I added links today in the following Wikipedia articles:
- Diamond theorem (disambiguation page)
- Miracle Octad Generator
- Binary Golay code
The links will probably soon be deleted,
but it seemed worth a try.
Monday, August 5, 2013
Wikipedia Updates
Comments Off on Wikipedia Updates
Saturday, July 13, 2013
Circles
A sort of poem
by Gauss and Weyl —
Click the circle for the context in Weyl's Symmetry .
For related remarks, see the previous post.
A literary excursus—
|
Brad Leithauser in a New Yorker post of July 11, 2013: If a poet determines that a poem should begin at point A and conclude at point D, say, the mystery of how to get there—how to pass felicitously through points B and C—strikes me as an artistic task both genuine and enlivening. There are fertile mysteries of transition, no less than of termination. And I’d like to suppose that Frost himself would recognize that any ingress into a poem is better than being locked out entirely. His little two-liner, “The Secret,” suggests as much: “We dance round in a ring and suppose / But the Secret sits in the middle and knows.” Most truly good poems might be said to contain a secret: the little sacramental miracle by which you connect, intimately, with the words of a total stranger. And whether you come at the poem frontward, or backward, or inside out—whether you approach it deliberately, word by word and line by line, or you parachute into it borne on a sudden breeze from the island of Serendip—surely isn’t the important thing. What matters is whether you achieve entrance into its inner ring, and there repose companionably beside the Secret. |
One should try, of course, to avoid repose in an inner circle of Hell .
Comments Off on Circles
Tuesday, July 9, 2013
Vril Chick
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 .
Comments Off on Vril Chick
Tuesday, July 2, 2013
Diamond Theorem Updates
My diamond theorem articles at PlanetMath and at
Encyclopedia of Mathematics have been updated
to clarify the relationship between the graphic square
patterns of the diamond theorem and the schematic
square patterns of the Curtis Miracle Octad Generator.
Comments Off on Diamond Theorem Updates
Tuesday, May 28, 2013
Codes
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):
Comments Off on Codes
Sunday, May 19, 2013
Priority Claim
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).
Comments Off on Priority Claim
Sunday, April 28, 2013
The Octad Generator
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M24 —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
Comments Off on The Octad Generator
Thursday, April 25, 2013
Rosenhain and Göpel Revisited
Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):
The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.
For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .
Comments Off on Rosenhain and Göpel Revisited
Saturday, April 6, 2013
Pascal via Curtis
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)
The 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirty-five 3-subsets of a 7-set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 354 Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines
(represented by 4-sets) that contain the 3-set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
Comments Off on Pascal via Curtis
Wednesday, February 13, 2013
Form:
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
- today's previous post on the Penrose diamond
- the remarks of Scott Aaronson on August 17, 2012
- the remarks in this journal on that same date
- the geometry of the 4×4 array in the context of M24.
Comments Off on Form:
Monday, December 24, 2012
All Over Again
"… the movement of analogy
begins all over once again."
See A Reappearing Number in this journal.
Illustrations:
Figure 1 —
Background: MOG in this journal.
Figure 2 —
Background —
Comments Off on All Over Again
Monday, November 19, 2012
Poetry and Truth
From today's noon post—
"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."
— Wallace Stevens at Bard College, March 30, 1951
Stevens also said at Bard that
"When Joan of Arc said:
Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.
these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."
There are those who would dispute this.
Some related material:
"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—
A poetic approach to geometry—
"A surface" and "the rock," from All Saints' Day, 2012—
— and from 1981—
Some mathematical background for poets in Purgatory—
"… the Klein correspondence underlies Conwell's discussion
of eight heptads. These play an important role in another
correspondence, illustrated in the Miracle Octad Generator
of R. T. Curtis, that may be used to picture actions
of the large Mathieu group M24."
Comments Off on Poetry and Truth
Speech
Speech by Wallace Stevens upon accepting
an honorary degree from Bard College in 1951—
Transcription of conclusion:
"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."
— Wallace Stevens at Bard College, March 30, 1951
Comments Off on Speech
Friday, August 17, 2012
Hidden
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)
Comments Off on Hidden
Monday, July 9, 2012
Taps for Fatso
A Story That Works
“There is the dark, eternally silent, unknown universe;
and lastly, there is lonely, story-telling, wonder-questing, – Fritz Leiber in “The Button Molder“ |
Comments Off on Taps for Fatso
Sunday, September 18, 2011
Midnight in Dostoevsky
"But Thou knewest not, it seems, that no sooner would man reject
miracle than he would reject God likewise, for he seeketh less
God than 'a sign' from Him." —The Grand Inquisitor
Update of 1:44 AM—
Seek and Ye Shall Find …
(Click images for larger context)
The Holy Office —
Comments Off on Midnight in Dostoevsky
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Saturday, August 6, 2011
Correspondences
Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, “Correspondances ”
From “A Four-Color Theorem”—
Figure 1
Note that this illustrates a natural correspondence
between
(A) the seven highly symmetrical four-colorings
of the 4×2 array at the left of Fig. 1, and
(B) the seven points of the smallest
projective plane at the right of Fig. 1.
To see the correspondence, add, in binary
fashion, the pairs of projective points from the
“points” section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)
A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—
Figure 2
| Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful. It yields, as shown, all of the 35 partitions of an 8-element set (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.
For some applications of the Curtis MOG, see |
Comments Off on Correspondences
Wednesday, July 6, 2011
Nordstrom-Robinson Automorphisms
A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—
"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."
— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22
A statement by Bierbrauer from 2004 has an error that doubles the above figure—
The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order
— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170
The error is corrected (though not detected) later in the same 2004 paper—
In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).
For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—
For some context, see the group of order 322,560 in Geometry of the 4×4 Square.
Comments Off on Nordstrom-Robinson Automorphisms
Sunday, June 19, 2011
The London Piracy Project
My work has been pirated by an artist in London.
An organization there, AND Publishing, sponsors what it calls
"The Piracy Project." The artist's piracy was a contribution
to the project.
The above material now reflects the following update:
|
UPDATE of June 21, 2011, 10:00 PM ET: The organization's weblog (a post for 19th June) In this weblog, changes have been made to correct my "AND Publishing is not sponsored by the art school. |
As this post originally stated…
The web pages from the site finitegeometry.org/sc that
the artist, Steve Richards, copied as part of his contribution to
the AND Publishing Piracy Project have had the author's name,
Steven H. Cullinane, and the date of composition systematically removed.
See a sample (jpg, 2.1 MB).
Here is some background on Richards.
Comments Off on The London Piracy Project
Abracadabra (continued)
Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator of R.T. Curtis.
A search tonight for related material on the Web yielded…
Weblog post by Steve Richards titled "The Search for Invariants:
The Diamond Theory of Truth, the Miracle Octad Generator
and Metalibrarianship." The artwork is by Steven H. Cullinane.
Richards has omitted Cullinane's name and retitled the artwork.
The author of the post is an artist who seems to be interested in the occult.
His post continues with photos of pages, some from my own work (as above), some not.
My own work does not deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.
The artist's post concludes with the following (note also the beginning of the preceding post)—
"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.
Comments Off on Abracadabra (continued)
Saturday, June 18, 2011
Ad Meld
Comments Off on Ad Meld
Sunday, June 5, 2011
Edifice Complex
"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."
— Wallace Stevens, "To an Old Philosopher in Rome"
The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…
"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology
The Kummer 166 Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)
For some background, see Configurations and Squares.
For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)
Related material— The Schwartz Notes of June 1.
Comments Off on Edifice Complex
Wednesday, June 1, 2011
The Schwartz Notes
A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)
The same search limited to results from only the past month yielded,
in addition, the following—
This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.
Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.
See Schwartz’s page 12, page 13, and page 14.
Compare to the web pages from today’s earlier search.
There are no references at the end of the Schwartz document,
but there is this at the beginning—
These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.
It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.
Comments Off on The Schwartz Notes
Thursday, May 26, 2011
Life’s Persistent Questions
This afternoon's online New York Times reviews "The Tree of Life," a film that opens tomorrow.
With disarming sincerity and daunting formal sophistication “The Tree of Life” ponders some of the hardest and most persistent questions, the kind that leave adults speechless when children ask them. In this case a boy, in whispered voice-over, speaks directly to God, whose responses are characteristically oblique, conveyed by the rustling of wind in trees or the play of shadows on a bedroom wall. Where are you? the boy wants to know, and lurking within this question is another: What am I doing here?
Persistent answers… Perhaps conveyed by wind, perhaps by shadows, perhaps by the New York Lottery.
For the nihilist alternative— the universe arose by chance out of nothing and all is meaningless— see Stephen Hawking and Jennifer Ouellette.
Update of 10:30 PM EDT May 26—
Today's NY Lottery results: Midday 407, Evening 756. The first is perhaps about the date April 7, the second about the phrase "three bricks shy"— in the context of the number 759 and the Miracle Octad Generator. (See also Robert Langdon and The Poetics of Space.)
Comments Off on Life’s Persistent Questions
Tuesday, May 24, 2011
Noncontinuous (or Non-Continuous) Groups
The web page has been updated.
An example, the action of the Mathieu group M24
on the Miracle Octad Generator of R.T. Curtis,
was added, with an illustration from a book cover—
Comments Off on Noncontinuous (or Non-Continuous) Groups
Wednesday, March 2, 2011
Labyrinth of the Line
“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”
“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”
Another single-line labyrinth—
Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—
Related material —
The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.
A search in the latter for miracle octad is updated below.
This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”
Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.
Comments Off on Labyrinth of the Line
Tuesday, January 25, 2011
For Your Consideration…
Early Nothing
Manohla Dargis on film director Fritz Lang in The New York Times (online Jan. 21, 2011, printed Jan. 23)—
"Hollywood endings can be beautiful fibs, but in Lang’s movies the glossy smiles and fade-outs feel forced. You can almost feel him pulling at them, trying to bring them back into the dark where they belong. The miracle of his Hollywood era is that, even when the screenplays tried to force his work in one direction, he managed to take them into richer, more complex realms with a style that was alternately baroque and stripped down and peopled with characters whose cynicism was earned. Every so often, though, he did strike screenwriting gold, notably in 'The Big Heat,' his 1953 crime masterwork. 'Say, I like this, early nothing,' a mink-swaddled Gloria Grahame says of a hotel room. Everyone really is a critic."
Comments Off on For Your Consideration…
Monday, October 25, 2010
The Embedding*
A New York Times "The Stone" post from yesterday (5:15 PM, by John Allen Paulos) was titled—
Stories vs. Statistics
Related Google searches—
"How to lie with statistics"— about 148,000 results
"How to lie with stories"— 2 results
What does this tell us?
Consider also Paulos's phrase "imbedding the God character." A less controversial topic might be (with the spelling I prefer) "embedding the miraculous." For an example, see this journal's "Mathematics and Narrative" entry on 5/15 (a date suggested, coincidentally, by the time of Paulos's post)—
* Not directly related to the novel The Embedding discussed at Tenser, said the Tensor on April 23, 2006 ("Quasimodo Sunday"). An academic discussion of that novel furnishes an example of narrative as more than mere entertainment. See Timothy J. Reiss, "How can 'New' Meaning Be Thought? Fictions of Science, Science Fictions," Canadian Review of Comparative Literature , Vol. 12, No. 1, March 1985, pp. 88-126. Consider also on this, Picasso's birthday, his saying that "Art is a lie that makes us realize truth…."
Comments Off on The Embedding*
Saturday, July 24, 2010
Sunday, July 4, 2010
Brightness at Noon (continued)
Today's sermon mentioned the phrase "Omega number."
Other sorts of Omega numbers— 24 and 759— occur
in connection with the set named Ω by R. T. Curtis in 1976—
— R. T. Curtis, "A New Combinatorial Approach to M24,"
Math. Proc. Camb. Phil. Soc. (1976), 79, 25-42
Comments Off on Brightness at Noon (continued)
Friday, June 11, 2010
Toward the Light
The title is a reference to yesterday's noon post.
For the late Vladimir Igorevich Arnold—
"All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical Mathematicks of the City of Heaven."
— Sir Thomas Browne, The Garden of Cyrus, Chapter V
Arnold's own mystical mathematics may be found in his paper
"Polymathematics: Is Mathematics a Single Science or a Set of Arts?"
Page 13–
"In mathematics we always encounter mysterious analogies, and our trinities [page 8] represent only a small part of these miracles."
Also from that paper—
Page 5, footnote 2–
"The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalization."
Arnold died in Paris on June 3. A farewell gathering was held there on June 8—
"Celles et ceux qui le souhaitent pourront donner un dernier adieu à Vladimir Igorevitch
mardi 8 juin, de 14h a 16h, chambre mortuaire de l'hopital Saint Antoine…."
An International Blue Diamond
In Arnold's memory— Here, in the Russian style, is a link to a "first nontrivial case" of a blue diamond— from this journal on June 8 (feast of St. Gerard Manley Hopkins). For those who prefer French style, here is a link to a blue diamond from May 18—
From French cinema—
"a 'non-existent myth' of a battle between
goddesses of the sun and the moon
for a mysterious blue diamond
that has the power to make
mortals immortal and vice versa"
Comments Off on Toward the Light
Saturday, May 15, 2010
Mathematics and Narrative continued…
Comments Off on Mathematics and Narrative continued…
Friday, May 14, 2010
Competing MOG Definitions
A recently created Wikipedia article says that “The Miracle Octad Generator [MOG] is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24-dimensional space….” (Clearly any array with 24 parts is so capable.) The article ignores the fact that the MOG, as defined by R.T. Curtis in 1976, is not an array of coordinates, but rather a picture of a correspondence between two sets, each containing 35 structures. (As a later commentator has remarked, this correspondence is a well-known one that preserves a certain incidence property. See Eightfold Geometry.)
From the 1976 paper defining the MOG—
“There is a correspondence between the two systems of 35 groups, which is illustrated in Fig. 4 (the MOG or Miracle Octad Generator).” —R.T. Curtis, “A New Combinatorial Approach to M24,” Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42
Curtis’s 1976 Fig. 4. (The MOG.)
The Wikipedia article, like a similar article at PlanetMath, is based on a different definition, from a book first published in 1988—
I have not seen the 1973 Curtis paper, so I do not know whether it uses the 35-sets correspondence definition or the 6×4 array definition. The remarks of Conway and Sloane on page 312 of the 1998 edition of their book about “Curtis’s original way of finding octads in the MOG [Cur2]” indicate that the correspondence definition was the one Curtis used in 1973—
Here the picture of “the 35 standard sextets of the MOG”
is very like (modulo a reflection) Curtis’s 1976 picture
of the MOG as a correspondence between two 35-sets.
A later paper by Curtis does use the array definition. See “Further Elementary Techniques Using the Miracle Octad Generator,” Proceedings of the Edinburgh Mathematical Society (1989) 32, 345-353.
The array definition is better suited to Conway’s use of his hexacode to describe octads, but it obscures the close connection of the MOG with finite geometry. That connection, apparent in the phrases “vector space structure in the standard square” and “parallel 2-spaces” (Conway and Sloane, third ed., p. 312, illustrated above), was not discussed in the 1976 Curtis paper. See my own page on the MOG at finitegeometry.org.
Comments Off on Competing MOG Definitions
Wednesday, April 28, 2010
Eightfold Geometry
Related web pages:
Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square
Related folklore:
"It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6
The Miracle Octad Generator may be regarded as illustrating the folklore.
Update of August 20, 2010–
For facts rather than folklore about the above bijection, see The Moore Correspondence.
Comments Off on Eightfold Geometry
Thursday, August 6, 2009
Thursday August 6, 2009
A Fisher of Men
Update: The above image was added
at about 11 AM ET Aug. 8, 2009.
Update: The above image was added
at about 11 AM ET Aug. 8, 2009.
From a webpage of the First United Methodist Church of Bloomington, Indiana–
Dr. Joe Emerson, April 24, 2005–
— Text: I Peter 2:1-9
Dr. Emerson falsely claims that the film "On the Waterfront" was based on a book by the late Budd Schulberg (who died yesterday). (Instead, the film's screenplay, written by Schulberg– similar to an earlier screenplay by Arthur Miller, "The Hook"– was based on a series of newspaper articles by Malcolm Johnson.)
"The movie 'On the Waterfront' is once more in rerun. (That’s when Marlon Brando looked like Marlon Brando. That’s the scary part of growing old when you see what he looked like then and when he grew old.) It is based on a book by Budd Schulberg."
Emerson goes on to discuss the book, Waterfront, that Schulberg wrote based on his screenplay–
"In it, you may remember a scene where Runty Nolan, a little guy, runs afoul of the mob and is brutally killed and tossed into the North River. A priest is called to give last rites after they drag him out."
New York Times today
Dr. Emerson flunks the test.
Dr. Emerson's sermon is, as noted above (Text: I Peter 2:1-9), not mainly about waterfronts, but rather about the "living stones" metaphor of the Big Fisherman.
My own remarks on the date of Dr. Emerson's sermon—
Those who like to mix mathematics with religion may regard the above 4×6 array as a context for the "living stones" metaphor. See, too, the five entries in this journal ending at 12:25 AM ET on November 12 (Grace Kelly's birthday), 2006, and today's previous entry.
Comments Off on Thursday August 6, 2009
Wednesday, May 20, 2009
Wednesday May 20, 2009
From Quilt Blocks to the
Mathieu Group M24
Mathieu Group M24
Click on illustrations for details.
The connection:
The four-diamond figure is related to the finite geometry PG(3,2). (See "Symmetry Invariance in a Diamond Ring," AMS Notices, February 1979, A193-194.) PG(3,2) is in turn related to the 759 octads of the Steiner system S(5,8,24). (See "Generating the Octad Generator," expository note, 1985.)
The relationship of S(5,8,24) to the finite geometry PG(3,2) has also been discussed in–
- "A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177-186.
Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."
- "A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223-228.
Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."
For the connection of S(5,8,24) with the Mathieu group M24, see the references in The Miracle Octad Generator.
Comments Off on Wednesday May 20, 2009
Tuesday, May 19, 2009
Tuesday May 19, 2009
Exquisite Geometries
"By far the most important structure in design theory is the Steiner system
— "Block Designs," 1995, by Andries E. Brouwer
"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."
— The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)
"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."
The 1931 paper of Carmichael is now available online from the publisher for $10.
Comments Off on Tuesday May 19, 2009
Tuesday, January 6, 2009
Tuesday January 6, 2009
Archetypes, Synchronicity,
and Dyson on Jung
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson's 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung's theory of archetypes:
"… we do not need to accept Jung’s theory as true in order to find it illuminating."
The same is true of Jung's remarks on synchronicity.
For example —
Yesterday's entry, "A Wealth of Algebraic Structure," lists two articles– each, as it happens, related to Jung's four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis's 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis's 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 1984-2003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
In Chapter 3, "Sutures of Structures," Taylor asks —
"What, then, is a frame, and what is frame work?"
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 "frame of reference" found in the above Cambridge University Press articles.
"Examples are the stained-glass
windows of knowledge."
— Vladimir Nabokov
windows of knowledge."
— Vladimir Nabokov
Comments Off on Tuesday January 6, 2009
Monday, January 5, 2009
Monday January 5, 2009
A Wealth of
Algebraic Structure
A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
— Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.
(Published online by Cambridge University Press 19 Dec 2008.)
In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)
For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):
A new combinatorial approach to M24, by R. T. Curtis. Abstract:
“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”
(Received June 15 1974)
— Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.
(Published online by Cambridge University Press 24 Oct 2008.)
* For instance:
Click for details.
Comments Off on Monday January 5, 2009
Saturday, December 27, 2008
Monday, November 24, 2008
Monday November 24, 2008
Frame Tale
Click on image for details.
Comments Off on Monday November 24, 2008
Saturday, October 25, 2008
Saturday October 25, 2008
Actual Being
The New York Times Book Review online today has a review by Sam Tanenhaus of a new John Updike book.
The title of the review (not the book) is "Mr. Wizard."
"John Updike is the great genial sorcerer of American letters. His output alone (60 books, almost 40 of them novels or story collections) has been supernatural. More wizardly still is the ingenuity of his prose. He has now written tens of thousands of sentences, many of them tiny miracles of transubstantiation whereby some hitherto overlooked datum of the human or natural world– from the anatomical to the zoological, the socio-economic to the spiritual– emerges, as if for the first time, in the completeness of its actual being."
Rolling Stone interview with Sting, February 7, 1991:
"'I was brought up in a very strong Catholic community,' Sting says. 'My parents were Catholic, and in the Fifties and Sixties, Catholicism was very strong. You know, they say, "Once a Catholic, always a Catholic." In a way I'm grateful for that background. There's a very rich imagery in Catholicism: blood, guilt, death, all that stuff.' He laughs."
RS 597, Feb. 7, 1991
Last night's 12:00 AM
Log24 entry:
From this date six years ago:
From this morning's newspaper,
a religious meditation I had not
seen last night:
Related material:
Juneteenth through
Midsummer Night, 2007and
Comments Off on Saturday October 25, 2008
Thursday, July 31, 2008
Thursday July 31, 2008
Symmetry in Review
“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 —
 |
Related material:
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.
Comments Off on Thursday July 31, 2008
Wednesday, July 2, 2008
Wednesday July 2, 2008
Sacerdotal Jargon
|
Wallace Stevens, from
"Credences of Summer" in Transport to Summer (1947):
"Three times the concentred
In memory of the former
Till Summer folds her miracle — |
Symbols of the
thrice concentred self:
The circular symbol is from July 1.
The square symbol is from June 24,
the date of death for the former
first lady of Brazil.
Wallace Stevens quotes Paul Klee:
"'… what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow…."
— "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)
Comments Off on Wednesday July 2, 2008
Monday, October 1, 2007
Monday October 1, 2007
Bright as Magnesium
"Definitive"
— The New York Times,
Sept. 30, 2007, on
Blade Runner:
The Final Cut
Institute for Advanced Study, Princeton, N.J.—
"The art historian Kirk Varnedoe died on August 14, 2003, after a long and valiant battle with cancer. He was 57. He was a faculty member in the Institute for Advanced Study’s School of Historical Studies, where he was the fourth art historian to hold this prestigious position, first held by the German Renaissance scholar Erwin Panofsky in the 1930s."
"His final lecture was an eloquent, prophetic flight of free association….
Varnedoe chose to introduce his final lecture with the less-quoted last words of the android Roy Batty (Rutger Hauer) in Ridley Scott's film Blade Runner: 'I've seen things you people wouldn't believe– attack ships on fire off the shoulder of Orion, bright as magnesium; I rode on the back decks of a blinker and watched C-beams glitter in the dark near the Tannhauser Gate. All those moments will be lost in time, like tears in the rain. Time to die.'"
Comments Off on Monday October 1, 2007
Thursday, June 28, 2007
Thursday June 28, 2007
Christianus
Cornelius Uhlenbeck
Cornelius Uhlenbeck
Oct. 18, 1866 – Aug. 12, 1951
“… born at Voorburg near The Hague in Holland, and studied philology at the University of Leiden…. Though he would actually have preferred to graduate in Basque, Uhlenbeck in 1888, when only 22 years old, took his doctor’s degree in Dutch. It must be here noted that for this degree the requirements in comparative philology were very considerable….” —International Journal of American Linguistics, Jan. 1953
From Uhlenbeck’s A Manual of Sanskrit Phonetics (1898):
From Uhlenbeck’s A Manual of Sanskrit Phonetics (1898):
“The Indogermanic family of languages. The great family of languages, in which Sanskrit belongs, is called the Indogermanic, Indoceltic or Aryan…. The word Indogermanic dates from a time, when it was not yet proved, that the Celtic dialects also make part of our family of languages, and indicates by the combined name of the utmost branches, Indian and Germanic, the whole territory of speech, to which they belong. Now that it is certain, that Celtic also is a member of our family, it would be accurate to replace the word Indogermanic by Indoceltic, because not Germanic, but Celtic is the utmost branch in the Occident. The name Indogermanic however is generally adopted and it would be impossible to supplant it by another. By the word Aryan is generally understood a certain subdivision of the Indogermanic family, viz. the Indo-Iranian, and therefore it would seem unsuitable to use this name also for the whole Indogermanic family.”
An unsuitable Santa:
A Santa understudy:
Transcript of
“Miracle on 34th Street”—
KRIS: Bye. Merry Christmas!
Well, young lady,
what's your name?
MOTHER: I'm sorry.
She doesn't speak English.
She's Dutch. She just came over.
She's been living
in an orphans home...
in Rotterdam ever since...
We've adopted her.
I told her you wouldn't
be able to speak to her...
but when she saw you
in the parade yesterday...
she said you were
"Sinter Claes"...
and you could talk to her.
I didn't know what to do.
KRIS: Hello. [Speaking Dutch]
[Speaking Dutch]
[Singing in Dutch]
DORIS: Now do you understand?
Related material:
Pope Approves Wider
Use of Latin Mass,
(Click on image for details),
and
Comments Off on Thursday June 28, 2007
Sunday, June 10, 2007
Sunday June 10, 2007
Like a Melody
An excerpt from
The Miracle of the Bells
quoted in
A Mass for Lucero—
The Miracle of the Bells
quoted in
A Mass for Lucero—
"'A pretty girl–
is like a melody—- !'
But that was always
Bill Dunnigan's
Song of Victory….
Thus thought the…
press agent for
'The Garden of the Soul.'"
"Ay que bonito es volar
A las dos de la mañana…."
— "La Bruja"
For a rendition by
Salma Hayek, click
on the picture below.
Related material:
Log24 entries for
May 18, 2007.
Comments Off on Sunday June 10, 2007
Wednesday, May 30, 2007
Wednesday May 30, 2007
Al Gore and the
Absence of Truth
"Evil is a negation, because
it is the absence of truth."
— Mary Baker Eddy,
founder of Christian Science,
in Science and Health
with Key to the Scriptures,
(Boston, 1906,
page 186, line 11)
M. Scott Peck on evil:
"There are quite popular
systems of thought these days,
such as Christian Science
or the Course in Miracles,
which define evil as unreality.
It is a half-truth. The spirit of evil
is one of unreality, but it itself
is real. It really exists."
"We must not fall back into Saint
Augustine's now discarded doctrine
of the 'privatio boni,' whereby evil
was defined as the absence of good.
Satan's personality cannot be
characterized simply by
an absence, a nothingness."
— People of the Lie:
The Hope for Healing Human Evil,
by Morgan Scott Peck, 1986.
(Touchstone paperback,
2nd ed., 1998, page 208)
Al Gore on M. Scott Peck:
|
Al Gore trains a global army – USATODAY.com
"Peck wrote that 'Evil is the absence
— www.usatoday.com/news/nation/ |
He did?
"The greatest trick the Devil ever pulled
was convincing the world he didn't exist."
— Verbal Kint in "The Usual Suspects"
Comments Off on Wednesday May 30, 2007
Monday, May 28, 2007
Monday May 28, 2007
and a Finite Model
Notes by Steven H. Cullinane
May 28, 2007
Part I: A Model of Space-Time
The following paper includes a figure illustrating Penrose’s model of “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”
 |
Click on picture to enlarge.
For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.
Part II: A Corresponding Finite Model
The Klein quadric also occurs in a finite model of projective 5-space. See a 1910 paper:
G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.
Conwell discusses the quadric, and the related Klein correspondence, in detail. This is noted in a more recent paper by Philippe Cara:
 |
As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads. These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M24.
Related material:
The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.
The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.
There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube. This correspondence leads to a natural way to generate the affine group AGL(6,2). This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.
“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game. Elder Brother laughed. ‘Go ahead and try,’ he exclaimed. ‘You’ll see how it turns out. Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”
— Hermann Hesse, The Glass Bead Game,
translated by Richard and Clara Winston
Saturday, May 26, 2007
Saturday May 26, 2007
A Baffled Reader
A reader this morning commented on my first Xanga entry (July 20, 2002):
“To set one up (which I have not done because I don’t want anyone to know what I think),” … William Safire regarding “blogs”.
I still don’t know what you think. Yet … I try, try, try.
Here’s one thing that I think– today– based on my “Hate Speech for Harvard,” on “Devil in the Details” (Log24, May 18 and 23), and, more recently, on
- last evening’s PA lottery number 005,
- the I Ching hexagram of the same number, and
- the New Yorker issue linked to at the end of the previous entry:
Revised version of the
New Yorker cover of 5/21/07
Commentary on the cover
by the PA lottery
on 5/25/07
in the form of the
evening number, 005.
In the I Ching, this
is the number of
by the PA lottery
on 5/25/07
in the form of the
evening number, 005.
In the I Ching, this
is the number of
HSU:
WAITING
(NOURISHMENT)
See also the previous entry
and Natalie Angier’s sneer
at a politician’s call for
prayer, which, she
said, involved the
“assumption that prayer is
some sort of miracle
Vicks VapoRub.”
Detail from the
5/21/07 New Yorker:
THE IMAGE
Clouds rise up to heaven:
The image of WAITING.
Thus the superior man
eats and drinks,
Is joyous and
of good cheer.
AMEN.
Comments Off on Saturday May 26, 2007
Friday, May 25, 2007
Friday May 25, 2007
Dance and the Soul
From Log24 on
this date last year:
"May there be an ennui
of the first idea?
What else,
prodigious scholar,
should there be?"
— Wallace Stevens,
"Notes Toward a
Supreme Fiction"
The Associated Press,
May 25, 2007–
Thought for Today:
"I hate quotations.
Tell me what you know."
— Ralph Waldo Emerson
This "telling of what
I know" will of course
mean little to those
who, like Emerson,
have refused to learn
through quotations.
For those less obdurate
than Emerson —Harold Bloom
on Wallace Stevens
and Paul Valery's
"Dance and the Soul"–
"Stevens may be playful, yet seriously so, in describing desire, at winter's end, observing not only the emergence of the blue woman of early spring, but seeing also the myosotis, whose other name is 'forget-me-not.' Desire, hearing the calendar hymn, repudiates the negativity of the mind of winter, unable to bear what Valery's Eryximachus had called 'this cold, exact, reasonable, and moderate consideration of human life as it is.' The final form of this realization in Stevens comes in 1950, in The Course of a Particular, in the great monosyllabic line 'One feels the life of that which gives life as it is.' But even Stevens cannot bear that feeling for long. As Eryximachus goes on to say in Dance and the Soul:
A cold and perfect clarity is a poison impossible to combat. The real, in its pure state, stops the heart instantaneously….[…] To a handful of ashes is the past reduced, and the future to a tiny icicle. The soul appears to itself as an empty and measurable form. –Here, then, things as they are come together, limit one another, and are thus chained together in the most rigorous and mortal* fashion….
O Socrates, the universe cannot for one instant endure to be only what it is.
Valery's formula for reimagining the First Idea is, 'The idea introduces into what is, the leaven of what is not.' This 'murderous lucidity' can be cured only by what Valery's Socrates calls 'the intoxication due to act,' particularly Nietzschean or Dionysiac dance, for this will rescue us from the state of the Snow Man, 'the motionless and lucid observer.'" —Wallace Stevens: The Poems of Our Climate
* "la sorte… la plus mortelle":
mortal in the sense
"deadly, lethal"
Other quotations
(from March 28,
the birthday of
Reba McEntire):
Logical Songs
Logical Song I
(Supertramp)
"When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they'd be singing so happily,
Joyfully, playfully watching me"
Logical Song II
(Sinatra)
"You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I'm such a happy in-
dividual….
You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots"
Comments Off on Friday May 25, 2007
Wednesday, March 28, 2007
Wednesday March 28, 2007
Logical Songs
Logical Song I
(Supertramp)
“When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they’d be singing so happily,
Joyfully, playfully watching me”
Logical Song II
(Sinatra)
“You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I’m such a happy in-
dividual….
You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots“
Comments Off on Wednesday March 28, 2007
Wednesday, February 28, 2007
Wednesday February 28, 2007
Elements
of Geometry
The title of Euclid’s Elements is, in Greek, Stoicheia.
From Lectures on the Science of Language,
by Max Muller, fellow of All Souls College, Oxford.
New York: Charles Scribner’s Sons, 1890, pp. 88-90 –
Stoicheia
“The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.
Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row….
Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:–
- stix, gen. stichos, a row, a line of soldiers
- stichos, a row, a line; distich, a couplet
- steicho, estichon, to march in order, step by step; to mount
- stoichos, a row, a file; stoichein, to march in a line
In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount….
Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”
Example:
The Miracle Octad Generator of R. T. Curtis
For the geometry of these stoicheia, see
The Smallest Perfect Universe and
Finite Geometry of the Square and Cube.
Comments Off on Wednesday February 28, 2007
Saturday, February 3, 2007
Saturday February 3, 2007
Catholic Schools Week,
Jan. 28 – Feb. 3, 2007,
concludes:
Jan. 28 – Feb. 3, 2007,
concludes:
For 373, see
Miracle.
For 401, see 4/01:
April 1 at Noon.
“Feel lucky?
Well, do you?”
Comments Off on Saturday February 3, 2007
Tuesday, December 5, 2006
Tuesday December 5, 2006
Today in History
(via The Associated Press)
|
On this date (Dec. 5): In 1776, the first scholastic fraternity in America, Phi Beta Kappa, was organized at the College of William and Mary in Williamsburg, Va. In 1791, composer Wolfgang Amadeus Mozart died in Vienna, Austria, at age 35. In 2006, author Joan Didion is 72. |
“We tell ourselves stories in order to live….
We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the ‘ideas’ with which we have learned to freeze the shifting phantasmagoria which is our actual experience.
Or at least we do for a while. I am talking here about a time when I began to doubt the premises of all the stories I had ever told myself, a common condition but one I found troubling.”
An Alternate History
(based on entries of
the past three days):
“A FAMOUS HISTORIAN:
England, 932 A.D. —
A kingdom divided….”
|
A Story That Works
|
Comments Off on Tuesday December 5, 2006
Tuesday, October 3, 2006
Tuesday October 3, 2006
Serious
"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."
— Charles Matthews at Wikipedia, Oct. 2, 2006
"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."
— G. H. Hardy, A Mathematician's Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
Comments Off on Tuesday October 3, 2006
Saturday, July 29, 2006
Saturday July 29, 2006
Big Rock
Thanks to Ars Mathematica, a link to everything2.com:
“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”
Another example:
Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis. See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:
The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.
This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts. (See the diamond theorem.)
This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.
For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.
For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.
Comments Off on Saturday July 29, 2006
Sunday, May 7, 2006
Sunday May 7, 2006
Bagombo Snuff Box
(in memory of
Burt Kerr Todd)
“Well, it may be the devil
or it may be the Lord
But you’re gonna have to
serve somebody.”
— “Bob Dylan”
(pseudonym of Robert Zimmerman),
quoted by “Bob Stewart”
on July 18, 2005
“Bob Stewart” may or may not be the same person as “crankbuster,” author of the “Rectangular Array Theorem” or “RAT.” This “theorem” is intended as a parody of the “Miracle Octad Generator,” or “MOG,” of R. T. Curtis. (See the Usenet group sci.math, “Steven Cullinane is a Crank,” July 2005, messages 51-60.)
“Crankbuster” has registered at Math Forum as a teacher in Sri Lanka (formerly Ceylon). For a tall tale involving Ceylon, see the short story “Bagombo Snuff Box” in the book of the same title by Kurt Vonnegut, who has at times embodied– like Martin Gardner and “crankbuster“– “der Geist, der stets verneint.”
Here is my own version (given the alleged Ceylon background of “crankbuster”) of a Bagombo snuff box:
Comments Off on Sunday May 7, 2006
Friday, April 28, 2006
Friday April 28, 2006
Exercise
Review the concepts of integritas, consonantia, and claritas in Aquinas:
"For in respect to beauty three things are essential: first of all, integrity or completeness, since beings deprived of wholeness are on this score ugly; and [secondly] a certain required design, or patterned structure; and finally a certain splendor, inasmuch as things are called beautiful which have a certain 'blaze of being' about them…."
— Summa Theologiae Sancti Thomae Aquinatis, I, q. 39, a. 8, as translated by William T. Noon, S.J., in Joyce and Aquinas, Yale University Press, 1957
Review the following three publications cited in a note of April 28, 1985 (21 years ago today):
(1) Cameron, P. J.,
Parallelisms of Complete Designs,
Cambridge University Press, 1976.
(2) Conwell, G. M.,
The 3-space PG(3,2) and its group,
Ann. of Math. 11 (1910) 60-76.
(3) Curtis, R. T.,
A new combinatorial approach to M24,
Math. Proc. Camb. Phil. Soc.
79 (1976) 25-42.
Discuss how the sextet parallelism in (1) illustrates integritas, how the Conwell correspondence in (2) illustrates consonantia, and how the Miracle Octad Generator in (3) illustrates claritas.
Comments Off on Friday April 28, 2006
Wednesday, April 12, 2006
Wednesday April 12, 2006
Eternal
Franklin Delano Roosevelt:
“Eternal truths will be neither true nor eternal unless they have fresh meaning for every new social situation.”
— AP, Today in History,
apparently quoted from an address
at the University of Pennsylvania,
Sept. 20, 1940
Related material:
Gravity’s Rainbow, the beginning of page 373*:
“white and geometric capital before the destruction”
Gravity’s Rainbow, the end of page 373*:
“Slothrop was going into high school when FDR was starting out in the White House. Broderick Slothrop professed to hate the man, but young Tyrone thought he was brave.”
See also the Log24 entry
for Dec. 20, 2003 —
White, Geometric, and Eternal —
and the entry for 8 PM on
the feast of St. John Paul II —
Miracle, April 2, 2006.
for Dec. 20, 2003 —
White, Geometric, and Eternal —
and the entry for 8 PM on
the feast of St. John Paul II —
Miracle, April 2, 2006.
Comments Off on Wednesday April 12, 2006
Sunday, April 2, 2006
Sunday April 2, 2006
Miracle
Looking for a Miracle:
The Beatification of John Paul II
Background:
| Preface:
Part I: Part II: |
Today’s lottery in the
State of Grace
(Kelly, of Philadelphia)–
Mid-day: 008
Evening: 373.
Done.
Comments Off on Sunday April 2, 2006
Wednesday, February 8, 2006
Wednesday February 8, 2006
For Grammy Night
From A Mass for Lucero:
“To the two gods of art, Apollo and Dionysus, we owe our recognition that… there is a tremendous opposition, as regards both origins and aims, between the Apolline art of the sculptor and the non-visual, Dionysiac art of music.”
— The Birth of Tragedy, by Friedrich Nietzsche, Penguin, 1993, page 14
“Melody, then, is both primary and universal.” (Author’s italics)
— Nietzsche, op. cit., page 33
“…in so far as he interprets music in images, he himself lies amidst the peaceful waves of Apolline contemplation….”
— Nietzsche, op. cit., page 35
From The Miracle of the Bells, by Russell Janney, Prentice-Hall, 1946, page 333–
“He was singing softly:
‘A pretty girl–
is like a melody—- !‘
But that was always
Bill Dunnigan’s
Song of Victory….
Thus thought the…
press agent for
‘The Garden of the Soul.'”
Comments Off on Wednesday February 8, 2006
Tuesday, January 3, 2006
Tuesday January 3, 2006
See also Step One —
“Happy Six,”*
and Step Two —
“Then a Miracle Occurs.”**
“Happy Six,”*
and Step Two —
“Then a Miracle Occurs.”**
The miracle occurred on the
Feast of the Transfiguration,
August 6, 1945, at 8:15 AM
in Hiroshima, Japan.
* This Jan. 1 entry links to a paper by Robert T. Curtis from The Arabian Journal for Science and Engineering (King Fahd University, Dhahran, Saudi Arabia), Volume 27, Number 1A, January 2002.
** This Jan. 2 entry discusses Einstein as, according to the New York Times, “a moral and even spiritual sage.”
Comments Off on Tuesday January 3, 2006
Monday, January 2, 2006
Monday January 2, 2006
Then a Miracle Occurs
The New York Times on Sunday,
New Year’s Day, 2006,
by John Horgan–
“Einstein Has Left the Building“–
“Down the hall from my office, Albert Einstein’s electric-haired visage beams from a poster for the ‘World Year of Physics 2005.’ The poster celebrates the centennial of the ‘miraculous year’ when a young patent clerk in Bern, Switzerland, revolutionized physics with five papers on relativity, quantum mechanics and thermodynamics. ‘Help make 2005 another Miraculous Year!’ the poster exclaims…. As 2005 wound down with no miracles in sight, the poster took on an increasingly poignant cast, like a Kerry/Edwards bumper sticker….”
From a debate:
KERRY: “I’m going to be a president who believes in science.”
KERRY: “I’m a Catholic – raised a Catholic. I was an altar boy. Religion has been a huge part of my life, helped lead me through a war, leads me today.”
BUSH: “Trying to decipher that.”
Click on “religion” in this quote to find out what Einstein really meant.
Here’s a bumper sticker for Horgan:
More from Horgan’s New Year’s Day sermon:
“We revere [Einstein] not only as a scientific genius but also as a moral and even spiritual sage….”
“What you mean ‘we,’ blue man?”
— The Red States
Wednesday, December 14, 2005
Wednesday December 14, 2005
From Here
to Eternity
to Eternity
For Loomis Dean
See also
For Rita Moreno
on Her Birthday
(Dec. 11, 2005)
|
Los Angeles Times
Tuesday, Dec. 13, 2005
OBITUARIES 
LOOMIS DEAN Loomis Dean, 88; By Jon Thurber, Times Staff Writer
Loomis Dean, a Life magazine photographer who made memorable pictures of the royalty of both Europe and Hollywood, has died. He was 88.
Dean died Wednesday [December 7, 2005] at Sonoma Valley Hospital in Sonoma, Calif., of complications from a stroke, according to his son, Christopher. In a photographic career spanning six decades, Dean's leading images included shirtless Hollywood mogul Darryl F. Zanuck trying a one-handed chin-up on a trapeze bar, the ocean liner Andrea Doria listing in the Atlantic and writer Ernest Hemingway in Spain the year before he committed suicide. One of his most memorable photographs for Life was of cosmopolitan British playwright and composer Noel Coward in the unlikely setting of the Nevada desert. Dean shot 52 covers for Life, either as a freelance photographer or during his two stretches as a staffer with the magazine, 1947-61 and 1966-69. After leaving the magazine, Dean found steady freelance work in magazines and as a still photographer on film sets, including several of the early James Bond movies starring Sean Connery. Born in Monticello, Fla., Dean was the son of a grocer and a schoolteacher. When the Dean family's business failed during the Depression, they moved to Sarasota, Fla., where Dean's father worked as a curator and guide at the John and Mable Ringling Museum of Art. Dean studied engineering at the University of Florida but became fascinated with photography after watching a friend develop film in a darkroom. He went off to what is now the Rochester Institute of Technology, which was known for its photography school. After earning his degree, Dean went to work for the Ringling circus as a junior press agent and, according to his son, cultivated a side job photographing Ringling's vast array of performers and workers. He worked briefly as one of Parade magazine's first photographers but left after receiving an Army Air Forces commission during World War II. During the war, he worked in aerial reconnaissance in the Pacific and was along on a number of air raids over Japan. His first assignment for Life in 1946 took him back to the circus: His photograph of clown Lou Jacobs with a giraffe looking over his shoulder made the magazine's cover and earned Dean a staff job. In the era before television, Life magazine photographers had some of the most glamorous work in journalism. Life assigned him to cover Hollywood. In 1954, the magazine published one of his most memorable photos, the shot of Coward dressed for a night on the town in New York but standing alone in the stark Nevada desert. Dean had the idea of asking Coward, who was then doing a summer engagement at the Desert Inn in Las Vegas, to pose in the desert to illustrate his song "Mad Dogs and Englishmen Go Out in the Midday Sun." As Dean recalled in an interview with John Loengard for the book "Life Photographers: What They Saw," Coward wasn't about to partake of the midday sun. "Oh, dear boy, I don't get up until 4 o'clock in the afternoon," Dean recalled him saying. But Dean pressed on anyway. As he related to Loengard, he rented a Cadillac limousine and filled the back seat with a tub loaded with liquor, tonic and ice cubes — and Coward. The temperature that day reached 119 as Coward relaxed in his underwear during the drive to a spot about 15 miles from Las Vegas. According to Dean, Coward's dresser helped him into his tuxedo, resulting in the image of the elegant Coward with a cigarette holder in his mouth against his shadow on the dry lake bed. "Splendid! Splendid! What an idea! If we only had a piano," Coward said of the shoot before hopping back in the car and stripping down to his underwear for the ride back to Las Vegas. In 1956, Life assigned Dean to Paris. While sailing to Europe on the Ile de France, he was awakened with the news that the Andrea Doria had collided with another liner, the Stockholm. The accident occurred close enough to Dean's liner that survivors were being brought aboard. His photographs of the shaken voyagers and the sinking Andrea Doria were some of the first on the accident published in a U.S. magazine. During his years in Europe, Dean photographed communist riots and fashion shows in Paris, royal weddings throughout Europe and noted authors including James Jones and William S. Burroughs. He spent three weeks with Hemingway in Spain in 1960 for an assignment on bullfighting. In 1989, Dean published "Hemingway's Spain," about his experiences with the great writer. In 1965, Dean won first prize in a Vatican photography contest for a picture of Pope Paul VI. The prize included an audience with the pope and $750. According to his son, it was Dean's favorite honor. In addition to his son, he is survived by a daughter, Deborah, and two grandsons. Instead of flowers, donations may be made to the American Child Photographer's Charity Guild (www.acpcg.com) or the Make-A-Wish Foundation. |
Related material:
The Big Time
(Log 24, July 29, 2003):
|
A Story That Works
|
Comments Off on Wednesday December 14, 2005
Wednesday, November 30, 2005
Wednesday November 30, 2005
For St. Andrew’s Day
“The miraculous enters…. When we investigate these problems, some fantastic things happen….”
— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)
The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.
A picture of the Miracle Octad Generator, with my comments, is available online.
Related material:
Mathematics and Narrative.
Comments Off on Wednesday November 30, 2005
Thursday, November 17, 2005
Thursday November 17, 2005
All the King’s Men
(See also Time and
All the King’s Horses.)
LEAR:
Now you better do some thinkin’
then you’ll find
You got the only daddy
that’ll walk the line.
FOOL:
I’ve always been different
with one foot over the line….
I’ve always been crazy
but it’s kept me from going insane.
For related material, see
and last night’s winner of
the National Book Award
for nonfiction, i.e.,
“all hard facts, all reality, with
no illusions and no fantasy.”
|
A Story That Works
|
Comments Off on Thursday November 17, 2005
Monday, August 22, 2005
Monday August 22, 2005
The Hole
Part I: Mathematics and Narrative
Apostolos Doxiadis on last month's conference on "mathematics and narrative"–
Doxiadis is describing how talks by two noted mathematicians were related to
"… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "
Doxiadis says both talks had "a big hole in the middle."
"Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts. [In one part] I’ll tell you a few things about proofs. [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
"And the hole?"
"This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’ But the similarity was not clearly explained: the hole, you see, the ‘miracle.’ Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
Part II: Possible Worlds
"At times, bullshit can only be countered with superior bullshit."
— Norman Mailer
Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:
"The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide….
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
"Il faut cultiver notre jardin."– Voltaire
Background:
Modal Logic in Wikipedia
Possible Worlds in Wikipedia
Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)
Part III: Modal Theology
"The lapis was thought of as a unity and therefore often stands for the prima materia in general."
"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
'…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
— Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)
"The lapis was thought of as a unity and therefore often stands for the prima materia in general."
— Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)
"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."
— The Stars My Destination, by Alfred Bester, 1956 (Vintage hardcover, July 1996, p. 216)
|
"We symbolize
logical necessity with the box  )and logical possibility with the diamond  ).
"The possibilia that exist,
— Michael Sudduth, |
Comments Off on Monday August 22, 2005
Tuesday, April 26, 2005
Tuesday April 26, 2005
The Ring of Falsehood
In memory of Philip Morrison, bombmaker,
Morrison
Scientific American columnist,
pioneer of the
Search for Extraterrestrial Intelligence (SETI)
and author of
The Ring of Truth.
From The Measure of a Life:
Does religion play a role in attitudes toward ETIs? Philip Morrison gave his considered opinion… “Well, it might, but I think that it’s just one of the permissive routes; it isn’t an essential factor. My parents were Jewish. Their beliefs were conventional but not very deep. They belonged to the Jewish community; they went to services infrequently, on special occasions—funerals and high holidays”….
Although Sagan did not believe in God, he nevertheless said this about SETI’s importance… “It touches deeply into myth, folklore, religion, mythology; and every human culture in some way or another has wondered about that type of question. It’s one of the most basic questions there is.” In fact, in Sagan’s novel/film Contact, described by Keay Davidson as “one of the most religious science-fiction tales ever written”… Ellie discovers that pi—the ratio of the circumference of a circle to its diameter—is numerically encoded in the cosmos and this is proof that a super-intelligence designed the universe…
The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle—another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe.
|
|
See also yesterday's entry Mathematical Style.
Extra credit: |
Wednesday, August 18, 2004
Wednesday August 18, 2004
Drunk Bird
T. Charles Erickson
Shizuo Kakutani
in the 1980’s
“A drunk man will find his way home, but a drunk bird may get lost forever.”
— Shizuo Kakutani, quoted by J. Chang in Stochastic Processes (ps), p. 1-19. Chang says the quote is from an R. Durrett book on probability.
Meaning:
A random walk in d dimensions is recurrent if d = 1 or d = 2, but transient if d is greater than or equal to 3.
From a web page on Kylie Minogue:
Turns out she’s a party girl
who loves Tequila:
“Time disappears with Tequila.
It goes elastic, then vanishes.”
Kylie sings
“Locomotion”
From a web page on Malcolm Lowry’s classic novel Under the Volcano:
The day begins with Yvonne’s arrival at the Bella Vista bar in Quauhnahuac. From outside she hears Geoffrey’s familiar voice shouting a drunken lecture this time on the topic of the rule of the Mexican railway that requires that “A corpse will be transported by express!” (Lowry, Volcano, p. 43).
For further literary details in memory of Shizuo Kakutani, Yale mathematician and father of book reviewer Michiko Kakutani, see
Of course, Kakutani himself would probably prefer the anti-Santa, Michael Shermer. For a refutation of Santa by this high priest of Scientism, see
(Scientific American, July 26, 2004).
Sunday, April 11, 2004
Sunday April 11, 2004
Good Friday and
Descartes’s Easter Egg
“The use of z, y, x . . . to represent unknowns is due to René Descartes, in his La géometrie (1637)…. In a paper on Cartesian ovals, prepared before 1629, x alone occurs as unknown…. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown.”
— Florian Cajori, A History of Mathematical Notations. 2 volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929. (Vol. 1, page 381)
This is from
http://members.aol.com/jeff570/variables.html.
Descartes’s Easter Egg is found at
EggMath: The Shape of an Egg —
Cartesian Ovals.
An Easter Meditation
on Humpty Dumpty
The following is excerpted from a web page headed “Catholic Way.” It is one of a series of vicious and stupid Roman Catholic attacks on Descartes. Such attacks have been encouraged by the present Pope, who today said “may the culture of life and love render vain the logic of death.”
The culture of life and love is that of the geometry (if not the philosophy) of Descartes. The logic of death is that of Karol Wojtyla, as was made very clear in the past century by the National Socialist Party, which had its roots in Roman Catholicism.
|
Humpty Dumpty sat on a wall. “In the century just completed, the human race found itself in a position not unlike the scrambled mess at the base of an imaginary English wall…. … we are heirs to a humanity that is broken, fractured, confused, unsure of what to make of itself…. … ‘postmodernism’ is merely the articulation of the fractured, dissipated state of the human being…. Without relating a history of modern philosophy, our unfortunate human shell has suffered a continual fragmentation for a period of roughly 500 years. (You philosophers out there will recognize immediately that I am referring to the legacy of René Descartes.) And this fragmentation has been a one-way street: one assault after another on the integrity and dignity of the human person until you have, well, the 20th Century. But now it’s the 21st Century. The beauty … the marvel … the miracle of our time is the possibility that gravity will reverse itself: Humpty Dumpty may be able, once again, to assume his perch.” — Ted Papa, |
Voilà.
The upper part
of the above icon
is from EggMath.
For the lower part,
see Good Friday.
Comments Off on Sunday April 11, 2004
Tuesday, January 6, 2004
Tuesday January 6, 2004
720 in the Book
Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.
An article titled On Mathematical Imagination concludes by looking forward to
“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”
Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.
Hmm.
The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:
Abel’s Proof: An Essay
on the Sources and Meaning
of Mathematical Unsolvability
by Peter Pesic,
MIT Press, 2003
From a review:
“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….
Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations. The reader is left with little clarity on this sequel to the story….”
— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244
Here, it seems, is my epiphany:
“Elliptic modular functions suffice to solve all polynomial equations.”
Incidental Remarks
on Synchronicity,
Part I
Those who seek a star
on this Feast of the Epiphany
may click here.
Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.
Just how such equations can be solved is a less familiar story. I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.
The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions. Modular functions are also distantly related, via the topic of “moonshine” and via the “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.
Incidental Remarks
on Synchronicity,
Part II
There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.
Here is what I was able to find on the Web about Pesic’s claim:
From Wolfram Research:
From Solving the Quintic —
“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”
From Siegel Theta Function —
“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”
From Polynomial —
“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable. Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”
Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.
King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.
Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999.
From Angel Zhivkov,
Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:
“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker: in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function…. Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist. This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions. In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”
— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants“
Incidental Remarks
on Synchronicity,
Part III
From Music for Dunne’s Wake:
“Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”
— Carrie Fisher,
Postcards from the Edge
|
|
“720 in |
“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.
Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.
For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see
For the relevance of the time
of this entry, 10:10, see
|
Related recreational reading:
|
Labyrinth |
|
Comments Off on Tuesday January 6, 2004
Friday, January 2, 2004
Friday January 2, 2004
“Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”
— Carrie Fisher,
Postcards from the Edge
|
|
“720 in |
Musical Note: A Star is Born
Natalie Wood played a six-year-old
in “Miracle on 34th Street,”
six factorial equals 720,
and Wood was born on 7/20, 1938.
“How I love music.”
— John O’Hara, Hope of Heaven, 1938
For related metaphors, see
Immortal Diamond,
The Diamond Archetype, and
the first log24.net entry…
for July 20, 2002.
Friday January 2, 2004
What, and Give Up Show Biz?
"Dying is easy. Comedy is hard."
— Saying attributed to Edmund Gwenn, star of "Miracle on 34th Street," and also attributed to "Noel Coward, David Garrick, William Holden, Edmund Kean, Marcel Marceau, Groucho Marx, and Oscar Wilde."
See also yesterday's entry on the Dark Lady. For more on Santa and the Dark Lady, see my archive for Aug.-Sept. 2002.
"Drink up, sweet. You gotta go some. How I love music. Frère Jacques, Cuernavaca, ach du lieber August. All languages. A walking Berlitz. Berlitz sounds like you with that champagne, my sweet, or how you're gonna sound."
— Hope of Heaven, by John O'Hara,
"another acidic writer to whom he
[John Gregory Dunne]
was often compared"
(Adam Bernstein, Washington Post)
For some context for the Hope of Heaven quotation, see Immortal Diamond: O'Hara, Hopkins, and Joyce, or click on the adding machine in yesterday's entry.
For more on miracles and the afterlife, see my archive for September 2002.
Comments Off on Friday January 2, 2004
Friday, August 22, 2003
Friday August 22, 2003
Mr. Holland’s Week
On Monday, August 18, 2003,
a New York Times editor wrote
the following headline
for a book review:
Bending Over Backward
for a Well-Known Lout.
The word “lout” here refers to
author John O’Hara, who often
wrote about his native Pennsylvania.
On Thursday, August 21, 2003,
the Pennsylvania Lottery
midday number was
162.
For some other occurrences of this number,
see my entries of August 19, written
in honor of the birthday of
Jill St. John.
The “three days” remark referred to above
is from another St. John (2:19), allegedly
the author of an account of the last days
of one Jesus of Nazareth.
Those who share Mel Gibson’s
taste for religious drama may
savor the following dialogue:
|
Dramatis Personae:
Narrator: Those who had been healed did not join in with the throng at Jesus’ crucifixion who cried, “Crucify Him, crucify Him.” ….
Voice of Doom: It was a different story for the guilty ones who had fled from the presence of Jesus. Group 1: The priests and rulers never forgot the feeling of guilt they felt that moment in the temple. Group 2: The Holy Spirit flashed into their minds the prophets’ writings concerning Christ. Would they yield to this conviction? Voice of Doom: Nope! They would have to repent first! They would not admit that they were wrong! They knew that they were dead wrong. But they would not repent of it! And because Jesus had discerned their thoughts, they hated Him. With hate in their hearts they slowly returned to the temple. Voice of Hope: They could not believe their eyes when they saw the people being healed and praising God! These guilty ones were convicted that in Jesus the prophecies of the Messiah were fulfilled. As much as they hated Jesus, they could not free themselves from the thought that He might be a prophet sent by God to restore the sacredness of the temple. Voice of Doom: So they asked Him a stupid question! “What miracle can you perform to show us that you have the right to do what you did?” Voice of Jesus: “Destroy this temple and in three days I will build it again.” Voice of Doom: Those guys couldn’t believe it! |
Philosophers ponder the idea of identity:
what it is to give something a name
on Monday
and have it respond to that name
on Friday….
— Bernard Holland, The New York Times,
Monday, May 20, 1996
“Ask a stupid question…”
For further details, see
Comments Off on Friday August 22, 2003
Thursday, July 31, 2003
Thursday July 31, 2003
Killer Radio
"See the girl with the diamond ring?
She knows how to shake that thing."
— Jerry Lee "Killer" Lewis on
KHYI 95.3 FM, Plano, Texas,
at about 5:12 PM EDT 7/31/03,
introduced by DJ Allen Peck Sr.
"And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."
— Blaise Pascal, De l'Esprit Géométrique
"When the light came she was sitting on the bed beside an open suitcase, toying with her diamond rings. She saw the light first in the depths of the largest stone."
— Paul Preuss, Broken Symmetries,
scene at Diamond Head, Oahu, Hawaii
Now playing (6:41 PM EDT) on Killer Radio:
"Jack of Diamonds, that's
a hard card to find."
"This Jack, joke, poor potsherd, patch, matchwood, immortal diamond…."
— Gerard Manley Hopkins, Society of Jesus
Perhaps Sam Phillips was twanged by a Hawaiian guitar. (See previous two entries.)
|
The Big Time
|
See also "Top Ten Most Overheard Comments by new KHYI listeners" at Miss Lana's Anything Page, entry for
Comments Off on Thursday July 31, 2003
Wednesday, July 30, 2003
Wednesday July 30, 2003
Transcendental Meditation
|
Transcendental Man |
|
The bicentennial of Ralph Waldo Emerson was on May 25, 2003. For a commemoration of Emerson on that date, click on the picture below of Harvard University’s Room 305, Emerson Hall.
This will lead you to a discussion of the properties of a 5×5 array, or matrix, with a symbol of mystical unity at its center. Although this symbol of mystical unity, the number “1,” is not, pace the Shema, a transcendental number, the matrix is, as perhaps a sort of Emersonian compensation, what postmodernists would call phallologocentric. It is possible that Emerson is a saint; if so, his feast day (i.e., date of death), April 27, might reveal to us the sort of miraculous fact hoped for by Fritz Leiber
in my previous entry. A check of my April 27 notes shows us, lo and behold, another phallologocentric 5×5 array, this one starring Warren Beatty. This rather peculiar coincidence is, perhaps, the sort of miracle appropriate to a saint who is, as this week’s politically correct New Yorker calls him, a Big Dead White Male.
Leiber’s fiction furnishes “a behind-the-scenes view of the time change wars.”
“It’s quarter to three…” — St. Frank Sinatra
Comments Off on Wednesday July 30, 2003
Tuesday, July 29, 2003
Tuesday July 29, 2003
The Big Time
|
|
“The place outside the cosmos where I and my pals do our nursing job I simply call the Place. A lot of my nursing consists of amusing and humanizing Soldiers fresh back from raids into time. In fact, my formal title is Entertainer….” — The Big Time, |
A Story That Works
- “There is the dark, eternally silent, unknown universe;
- there are the friend-enemy minds shouting and whispering their tales and always seeking the three miracles —
- that minds should really touch, or
- that the silent universe should speak, tell minds a story, or (perhaps the same thing)
- that there should be a story that works, that is all hard facts, all reality, with no illusions and no fantasy;
- and lastly, there is lonely, story-telling, wonder-questing, mortal me.”
— Fritz Leiber in “The Button Molder“
Comments Off on Tuesday July 29, 2003
Monday, May 19, 2003
Monday May 19, 2003
DAY OF THE MOTHER SHIP
Part II: A Mighty Wind
I just saw the John Travolta film “Phenomenon” for the first time. (It was on the ABC Family Channel from 8 to 11.)
Why is it that tellers of uplifting stories (like Zenna Henderson, in “Day of the Mother Ship, Part I,” or the authors of “Phenomenon” or the Bible) always feel they have to throw in some cockamamie and obviously false miracles to hold people’s attention?
On May 11 (Mother’s Day), Mother Nature got my attention with a mighty wind waving the branches of nearby trees, just before a tornado watch was issued for the area I was in. This made me recall a Biblical reference I had come across in researching references to “Our Lady of the Woods” for my Beltane (May 1) entry.
…And his heart was moved, and the heart of his people, as the trees of the wood are moved with the wind.
This is what I thought of on May 11 watching branches swaying in the wind on Mother’s Day — which some might regard as a festival of Our Lady of the Woods. John Travolta in “Phenomenon” sees a very similar scene partway through the picture; then, at the end, explains to his girlfriend how the swaying branches made him feel — without mentioning the branches — by asking her to describe how she would cradle and rock a child in her arms. At the very end of the film, she herself is reminded of his question by the swaying branches of another tree.
Events like these are miracle enough for me.
Monday, April 28, 2003
Monday April 28, 2003
ART WARS:
Toward Eternity
April is Poetry Month, according to the Academy of American Poets. It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."
Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:
"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "
Consider the following journal entries from April 7, 2003:
|
Math Awareness Month April is Math Awareness Month.
|
|
An Offer He Couldn't Refuse Today's birthday: Francis Ford Coppola is 64.
From a note on geometry of April 28, 1985:
|
The "horse's head" figure above is from a note I wrote on this date 18 years ago. The following journal entry from April 4, 2003, gives some details:
|
The Eight Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight. Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight. Consider eight rectangular cells arranged in an array of four rows and two columns. Let us label these cells with coordinates, then apply a permutation.
The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight. This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact. It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month. The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions. See also "Pieces of Eight," by Robert L. Griess. |
Comments Off on Monday April 28, 2003
Friday, April 4, 2003
Friday April 4, 2003
The Eight
Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight. Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight. Consider eight rectangular cells arranged in an array of four rows and two columns. Let us label these cells with coordinates, then apply a permutation.
|
|
|
|
|
The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight. This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact. It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month. (See the 4:36 PM entry.)
The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions. See also "Pieces of Eight," by Robert L. Griess.
Comments Off on Friday April 4, 2003
Sunday, September 29, 2002
Sunday September 29, 2002
New from Miracle Pictures
– IF IT’S A HIT, IT’S A MIRACLE! –
Pi in the Sky
for Michaelmas 2002
“Fear not, maiden, your prayer is heard.
Michael am I, guardian of the highest Word.”
Contact, by Carl Sagan:
Chapter 1 – Transcendental Numbers
In the seventh grade they were studying “pi.” It was a Greek letter that looked like the architecture at Stonehenge, in England: two vertical pillars with a crossbar at the top. If you measured the circumference of a circle and then divided it by the diameter of the circle, that was pi. At home, Ellie took the top of a mayonnaise jar, wrapped a string around it, straightened the string out, and with a ruler measured the circle’s circumference. She did the same with the diameter, and by long division divided the one number by the other. She got 3.21. That seemed simple enough.
The next day the teacher, Mr. Weisbrod, said that pi was about 22/7, about 3.1416. But actually, if you wanted to be exact, it was a decimal that went on and on forever without repeating the pattern of numbers. Forever, Ellie thought. She raised her hand. It was the beginning of the school year and she had not asked any questions in this class.
“How could anybody know that the decimals go on and on forever?”
“That’s just the way it is,” said the teacher with some asperity.
“But why? How do you know? How can you count decimals forever?”
“Miss Arroway” – he was consulting his class list – “this is a stupid question. You’re wasting the class’s time.”
No one had ever called Ellie stupid before and she found herself bursting into tears….
After school she bicycled to the library at the nearby college to look through books on mathematics. As nearly as she could figure out from what she read, her question wasn’t all that stupid. According to the Bible, the ancient Hebrews had apparently thought that pi was exactly equal to three. The Greeks and Romans, who knew lots of things about mathematics, had no idea that the digits in pi went on forever without repeating. It was a fact that had been discovered only about 250 years ago. How was she expected to know if she couldn’t ask questions? But Mr. Weisbrod had been right about the first few digits. Pi wasn’t 3.21. Maybe the mayonnaise lid had been a little squashed, not a perfect circle. Or maybe she’d been sloppy in measuring the string. Even if she’d been much more careful, though, they couldn’t expect her to measure an infinite number of decimals.
There was another possibility, though. You could calculate pi as accurately as you wanted. If you knew something called calculus, you could prove formulas for pi that would let you calculate it to as many decimals as you had time for. The book listed formulas for pi divided by four. Some of them she couldn’t understand at all. But there were some that dazzled her: pi/4, the book said, was the same as 1 – 1/3 + 1/5 – 1/7 + …, with the fractions continuing on forever. Quickly she tried to work it out, adding and subtracting the fractions alternately. The sum would bounce from being bigger than pi/4 to being smaller than pi/4, but after a while you could see that this series of numbers was on a beeline for the right answer. You could never get there exactly, but you could get as close as you wanted if you were very patient. It seemed to her
a miracle
that the shape of every circle in the world was connected with this series of fractions. How could circles know about fractions? She was determined to learn
The book said something else: pi was called a “transcendental” number. There was no equation with ordinary numbers in it that could give you pi unless it was infinitely long. She had already taught herself a little algebra and understood what this meant. And pi wasn’t the only transcendental number. In fact there was an infinity of transcendental numbers. More than that, there were infinitely more transcendental numbers that ordinary numbers, even though pi was the only one of them she had ever heard of. In more ways than one, pi was tied to infinity.
She had caught a glimpse of something majestic.
Chapter 24 – The Artist’s Signature
The anomaly showed up most starkly in Base 2 arithmetic, where it could be written out entirely as zeros and ones. Her program reassembled the digits into a square raster, an equal number across and down. Hiding in the alternating patterns of digits, deep inside the transcendental number, was a perfect circle, its form traced out by unities in a field of noughts.
The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover
— another circle, drawn kilometers downstream of the decimal point. There would be richer messages farther in. It doesn’t matter what you look like, or what you’re made of, or where you come from. As long as you live in this universe, and have a modest talent for mathematics, sooner or later you’ll find it. It’s already here. It’s inside everything. You don’t have to leave your planet to find it. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons… there is an intelligence that antedates the universe. The circle had closed. She found what she had been searching for.
Song lyric not in Sagan’s book:
Will the circle be unbroken
by and by, Lord, by and by?
Is a better home a-waitin’
in the sky, Lord, in the sky?
“Contact,” the film:
 |
 |
|
Recording: |
Columbia 37669 |
|
Date Issued: |
Unknown |
|
Side: |
A |
|
|
|
|
Title: |
Can The Circle Be Unbroken |
|
Artist: |
Carter Family |
|
Recording Date: |
May 6, 1935 |
|
Listen: |
Realaudio |
|
|
|
The above conclusion to Sagan’s book is perhaps the stupidest thing by an alleged scientist that I have ever read. As a partial antidote, I offer the following.
| Today’s birthday: Stanley Kramer, director of “On the Beach.”
From an introduction to a recording of the famous Joe Hill song about Pie in the Sky: “They used a shill to build a crowd… You know, a carny shill.” |
|
Friday, September 6, 2002
Friday September 6, 2002
Santa’s Wit
Edmund Gwenn, actor, died on September 6, 1959.
When asked if he thought dying was tough, Gwenn reportedly said,
“Yes, it’s tough, but not as tough as doing comedy.”
This may or may not be true; if it is, Gwenn may be the true source of a quotation variously attributed to Edmund Kean, Edwin Booth, David Garrick, Donald Wolfit, William Holden, and Groucho Marx, Marcel Marceau, Noel Coward, and Oscar Wilde:
“Dying is easy. Comedy is hard.”
A very dubious version of the Gwenn story attributes the “comedy is hard” part to Jack Lemmon:
The lesson is best illustrated in a story involving Jack Lemmon, whose best work was in comedy. He visited the British actor Edmund Gwenn, suffering in a hospital. Gwenn is said to have lifted the flap on the oxygen tent and said, ”It’s really tough to die.” And Lemmon responded, ”It’s not as tough as doing comedy.”
— Elvis Mitchell in The New York Times Week in Review, Sunday, August 25, 2002
David Bruce, an English instructor at Ohio University, supplies another version of the Gwenn story, from Movie Anecdotes, by Peter Hay. New York: Oxford University Press, 1990:
Edmund Gwenn won an Oscar playing Santa Claus in the movie Miracle on 34 Street. As he lay dying, Jack Lemmon visited him and asked if dying was dead. [sic] Gwenn replied, “Oh, it’s hard, very hard indeed. But not as hard as doing comedy.”
Santa might appreciate the above misprint, as would Vladimir Nabokov…
“Life Everlasting–based on a misprint!”
— Pale Fire
and John Donne…
“And death shall be no more, Death, thou shalt die.”
— Holy Sonnets
Comments Off on Friday September 6, 2002
Saturday, July 20, 2002
Saturday July 20, 2002
|
|
||||
|
Example:
|
||||
|
|
||||
|
|
Initial Xanga entry. Updated Nov. 18, 2006.