Log24

See Galois Ambiguity in this  journal.

"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010

Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—

Cover Illustration Number 1 (1976):

Cover Illustration Number 2 (1991):

   The Shrikhande Graph

______________________________________________________________________________

This post was prompted by two remarks…

1.  In a different weblog, also on August 26, 2010—

    The Accidental Mathematician— "The Girl Who Played with Fermat's Theorem."

"The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire….

Salander is fascinated by a theorem on perfect numbers—
one can verify it for as many numbers as one wishes, and it never fails!—
and then advances through 'Archimedes, Newton, Martin Gardner,*
and a dozen other classical mathematicians,' all the way to Fermat’s last theorem."

2.  "The fact that the pattern retains its symmetry when you permute the rows and columns
     is very well known to combinatorial theorists who work with matrices."
     [My italics; note resemblance to the Brualdi-Ryser title above.]

     –Martin Gardner in 1976 on the diamond theorem

* Compare Eric Temple Bell (as quoted at the MacTutor history of mathematics site)—

    "Archimedes, Newton, and Gauss, these three, are in a class by themselves
     among the great mathematicians, and it is not for ordinary mortals
     to attempt to range them in order of merit."

     This is from the chapter on Gauss in Men of Mathematics .

"… I know for sure that my best insights (those which 
are not just routine calculations) are pre-linguistic, and
I struggle to put them into words . . . ."

— Peter J. Cameron today

See also "George Steiner" + Language in this  journal.

A related figure —

    Click the above image for the IMDb page from which it is taken.

Peter J. Cameron today:

"Course material associated with Jack Edmonds’ lectures
can be found here."

See also High Concept (Jan. 21, 2015) —

Image from the end of the 2012 film "Travelling Salesman" 

— and another eye in a triangle …

From the Log24 post Prize (June 7, 2015)

An image that led off the year-end review yesterday in
the weblog of British combinatorialist Peter J. Cameron:

See also this  weblog's post final post of 2014,
with a rectangular array illustrating the six faces
of a die, and Cameron's reference yesterday to
a die-related post…

"The things on my blog that seem to be
of continuing value are the expository
series like the one on the symmetric group
(the third post in this series was reblogged
by Gil Kalai last month, which gave it a new
lease of life)…."

A tale from an author of Prague:

See also a passage quoted in this  weblog on the original
date of Cameron's Prague image, July 26, 2014 —

"The philosopher Graham Harman is invested in
re-thinking the autonomy of objects and is part 
of a movement called Object-Oriented-Philosophy
(OOP)." — From “The Action of Things,” a 2011
M.A. thesis at the Center for Curatorial Studies,
Bard College, by Manuela Moscoso 

— in the context of a search here for the phrase
     "structure of the object." An image from that search:

Continued from November 30, 2014

"Number right → Everything right." — Burkard Polster. 

See also the six  posts of November 30, St. Andrew's Day.

Related material —

Peter J. Cameron today discussing Julia Kristeva on poetry …

"This seems to be saying that the Kolmogorov
complexity of poetry is very low: the entire poem
can be generated from a small amount of information."

… and this  journal on St. Andrew's day :

From "A Piece of the Storm,"
by the late poet Mark Strand —

A snowflake, a blizzard of one….

"Take a cube, and write the numbers 1,…,6 on its faces.
Now the pairs of numbers on opposite faces
form a syntheme. (Standard dice, for example, represent
the syntheme 12|34|56.) "
— Peter J. Cameron, weblog post of May 11, 2010 

"For every kind of vampire, there is a kind of cross."
— Gravity's Rainbow

From a Sept. 17, 2010, post of Peter J. Cameron
that was linked to here at 8 AM ET today—

This should read instead…

Related material from this journal in 2003—

Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.

See Gowers and Cameron.

My own concern is more with the so-called "Non-Euclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).

A 1976 document relevant to the concerns in the Trudeau book—

Though not as well known as another document discussing
"self-evident" truths, Cameron's remarks are also of some
philosophical interest.

They apply to finite  geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations  
of mathematics.

"The hand of the creative artist, laid upon the major premise,
 rocks the foundations of the world." — Dorothy Sayers

Tue Oct 25, 2011 08:26 AM [London time]
from the weblog of Peter Cameron—

Today is Évariste Galois’ 200th birthday.

The event will be celebrated with the publication of a new transcription
and translation of Galois’ works (edited by Peter M. Neumann)
by the European Mathematical Society. The announcement is here.

Cameron's further remarks are also of interest.

The following was suggested by a link within this evening's earlier Kane site link.

Peter J. Cameron's weblog on August 26, 2010—

A Latin square  of order n  is a n × n  array with entries from the symbol set {1, 2, …, n }, such that each symbol occurs once in each row and once in each column. Now it is not hard to show that, up to permutations of the rows, columns and symbols, there are only two Latin squares of order 4:

 Some related literary remarks—

Proginoskes and Latin Squares.

See also "It was a perfectly ordinary night at Christ's high table…."

Two pictures suggested by recent comments on
Peter J. Cameron's Sept. 17 post about T.S. Eliot—

For some further background, see Symmetry of Walsh Functions.

Or— Childhood's Rear End

This post was suggested by…

1. Today's New York Times—
   "For many artists Electric Lady has become a home away from home…. For Jimmy Page the personal imprimaturs of Hendrix and Mr. Kramer made all the difference when Led Zeppelin mixed parts of 'Houses of the Holy' there in 1972."
2. The album cover pictures for "Houses of the Holy"
3. Boleskine House, home to Aleister Crowley and (occasionally) to Jimmy Page.

Related material:

The Zeppelin album cover, featuring rear views of nude children, was shot at the Giant's Causeway.

From a page at led-zeppelin.org—

See also Richard Rorty on Heidegger—

Safranski, the author of ''Schopenhauer and the Wild Years of Philosophy,'' never steps back and pronounces judgment on Heidegger, but something can be inferred from the German title of his book: ''Ein Meister aus Deutschland'' (''A Master From Germany''). Heidegger was, undeniably, a master, and was very German indeed. But Safranski's spine-chilling allusion is to Paul Celan's best-known poem, ''Death Fugue.'' In Michael Hamburger's translation, its last lines are:

death is a master from Germany his eyes are blue
he strikes you with leaden bullets his aim is true
a man lives in the house your golden hair Margarete
he sets his pack on us he grants us a grave in the air
he plays with the serpents and daydreams death is a master from Germany

your golden hair Margarete
your ashen hair Shulamith.

No one familiar with Heidegger's work can read Celan's poem without recalling Heidegger's famous dictum: ''Language is the house of Being. In its home man dwells.'' Nobody who makes this association can reread the poem without having the images of Hitler and Heidegger — two men who played with serpents and daydreamed — blend into each other. Heidegger's books will be read for centuries to come, but the smell of smoke from the crematories — the ''grave in the air'' — will linger on their pages.

Heidegger is the antithesis of the sort of philosopher (John Stuart Mill, William James, Isaiah Berlin) who assumes that nothing ultimately matters except human happiness. For him, human suffering is irrelevant: philosophy is far above such banalities. He saw the history of the West not in terms of increasing freedom or of decreasing misery, but as a poem. ''Being's poem,'' he once wrote, ''just begun, is man.''

For Heidegger, history is a sequence of ''words of Being'' — the words of the great philosophers who gave successive historical epochs their self-image, and thereby built successive ''houses of Being.'' The history of the West, which Heidegger also called the history of Being, is a narrative of the changes in human beings' image of themselves, their sense of what ultimately matters. The philosopher's task, he said, is to ''preserve the force of the most elementary words'' — to prevent the words of the great, houses-of-Being-building thinkers of the past from being banalized.

Related musical meditations—

Shine On (Saturday, April 21, 2007), Shine On, Part II, and Built (Sunday, April 22, 2007).

Related pictorial meditations—

The Giant's Causeway at Peter J. Cameron's weblog

and the cover illustration for Diamond Theory (1976)—

The connection between these two images is the following from Cameron's weblog today—

… as we saw, there are two different Latin squares of order 4;
one, but not the other, can be extended to a complete set
of 3 MOLS [mutually orthogonal Latin squares].

The underlying structures of the square pictures in the Diamond Theory cover are those of the two different Latin squares of order 4 mentioned by Cameron.

Connection with childhood—

The children's book A Wind in the Door, by Madeleine L'Engle. See math16.com. L'Engle's fantasies about children differ from those of Arthur C. Clarke and Led Zeppelin.

From a post of Peter J. Cameron today —

"… I want to consider the question: What is the role of the symmetric group in mathematics? "

Cameron's examples include, notably, parallelisms of lines in affine spaces over GF(2).

Where Entertainment is God, continued—

The following paragraphs are from a review by Piotr Siemion of Infinite Jest, a novel by David Foster Wallace. Illustrations have been added.

"Wallace was somehow able to twist together three yarns…. …there's a J.D Salinger for those who like J.D. Salinger. There's William Burroughs for those hardy souls who like some kick in their prose. And there's a dash of Kurt Vonnegut too. All three voices, though, are amplified in Infinite Jest beyond mere distortion and then projected onto Wallace's peculiar own three-ring circus….

… there's entertainment. Make it a capital E.

Illustration by Clint Eastwood
from Log24 post "E is for Everlast"

Infinite Jest revolves, among its many gyrations, around the story of the Entertainment, a film-like creation going by the title of 'Infinite Jest' and created shortly before his suicidal death by the young tennis star's father. The Entertainment's copies are now being disseminated clandestinely all over Wallace's funny America. Problem is, of course, that the film is too good. Anybody who gets to watch it becomes hooked instantly and craves only to watch it again, and again, and again, until the audience drops dead of exhaustion and hunger. Why eat when you're entertained by such a good movie? Wallace's premise brings you back to that apocryphal lab experiment in which rats were treated to a similar choice. When the rat pushed one button, marked FOOD, it would get a food pellet. The other button, marked FUN, would fire up an electrode rigged right into the orgasm center somewhere in the rat's cortex. Needless to add, one rat after another would drop dead from hunger, still twitching luridly and trying to finesse one last push of the button. Same thing in Wallace's story, especially that even those characters who have not seen the Entertainment yet, keep on entertaining themselves by different means."

The title of the Entertainment, "Infinite Jest," might also be applied to a BBC program featuring mathematician Peter J. Cameron. The program's actual title was "To Infinity and Beyond." It was broadcast the night of Feb. 10 (the date of this journal's previous post).

Few, however, are likely to find the Infinity program addictive. For closer approaches to Wallace's ideal Entertainment, see instead Dante (in the context of this journal's Feb. 4 posts on Cameron and the afterlife) and the BBC News.