleine L'Engle
Today's previous post recalled a post
from ten years before yesterday's date.
The subject of that post was the
Galois tesseract.
Here is a post from ten years before
today's date.
The subject of that post is the Halmos
tombstone:
"The symbol 
is used throughout the entire book
in place of such phrases as 'Q.E.D.' or 'This
completes the proof of the theorem' to signal
the end of a proof."
— Measure Theory (1950)
For exact proportions, click on the tombstone.
For some classic mathematics related
to the proportions, see September 2003.
"Forget about your rainbow schemes,
Spin a little web of dreams."
Related material:
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.
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 was suggested by an ad for a film that opens
at 10 PM EST today: "Hansel & Gretel: Witch Hunters."
Related material: Grimm Day 2012, as well as
Amy Adams in Raiders of the Lost Tesseract
and in a Film School Rejects page today.
See also some Norwegian art in
Trish Mayo's Photostream today and in
Omega Point (Log24, Oct. 15, 2012)—
Monday, October 15, 2012
|
For the 2013 Joint Mathematics Meetings in San Diego,
which start today, a cartoon by Andrew Spann—
(Click for larger image.)
Related remarks:
This journal on the Feast of Epiphany, 2013—
"The Tesseract is where it belongs: out of our reach."
— The Avengers' Nick Fury, played by Samuel L. Jackson
"You never know what could happen.
If you have Sam, you’re going to be cool."
— The late David R. Ellis, film director
If anyone in San Diego cares about the relationship
of Spann's plane to Fury's Tesseract, he or she may
consult Finite Geometry of the Square and Cube.
For the Feast of Epiphany:
A trip back to December 1955—
Meditations for Three Kings Day (Feast of Epiphany)—
"Show me all the blueprints." — Leonardo DiCaprio as Howard Hughes
"The Tesseract is where it belongs: out of our reach." — Samuel L. Jackson as Nick Fury
"Here was finality indeed, and cleavage!" — Malcolm Lowry's Under the Volcano (1947)
Click images for some background.
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
A New Yorker weblog post from yesterday, All Souls' Day—
"As the mathematician Terence Tao has written,
math study has three stages:
the 'pre-rigorous,' in which basic rules are learned,
the theoretical 'rigorous' stage, and, last and most intriguing,
'the post-rigorous,' in which intuition suddenly starts to play a part."
Related material—
Rigor in a Log24 post of Sunday evening, May 25, 2008: "Hall of Mirrors."
Note in that post the tesseract viewed as the lattice of
the 16 subsets of a 4-element set.
Some further material related to tesseracts and time, in three stages
(roughly corresponding to Tao's, but not in chronological order):
See also a recent Log24 post on remarks from Four Quartets .
(The vertices of a tesseract form, in various natural ways, four quartets.)
From this journal yesterday (All Saints' Day)—
"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."
— New Yorker profile of tesseract
author Madeleine L'Engle
For a discussion of this issue in greater depth—
"Truth and fact are not the same thing."
— see a 1998 award acceptance speech by L'Engle.
See also a Log24 post of March 1st, 2008, on the soul.
A review of two theories of truth described
by a clergyman, Richard J. Trudeau, in
The Non-Euclidean Revolution—
"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."
— New Yorker profile of tesseract
author Madeleine L'Engle
(Click image for some background.)
See also the links on a webpage at finitegeometry.org.
(Continued from previous TARDIS posts)
Summary: A review of some posts from last August is suggested by the death,
reportedly during the dark hours early on October 30, of artist Lebbeus Woods.
An (initially unauthorized) appearance of his work in the 1995 film
Twelve Monkeys …
… suggests a review of three posts from last August.
Wednesday, August 1, 2012Defining FormContinued from July 29 in memory of filmmaker Chris Marker, See Slides and Chanting†and Where Madness Lies. See also Sherrill Grace on Malcolm Lowry. * Washington Post. Other sources say Marker died on July 30. † These notably occur in Marker's masterpiece |
Wednesday, August 1, 2012Triple Feature
For related material, see this morning's post Defining Form. |
Sunday, August 12, 2012Doctor WhoOn Robert A. Heinlein's novel Glory Road— "Glory Road (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road was influenced by the debut of the science fiction television series Doctor Who on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out." — Todd, Tesseract article at exampleproblems.com From the same exampleproblems.com article— "The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square." For further details, see today's new page on vertex adjacency at finitegeometry.org. |
"It was a dark and stormy night."— A Wrinkle in Time
For Sergeant-Major America—
The image is from posts of Feb. 20, 2011, and Jan. 27, 2012.
This instance of the omega point is for a sergeant major
who died at 92 on Wednesday, October 10, 2012.
See also posts on that date in this journal—
Midnight, Ambiguation, Subtitle for Odin's Day, and
Melancholia, Depression, Ambiguity.
Occurrences of the phrase "magic square" in Lowe-Porter's translation of the Thomas Mann novel Doctor Faustus—
"On the wall above the piano was an arithmetical diagram fastened with drawing-pins, something he had found in a second-hand shop: a so-called magic square, such as appears also in Dürer's Melancolia , along with the hour-glass, the circle, the scale, the polyhedron, and other symbols. Here as there, the figure was divided into sixteen Arabic-numbered fields, in such a way that number one was in the right-hand lower corner, sixteen in the upper left; and the magic, or the oddity, simply consisted in the fact that the sum of these numerals, however you added them, straight down, crosswise, or diagonally, always came to thirty-four. What the principle was upon which this magic uniformity rested I never made out, but by virtue of the prominent place Adrian had given it over the piano, it always attracted the eye, and I believe I never visited his room without giving a quick glance, slanting up or straight down and testing once more the invariable, incredible result."
….
"Adrian kept without changing during the whole four and a half years he spent in Leipzig his two-room quarters in Peterstrasse near the Collegium Beatae Virginis, where he had again pinned the magic square above his cottage piano."
….
" 'The decisive factor is that every note, without exception, has significance and function according to its place in the basic series or its derivatives. That would guarantee what I call the indifference to harmony and melody.'
'A magic square,' I said. 'But do you hope to have people hear all that?' "
….
" 'Extraordinarily Dürerish. You love it. First "how will I shiver after the sun"; and then the houre-glasse of the Melancolia . Is the magic square coming too?' "
….
"Here I will remind the reader of a conversation I had with Adrian on a long-ago day, the day of his sister's wedding at Buchel, as we walked round the Cow Trough. He developed for me— under pressure of a headache— his idea of the 'strict style,' derived from the way in which, as in the lied 'O lieb Madel, wie schlecht bist du ' melody and harmony are determined by the permutation of a fundamental five-note motif, the symbolic letters h, e, a, e, e-flat. He showed me the 'magic square' of a style of technique which yet developed the extreme of variety out of identical material and in which there is no longer anything unthematic, anything that could not prove itself to be a variation of an ever constant element. This style, this technique, he said, admitted no note, not one, which did not fulfil its thematic function in the whole structure— there was no longer any free note."
Review of related material—
Last night's midnight post (disambiguation), the followup 1 AM post (ambiguation), today's noon post (ambiguity), and Dürer in this journal.
The tesseracts of the noon post are related to the Dürer magic square by a well-known adjacency property.
"… the once stable 'father's depression' has been transmuted into a shifting reality that shimmered in a multiplicity of facets."
— Haim Omer, Tel-Aviv University, on Milanese ambiguation therapy,
p. 321 in "Three Styles of Constructive Therapy,"
Constructive Therapies, Vol. 2 , pp. 319-333,
ed. by Michael F. Hoyt (Guilford Press paperback, 1998)
The subtitle of Jack Kerouac's novel Doctor Sax
is Faust Part Three.
Related material—
Types of Ambiguity— Galois Meets Doctor Faustus
(this journal, December 14, 2010).
Today's Harvard Crimson—
Students Discuss Mental Health
In an effort to break the silence on an often-stigmatized topic,
members of the Harvard community gathered to share
experiences with mental illness through spoken word,
interpretive dance, and candid conversations at Friday night’s
second-annual “Words on the Mind” open mic night.
Related material from this journal on Friday—
The Hallowed Crucible—
Some related symbolism (headings added Oct. 7)—
|
Words:
Applied Mathematics |
The Mind:
Pure Mathematics |
Today's (Sunday, Oct. 7, 2012) Google Doodle for Bohr's birthday—
Review (See also Faust in Copenhagen in this journal)—
» more
The Hallowed Crucible—
Some related symbolism—
|
Applied Mathematics |
Pure Mathematics |
See also Stallion Gate (a novel) in this journal.
For some related nonfiction, see interviews with
Los Alamos physicist Robert F. Christy, who died
at 96 on Wednesday, October 3, 2012.
From an obituary for Helen Nicoll, author
of a popular series of British children's books—
"They feature Meg, a witch whose spells
always seem to go wrong, her cat Mog,
and their friend Owl."
For some (very loosely) related concepts that
have been referred to in this journal, see…
See, too, "Kids grow up" (Feb. 13, 2012).
|
From French cinema—
"a 'non-existent myth' of a battle between |
"Moon River, wider than a mile…"
|
The most damaging and obstructive Like “genius.” And “sincerity.” And “inspiration.” Distrust these words.
They stand for cherished myths,
— Verlyn Klinkenborg, |
"All she had to do was kick off and flow."
"I'se so silly to be flowing but I no canna stay."
Part of a New York Times banner ad last night—
(Fashion week dates 2012 —
New York Sept. 6-13, London Sept. 14-18,
Milan Sept. 19-25, Paris Sept. 25-Oct. 3.)
Some related prose suggested by a link in
last night's Log24 post—
The theory, he had explained, was that the persona
was a four-dimensional figure, a tessaract in space,
the elementals Fire, Earth, Air, and Water permutating
and pervolving upon themselves, making a cruciform
(in three-space projection) figure of equal lines and
ninety degree angles.
— The Gameplayers of Zan , a novel by M. A. Foster
See also, if you can find a copy, Jeff Riggenbach's
"Science Fiction as Will and Idea," Riverside Quarterly
Vol. 5, No. 3 (whole number 19, August 1972, ed. by
Leland Sapiro et al.), 168-177.
Some background—
Tuesday's Simple Skill and 4D Ambassador,
as well as Now What? from May 23, 2012.
Today's previous post, "For Odin's Day," discussed
a mathematical object, the tesseract, from a strictly
narrative point of view.
In honor of George Balanchine, Odin might yield the
floor this evening to Apollo.
From a piece in today's online New York Times titled
"How a God Finds Art (the Abridged Version)"—
"… the newness at the heart of this story,
in which art is happening for the first time…."
Some related art—
and, more recently—
This more recent figure is from Ian Stewart's 1996 revision
of a 1941 classic, What Is Mathematics? , by Richard Courant
and Herbert Robbins.
Apollo might discuss with Socrates how the confused slave boy
of Plato's Meno would react to Stewart's remark that
"The number of copies required to double an
object's size depends on its dimension."
Apollo might also note an application of Socrates' Meno diagram
to the tesseract of this afternoon's Odin post—
.
(Mathematics and Narrative, continued)
| "My dad has a great expression," Steve Sabol told USA TODAY Sports last year. "He always says, 'Tell me a fact, and I'll learn. Tell me the truth, and I believe. But tell me a story, and it will live in my heart forever.' " |
Fact—
Truth—
An art gallery in Oslo is exhibiting a tesseract.
Story—
The Jewel of Odin's Treasure Room
* I.e., Wednesday. For some apt Nordic spirit,
see Odin's Day 2012 Trailer.
Wikipedia (links added)—
"Hubbard coined Dianetics from the Greek stems dia ,
meaning through, and nous , meaning mind."
"The snow kept falling on the world,
big white flakes like white gloves."
— Frederick Seidel, "House Master,"
poem in The New Yorker of Sept. 3, 2012
Detail of Aug. 30 illustration, with added arrow—
The part of the illustration at upper right is from a post of
Friday, July 13th, 2012, on the death of producer Richard Zanuck.
"Pay no attention to the shadow behind the curtain."
Last night's 10 PM post linked to an April 7, 2012,
post that through a series of further links leads
to Columbia Film Theory .
For other film-related remarks, by a
Columbia alumnus,* see last night's post.
See also the 1.3 MB image from Aug. 16, the night
of Elvis's Wrap Party. An excerpt from that image
stars Amy Adams—
For Amy, from the current New Yorker—
The Master—
* N.O.C.D.
"Plato's cave was brought up to date in 1978…."
— Keith Devlin in Mathematics: The Science of Patterns
Related material from yesterday: Touchy-Feely and Plan 9.
"Plan 9 deals with the resurrection of the dead."
For a rather different approach to Plato, see three posts of August 16, 2012—
A remark by the late William P. Thurston—
Please note: I'm not advocating that
we turn mathematics into a touchy-feely subject.
Noted. But see this passage—
|
The Mathematical Experience , by Philip J. Davis and Reuben Hersh (1981), updated study edition, Springer, 2011— From the section titled "Four-Dimensional Intuition," pages 445-446: "At Brown University Thomas Banchoff, a mathematician, and Charles Strauss, a computer scientist, have made computer-generated motion pictures of a hypercube…. … at the Brown University Computing Center, Strauss gave me a demonstration of the interactive graphic system which made it possible to produce such a film…. … Strauss showed me how all these controls could be used to get various views of three-dimensional projections of a hypercube. I watched, and tried my best to grasp what I was looking at. Then he stood up, and offered me the chair at the control. I tried turning the hypercube around, moving it away, bringing it up close, turning it around another way. Suddenly I could feel it!. The hypercube had leaped into palpable reality, as I learned how to manipulate it, feeling in my fingertips the power to change what I saw and change it back again. The active control at the computer console created a union of kinesthetics and visual thinking which brought the hypercube up to the level of intuitive understanding." |
Thanks to the Web, a version of this experience created by Harry J. Smith
has been available to non-academics for some time.
On the author of a novel published August 14th,
"Where'd You Go, Bernadette"—
"Semple moved to the Pacific Northwest several years ago
seeking refuge from Los Angeles, but that doesn't mean
that the Emerald City gets a free pass from Semple's
sharp, satirical eye."
— Stewart Oksenhorn yesterday in The Aspen Times
See also a detail from Thursday's 1.3 MB image
"Search for the Lost Tesseract"—
Update of 9 PM EDT (6 PM LA time) the same day, Saturday, Aug. 18—
Detail from last night's 1.3 MB image
"Search for the Lost Tesseract"—
The lost tesseract appears here on the cover of Wittgenstein's
Zettel and, hidden in the form of a 4×4 array, as a subarray
of the Miracle Octad Generator on the cover of Griess's
Twelve Sporadic Groups and in a figure illustrating
the geometry of logic.
Another figure—
Gligoric died in Belgrade, Serbia, on Tuesday, August 14.
From this journal on that date—
"Visual forms, he thought, were solutions to
specific problems that come from specific needs."
— Michael Kimmelman in The New York Times
obituary of E. H. Gombrich (November 7th, 2001)
On Robert A. Heinlein's novel Glory Road—
"Glory Road (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road was influenced by the debut of the science fiction television series Doctor Who on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out."
— Todd, Tesseract article at exampleproblems.com
From the same exampleproblems.com article—
"The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square."
For further details, see today's new page on vertex adjacency at finitegeometry.org.
Euclidean square and triangle—
Galois square and triangle—
Background—
This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.
Today's previous post was "Midnight in Oslo (continued)."
The link "a 4-element set" in "Midnight"
was to a more elaborate structure in a post titled "Tesseract."
In memory of an Oslo "hero of midnight"
(a phrase quoted here last September 1)—
A search for material that is more entertaining—
Odin 's Tesseract.
See also a related Hollywood story in The Washington Post .
Background— George Steiner in this journal
and elsewhere—
"An intensity of outward attention —
interest, curiosity, healthy obsession —
was Steiner’s version of God’s grace."
— Lee Siegel in The New York Times ,
March 12, 2009
(See also Aesthetics of Matter in this journal on that date.)
Steiner in 1969 defined man as "a language animal."
Here is Steiner in 1974 on another definition—
Related material—
Also related — Kantor in 1981 on "exquisite finite geometries," and The Galois Tesseract.
A followup to this morning's post Stolen Glory—
Columbia's Butler Library "plays a role in
Paul Auster's 2009 novel Invisible ,
where the novel's main protagonist, Adam Walker,
takes a job as a 'page' in the library's stacks." —Wikipedia
Part I (from Feb. 24)—
Part II— (Click to enlarge)
For the page's source, see Butler Library.
Wednesday, February 1, 2012
Politics
|
Related material—
See also At the Still Point (a post in memory of film editor Sally Menke).
"Should we arbitrate life and death
at a round table or a square one?"
See also the two previous posts,
Disturbing Archimedes and Tesseract.
Update—
The following passage by Tolkien was suggested by a copy of next Sunday's New York Times Book Review that arrived in the mail today. (See Orson Scott Card's remarks on page 26— "Uncle Orson"— and the Review 's concluding essay "Grand Allusion.")
"Lastly, tengwesta [system or code of signs] has also become an impediment. It is in Incarnates clearer and more precise than their direct reception of thought. By it also they can communicate easily with others, when no strength is added to their thought: as, for example, when strangers first meet. And, as we have seen, the use of 'language' soon becomes habitual, so that the practice of ósanwe (interchange of thought) is neglected and becomes more difficult. Thus we see that the Incarnate tend more and more to use or to endeavour to use ósanwe only in great need and urgency, and especially when lambe is unavailing. As when the voice cannot be heard, which comes most often because of distance. For distance in itself offers no impediment whatever to ósanwe . But those who by affinity might well use ósanwe will use lambe when in proximity, by habit or preference. Yet we may mark also how the 'affine' may more quickly understand the lambe that they use between them, and indeed all that they would say is not put into words. With fewer words they come swifter to a better understanding. There can be no doubt that here ósanwe is also often taking place; for the will to converse in lambe is a will to communicate thought, and lays the minds open. It may be, of course, that the two that converse know already part of the matter and the thought of the other upon it, so that only allusions dark to the stranger need be made; but this is not always so. The affine** will reach an understanding more swiftly than strangers upon matters that neither have before discussed, and they will more quickly perceive the import of words that, however numerous, well-chosen, and precise, must remain inadequate."
* "If a poem catches a student's interest at all, he or she should damned well be able to look up an unfamiliar word in the dictionary…."
— Elizabeth Bishop, quoted in the essay "Grand Allusion" mentioned above. For a brief dictionary of most of the unfamiliar words in this post's title and in the above passage, see Vinyar Tengwar 39 (July 1998). This is copyrighted but freely available on the Web.
** The word "affine" has connotations not intended by Tolkien. See that word in this journal. See also page 5 of next Sunday's Times Book Review , which contains a full-page ad for the 50th anniversary edition of A Wrinkle in Time . "There is such a thing as a tesseract."
Princeton University Press on a book it will publish in March—
Circles Disturbed: The Interplay of Mathematics and Narrative
"Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier— 'Don't disturb my circles'— words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds–stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities."
Timeline of the Marvel Cinematic Universe — Norway, March 1942—
"The Red Skull finds the Tesseract, a cube of strange power,
said to be the jewel of Odin’s treasure room, in Tonsberg Norway.
(Captain America: The First Avenger)"
Tesseracts Disturbed — (Click to enlarge)
Detail of Tesseracts Disturbed —
Narrative of the detail—
See Tesseract in this journal and Norway, May 2010—
"Debates about canonicity have been raging in my field
(literary studies) for as long as the field has been
around. Who's in? Who's out? How do we decide?"
— Stephen Ramsay, "The Hermeneutics of Screwing Around"
An example of canonicity in geometry—
"There are eight heptads of 7 mutually azygetic screws, each consisting of the screws having a fixed subscript (from 0 to 7) in common. The transformations of LF(4,2) correspond in a one-to-one manner with the even permutations on these heptads, and this establishes the isomorphism of LF(4,2) and A8. The 35 lines in S3 correspond uniquely to the separations of the eight heptads into two complementary sets of 4…."
— J.S. Frame, 1955 review of a 1954 paper by W.L. Edge,
"The Geometry of the Linear Fractional Group LF(4,2)"
Thanks for the Ramsay link are due to Stanley Fish
(last evening's online New York Times ).
For further details, see The Galois Tesseract.
J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—
For earlier (and later) discussions of the geometry (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.
"Design is how it works." — Steve Jobs
From a commercial test-prep firm in New York City—
From the date of the above uploading—
|
|
From a New Year's Day, 2012, weblog post in New Zealand—
From Arthur C. Clarke, an early version of his 2001 monolith—
"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."
The numerical (not crystal) pyramid above is related to a sort of
mathematical block design known as a Steiner system.
For its relationship to the graphic block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M24," which contains the following
version of the above numerical pyramid—
For graphic block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.
For the barbed tail of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.
Mathematics —
(Some background for the Galois tesseract )
Narrative —
An essay on science and philosophy in the January 2012
Notices of the American Mathematical Society .
Note particularly the narrative explanation of the double-slit experiment—
"The assertion that elementary particles have
free will and follow Quality very closely leads to
some startling consequences. For instance, the
wave-particle duality paradox, in particular the baffling
results of the famous double slit experiment,
may now be reconsidered. In that experiment, first
conducted by Thomas Young at the beginning
of the nineteenth century, a point light source
illuminated a thin plate with two adjacent parallel
slits in it. The light passing through the slits
was projected on a screen behind the plate, and a
pattern of bright and dark bands on the screen was
observed. It was precisely the interference pattern
caused by the diffraction patterns of waves passing
through adjacent holes in an obstruction. However,
when the same experiment was carried out much
later, only this time with photons being shot at
the screen one at a time—the same interference
pattern resulted! But the Metaphysics of Quality
can offer an explanation: the photons each follow
Quality in their actions, and so either individually
or en masse (i.e., from a light source) will do the
same thing, that is, create the same interference
pattern on the screen."
This is from "a Ph.D. candidate in mathematics at the University of Calgary."
His essay is titled "A Perspective on Wigner’s 'Unreasonable Effectiveness
of Mathematics.'" It might better be titled "Ineffective Metaphysics."
Mathematics and Narrative, continued
"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"
— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th
"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."
— Dan Conover on September 1st, 2011
"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'
I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."
— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996
"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"
— Stanford Encyclopedia of Philosophy
"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."
— Interviewer for the National Association of Scholars
From that conversation—
Wolfe : "People in academia should start insisting on objective scholarship, insisting on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"
As for "mis-imagined quantum pins"…
This journal on the date of the above interview— February 28, 2008—
Illustration from a Perimeter Institute talk given on July 20, 2005
The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by Conover, "The Last Epiphany," and four posts from September 1st, 2011—
Boundary, How It Works, For Thor's Day, and The Galois Tesseract.
Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.
"There is such a thing as a tesseract." —A Wrinkle in Time
Today is day 256 of 2011, Programmers' Day.
Yesterday, Monday, R. W. Barraclough's website pictured the Octad of the Week—
" X never, ever, marks the spot."
See also The Galois Tesseract.
Betty Skelton, "the First Lady of Firsts," died on the last day of August.
From this journal on August thirty-first—
"The Tesseract was the jewel of Odin's treasure room."
Hugo Weaving also played Agent Smith
in The Matrix Trilogy .
For Cynthia Zarin, biographer of Madeleine L'Engle—
"There is such a thing as a tesseract."
— A Wrinkle in Time
It is now midnight. Yesterday was Odin's Day. Today is Thor's Day.
From a weblog post on Captain America and Thor—
"While all this [Captain America] is happening an SS officer, Johann Schmidt (Hugo Weaving), has found a religious artefact called the Tesseract which Schmidt describes as 'the jewel of Odin’s treasure room,' linking it in with the Thor storyline."
— That's Entertainment weblog, August 14, 2011
From Wallace Stevens, "An Ordinary Evening in New Haven," Canto III—
The point of vision and desire are the same.
It is to the hero of midnight that we pray
On a hill of stones to make beau mont thereof.
Captain America opened in the United States on Friday, July 22, 2011.
Thor opened in the United States on Friday, May 6, 2011.
"There is such a thing as a tesseract." —A Wrinkle in Time
* Continued from August 30.
Today is Wednesday.
O.E. Wodnesdæg "Woden's day," a Gmc. loan-translation of L. dies Mercurii "day of Mercury" (cf. O.N. Oðinsdagr , Swed. Onsdag , O.Fris. Wonsdei , M.Du. Wudensdach ). For Woden , see Odin . — Online Etymology Dictionary
Above: Anthony Hopkins as Odin in the 2011 film "Thor"
Hugo Weaving as Johann Schmidt in the related 2011 film "Captain America"—
"The Tesseract* was the jewel of Odin's treasure room."
Weaving also played Agent Smith in The Matrix Trilogy.
The figure at the top in the circle of 13** "Thor" characters above is Agent Coulson.
"I think I'm lucky that they found out they need somebody who's connected to the real world to help bring these characters all together."
— Clark Gregg, who plays Agent Coulson in "Thor," at UGO.com
For another circle of 13, see the Crystal Skull film implicitly referenced in the Bright Star link from Abel Prize (Friday, Aug. 26, 2011)—
Today's New York Times has a quote about a former mathematician who died on that day (Friday, Aug. 26, 2011)—
"He treated it like a puzzle."
Sometimes that's the best you can do.
* See also tesseract in this journal.
** For a different arrangement of 13 things, see the cube's 13 axes in this journal.
A search for some background on Dmitri Tymoczko, the subject of yesterday's evening entry on music theory, shows that his name and mine once both appeared in the same web page— "This Week's Finds in Mathematical Physics (Week 234)," by John Baez, June 12, 2006 (linked to by the Wikipedia article on transformational music theory).
In that page, Baez speculates on the possibility of a connection between music theory and Mathieu groups and says—
"For a pretty explanation of M24, also try this:
Steven H. Cullinane, Geometry of the 4 × 4 square, http://finitegeometry.org/sc/16/geometry.html."
I know of no connection* between the groups I discussed there and music theory. For some background on Tymoczko's work, see the helpful survey "Exploring Musical Space," by Julian Hook (Science magazine, 7 July 2006).
* Apart, that is, from the tesseract (see Geometry of the 4 × 4 Square) shown by Tymoczko in a 2010 lecture—
This is perhaps "Chopin's tesseract" from section 8.5 of Tymoczko's new book
A Geometry of Music (Oxford University Press, 2011).
A Year of Magical Thinking
In memory of Theodore Chaikin Sorensen, who died at noon in New York on Halloween —
Two posts from All Saints' Day, 2009 —
October Endgame and Indignation and Laughter in Toronto.
Related material: New York Lottery on All Hallows' Eve this year —
Midday 896, Evening 384.
"Man is a system that transforms itself." (Paul Valéry, Cahiers , Vol. 2, page 896)
"There is such a thing as a tesseract." (Madeleine L'Engle. See 384 on Halloween 2006.)
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the four-diamond figure
with the Pythagorean theorem.
A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
Four-Part Tesseract Divisions—
The above figure shows how four-part partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to four-part partitions
of the 16 points in a finite Galois space
|
Euclidean spaces versus Galois spaces in a larger context—
Infinite versus Finite The central aim of Western religion —
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy —
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy." |
Another picture related to philosophy and religion—
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
|
Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896). O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60) C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61) |
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
|
… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.” If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect. * That is, uses multi-dimensional symbols beyond our grasp. |
Related material:
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Some context by a British mathematician —
|
Imago by Wallace Stevens Who can pick up the weight of Britain, Who can move the German load Or say to the French here is France again? Imago. Imago. Imago. It is nothing, no great thing, nor man Of ten brilliancies of battered gold And fortunate stone. It moves its parade Of motions in the mind and heart, A gorgeous fortitude. Medium man In February hears the imagination's hymns And sees its images, its motions And multitudes of motions And feels the imagination's mercies, In a season more than sun and south wind, Something returning from a deeper quarter, A glacier running through delirium, Making this heavy rock a place, Which is not of our lives composed . . . Lightly and lightly, O my land, Move lightly through the air again. |
Photo caption in NY Times today— a pianist "preforming" in 1967. (See today's previous post.)
The pianist's life story seems in part to echo that of Juliette Binoche in the film "Bleu." Binoche appeared in this journal yesterday, before I had seen the pianist in today's Times obituaries. The Binoche appearance was related to the blue diamond in the film "Duelle " (Tuesday morning's post) and the saying of Heraclitus "immortals mortal, mortals immortal" (Tuesday afternoon's post).
This somewhat uncanny echo brings to mind Nabokov—
Life Everlasting—based on a misprint!
I mused as I drove homeward: take the hint,
And stop investigating my abyss?
But all at once it dawned on me that this
Was the real point, the contrapuntal theme;
Just this: not text, but texture; not the dream
But topsy-turvical coincidence,
Not flimsy nonsense, but a web of sense.
Whether sense or nonsense, the following quotation seems relevant—
"Archetypes function as living dispositions, ideas in the Platonic sense, that preform and continually influence our thoughts and feelings and actions." –C.G. Jung in Four Archetypes: Mother, Rebirth, Spirit, Trickster, the section titled "On the Concept of the Archetype."
That section is notable for its likening of Jungian archetypes to Platonic ideas and to axial systems of crystals. See also "Cubist Tune," March 18 —
Paul Robeson in
"King Solomon's Mines," 1937—
The image above is an illustration from
"Romancing the Hyperspace," May 4, 2010.
This illustration, along with Georgia Brown's
song from "Cabin in the Sky"—
"There's honey in the honeycomb"—
suggests the following picture.
"What might have been and what has been
Point to one end, which is always present."
— Four Quartets
The Unfolding
A post for Florencio Campomanes,
former president of the World Chess Federation.
Campomanes died at 83 in the Philippines
at 1:30 PM local time (1:30 AM Manhattan time)
on Monday, May 3, 2010.
From this journal on the date of his death —
"There is such a thing as a tesseract."
– Madeleine L'Engle
Image by Christopher Thomas at Wikipedia —
Unfolding of a hypercube and of a cube —
Related material from a story of the Philippines —
“…geometrically organized, with the parts labeled”
— Ursula K. Le Guin on what she calls “the Euclidean utopia”
“There is such a thing as a tesseract.”
Related material– Diamond Theory, 1937
From the September 1953 Bulletin of the American Mathematical Society—
Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."
Art Versus Chaos
From an exhibit,
"Reimagining Space"
The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."
For related mathematical details see
Diamond Theory in 1937.
"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
— Fritz Leiber, 1959
See also the Log24 entry for
Nov. 26, 2009, the date that
Forakis died.
"There is such a thing
as a tesseract."
— Madeleine L'Engle, 1962
|
Non-Euclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads….  Tesseract
"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Hardening of the thought-arteries," Jane interjected. Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–" "Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–" "Poor kid," Jane said. Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–" "Blocks? What kind?" Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid." — "Mimsy Were the Borogoves," Lewis Padgett, 1943 |
For the intuitive basis of one type of non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…
Friedrich Froebel
(1782-1852), who
invented kindergarten.
His "third gift" —
© 2005 The Institute for Figuring|
OCODE
"The first credential
— Ezra Pound, "OCR is a field of research in pattern recognition, artificial intelligence and machine vision." — Wikipedia "I named this script ocode and chmod 755'd it to make it executable…" — Software forum post on the OCR program Tesseract
Wednesday, Dec. 3, 2008: 
|
"Like so many other heroes
who have seen the light
of a higher order…."
For further backstory,
click on the mouse.
From Braque's birthday, 2006:
"The senses deform, the mind forms. Work to perfect the mind. There is no certitude but in what the mind conceives."
— Georges Braque,
Reflections on Painting, 1917
Those who wish to follow Braque's advice may try the following exercise from a book first published in 1937:
For a different view
of the square and cube
see yesterday's entry
Abstraction and Faith.
“Put bluntly, who is kidding whom?”
— Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.
Good question.
Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —
 |
1. The performance of a work by
Richard Strauss,
“Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008
2. Headline of a music review
in today’s New York Times:
Welcoming a Fresh Season of
Transformation and Death
3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:
4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”
5. Symmetry from Plato to
the Four-Color Conjecture
7. Yesterday’s entry,
“Theories of Everything“
Coda:
as a tesseract.“
— Madeleine L’Engle
For a profile of
L’Engle, click on
the Easter eggs.
David Brooks in
today’s New York Times:
“The mind seems to have
the ability to transcend itself
and merge with a larger
presence that feels more real.”
|
Singing-Masters Come from the holy fire, perne in a gyre,
|
Last Sunday night (May 11),
Turner Classic Movies
showed a film featuring
Jimmy Durante as a
singing-master of
Frank Sinatra:
|
A Diploma for Frank from… The Old School
These little town blues…
“… all good things — trout as well as eternal salvation — come by grace and grace comes by art and art does not come easy.” — A River Runs Through It |
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