The first 12 pages of my 1976 preprint "Diamond Theory" are
now scanned and uploaded. See a slideshow.
For downloading, all 12 pages are combined in a PDF.
The first 12 pages of my 1976 preprint "Diamond Theory" are
now scanned and uploaded. See a slideshow.
For downloading, all 12 pages are combined in a PDF.
A figure related to the general connecting theorem of Koen Thas —
See also posts tagged Dirac and Geometry in this journal.
Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."
William Grimes in The New York Times this evening —
"Clifford Irving, who perpetrated one of the biggest literary hoaxes
of the 20th century in the early 1970s when he concocted a
supposedly authorized autobiography of the billionaire Howard Hughes
based on meetings and interviews that never took place, died on Tuesday
at a hospice facility near his home in Sarasota, Fla. He was 87."
A figure reproduced here on Tuesday —
A related figure —
See too the 1973 Orson Welles film "F for Fake."
Some background on the second figure above —
posts tagged April 811, 2016.
Some background on the first figure above —
today's previous post, January 2018 AMS Notices.
The word "mythologem" on page 55 of The Burning Fountain
by Philip Wheelwright, revised edition of 1968 (p. 91 in the 1954
edition), suggests a Web search for that word. It was notably often
used in the 1998 English translation of a book by Eleazar Meletinsky
first published in Russian in 1976 —
Meletinsky reportedly died on December 17, 2005.
In his memory, Log24 posts from that date are now tagged Mythologem Day.
"And we may see the meadow in December,
icy white and crystalline" — Johnny Mercer
From the American Mathematical Society homepage today —
From concinnitasproject.org —
"Concinnitas is the title of a portfolio of fine art prints. . . .
The portfolio draws its name from a word famously used
by the Renaissance scholar, artist, architect, and philosopher
Leon Battista Alberti (14041472) to connote the balance of
number, outline, and position (in essence, number, geometry,
and topology) that he believed characterize a beautiful work of art."
The favicon of the Concinnitas Project —
The structure of the Concinnitas favicon —
This structure is from page 15 of
"Diamond Theory," a 1976 preprint —
The page preceding that of yesterday's post Wheelwright and the Wheel —
See also a Log24 search for
"Four Quartets" + "Four Elements".
A graphic approach to this concept:
"The Bounded Space" —
"The Fire, Air, Earth, and Water" —
The previous post suggests a review of
the philosophical concept of universals —
A part of the abovementioned 2011 "Saturday evening's post" that is
relevant to the illustration at the end of today's previous post —
Note the whatness of Singer's dagger definitions —
Two readings by James Parker —
From next year's first Atlantic issue
From last month's Atlantic issue
"Let’s return to that hillside where Clayton exited his Mercedes.
In the gray light, he climbs the pasture. Halfway up the slope,
three horses are standing: sculpturally still, casually composed
in a perfect triptych of horsitude."
— James Parker in The Atlantic , Nov. 2017 issue
Logosrelated material
The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com —
For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .
See also Symplectic in this journal.
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 5979 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
The above symplectic figure appears in remarks on
the diamondtheorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of linear (or line ) complex
in the finite projective space PG(3,2) —
Part I: Black Magician
"Schools of criticism create their own canons, elevating certain texts,
discarding others. Yet some works – Malcolm Lowry’s Under the Volcano
is one of them – lend themselves readily to all critical approaches."
— Joan Givner, review of
A Darkness That Murmured: Essays on Malcolm Lowry and the Twentieth Century
by Frederick Asals and Paul Tiessen, eds.
The AsalsTiessen book (U. of Toronto Press, 2000) was cited today
by Margaret Soltan (in the link below) as the source of this quotation —
"When one thinks of the general sort of snacky
underearnest writers whose works like windchimes
rattle in our heads now, it is easier to forgive Lowry
his pretentious seriousness, his oldfashioned ambitions,
his Proustian plans, [his efforts] to replace the reader’s
consciousness wholly with a black magician’s."
A possible source, Perle Epstein, for the view of Lowry as black magician —
Part II: Mythos and Logos
Part I above suggests a review of Adam Gopnik as black magician
(a figure from Mythos ) —
Tuesday, November 7, 2017
Polarities and Correlation

— and of an opposing figure from Logos ,
Paul B. Yale, in the references below:
In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —
Update from the morning of December 2 —
The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:
.
See as well Tsimtsum in this journal.
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss, 1976
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
From a Log24 post of June 2627, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.
Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053.
http://scholarworks.sjsu.edu/etd_theses/4053.
See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.
The Matrix —
The Grid —
^{ Picturing the Witt Construction —}
"Read something that means something." — New Yorker ad
Bernd Sturmfels to Receive 2018
George David Birkhoff Prize in Applied Mathematics
— American Mathematical Society on
Monday, November 20th, 2017
See also Sturmfels and Birkhoff + Geometry in this journal.
This is a sequel to yesterday's post Cube Space Continued.
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 19842003 (Oct. 24, 2008).
Sunday, October 29, 2017
File System… Unlocked

See as well Chloë Grace Moretz portraying a schoolgirl problem.
From Cambridge Core, suggested by a reference to
that website in the previous post and by the following
bibliographic data . . .
https://doi.org/10.1017/fmp.2016.5
Downloaded from https://www.cambridge.org/core
on 10 Nov 2017 at 19:06:19
See Conwell + Princeton in this journal.
Related art —
Figures from a search in this journal for Springer Knight
and from the All Souls' Day post The Trojan Pony —
For those who prefer pure abstraction to the quasifigurative
1985 sevencycle above, a different 7cycle for M_{24} , from 1998 —
Compare and contrast with my own "knight" labeling
of a 4row 2column array (an M_{24} octad, in the system
of R. T. Curtis) by the 8 points of the projective line
over GF(7), from 2008 —
From a search in this journal for Springer Knight —
Related material from Academia —
See also Log24 posts from the above "magic" date,
December 4, 2014, now tagged The Pony Argument.
From Tony Phillips's American Mathematical Society column
for November 2017 —
" It is significant that the authors chose to place this announcement
of their results not in a mathematics journal but in one aimed at a
much larger scientific audience; their writing is appropriately
expository, especially in the introduction. Nature itself ran an
assessment of the paper in their 'News and Views' section,
October 4: 'Mathematics: A pariah finds a home,' by Terry Gannon.
Gannon sets the stage, again in terms suitable for wide consumption,
and sketches out the story. He ends 'It is always difficult to gauge
the importance of a mathematical result without the hindsight
that many years brings. Nevertheless, Duncan et al. have shown us
a door. Whether it is to a new closet, house or world, we cannot yet say,
but the results are certainly unexpected, and no one will think of
the pariahs in the same way again.' "
See as well Log24 on the above date — Text and Context.
The previous post's Lewis Carroll cover,
modified to illustrate Plato's diamond —
See also "To Forge
a Head" (Oct. 27).
The passage from Lewis Carroll's Euclid and His Modern Rivals
in the previous post suggests two illustrations —
Click the Trudeau book for related Log24 posts.
Logo from the above webpage —
See also the similar structure of the eightfold cube, and …
Related dialogue from the new film "Unlocked" —
1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.
1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.
A sequel to the post CP is for Consolation Prize (Sept. 3, 2016)
An image from Log24 on this date last year:
A recent comment on a discussion of CP symmetry —
The New Yorker on the recent film "The Square" —
"It’s an aesthetic that presents,
so to speak, just the facts,
as if the facts themselves weren’t
deeply layered with living history
and crisscrossed with vectors
of divergent ideas and ideals."
— Richard Brody, Thursday, Oct. 26, 2017
For other images deeply layered and crisscrossed ,
see Geometry of the I Ching.
The title was suggested by a 2014 Vanity Fair piece
by James Toback (Harvard '66).
"He squinted at this vision of a Qualityless world for a while,
conjured up more details, thought about it, and then squinted
some more and thought some more and then finally circled
back to where he was before.
Squareness.
That's the look. That sums it. Squareness. When you subtract
quality you get squareness. Absence of Quality is the essence
of squareness."
— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance
And when you add quality?
A related Zen joke from Final Club (June 19, 2017) —
^{.}
This post was suggested by a New York Times obituary this evening —
"Tom Mathews, Promoter of Liberal Causes and Candidates, Dies at 96."
Mathews reportedly died on October 14, 2017.
"Mr. Mathews and his business partner Roger Craver 'dreamed for years
of finding the perfect citizencandidate,' the authors wrote, 'a man or
woman of the centerleft with a feel for issues, a history of independence,
a winning television manner and, most important of all, a center — a core
of beliefs more important to him or her than getting elected.'
Dream on.
From the date of Mathews's death:
Posts now tagged A Center for Krauss —
From Stanford — The death on October 9, 2017, of a man who
"always wanted to be at the most cutting of cuttingedge technology."
Related material from Log24 on April 26, 2017 —
A sketch, adapted from Girl Scouts of Palo Alto —
Click the sketch for further details.
The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017 —
A visually related concept — See Solomon's Cube in this journal.
Chronologically related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem —
Click for some background —
Another approach, for Dan Brown fans —
In the following passage, Brown claims that an eightray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy." Brown may have first encountered this symbol at a
questionable "Sacred Science" website. Wikipedia discusses
some even less respectable uses of the symbol.
"Category theory has become the central gateway
through which to learn pure mathematics."
— David Spivak, Harvard Math Table, Oct. 24, 2017
— The New Yorker , issue of October 23, 2017
See as well posts tagged Death Warmed Over.
The elementary shapes at the top of the figure below mirror
the lookingglass property of the classical Lo Shu square.
The nine shapes at top left* and their lookingglass reflection
illustrate the lookingglass reflection relating two orthogonal
Latin squares over the three digits of modulothree arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 19982008.
** A wellknown construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
An earlier post today, now tagged "Three Small Magic Squares,"
suggests a review of a post from October 25 three years ago
that contains the following figure —
Fans of the October Revolution may enjoy a passage
by Rosalind Krauss on grids:
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Twopart decomposition of base4 array
as two (nonLatin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Fourpart decomposition of base2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a wellknown construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
The title is a phrase by Octavio Paz from today's post
"Status Symbols."
Other phrases from a link target in Sunday's post
The Strength at the Centre —
… a single world
In which he is and as and is are one.
See also Four Dots in this journal.
"Status: Defunct" …
As is now its owner, who reportedly
died at 80 on Sunday, October 15, 2017.
In memoriam —
Excerpts from Log24 posts on Sunday night
and yesterday evening —
.
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Some literary background —
"At the point of convergence by Octavio Paz, translated by Helen Lane

"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and ConwayNortonRyba in this journal.
Some remarks on an orderfive magic square over GF(5^{2}):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2space over GF(5)
(or the vector 1space over GF(5^{2})) …
All vector row sums = (0, 0) (or 0, over GF(5^{2})).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
Hillel Italie at AP News —
"Richard Wilbur, the Pulitzer Prizewinning poet and translator
who intrigued and delighted generations of readers and theatergoers
through his rhyming editions of Moliere and his own verse on memory,
writing and nature, died. He was 96.
Wilbur died Saturday night [Oct. 14, 2017] in Belmont, Massachusetts,
with his family by his side, according to friend and fellow poet, Dana Gioia."
Images from the post "Center" in this journal on Saturday afternoon —
.
"Things fall apart; the centre cannot hold"
The title refers to today's earlier post "The 35Year Wait."
A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude LéviStrauss.
My "inscape" formula, from a note of Sept. 22, 1982 —
S = f ( f ( X ) ) .
Some mathematics from last year related to the 1982 formula —
See also Inscape in this journal and posts tagged Dirac and Geometry.
Continued from the previous post and from posts
now tagged Dueling Formulas —
The fourdiamond formula of Jung and
the fourdot "as" of Claude LéviStrauss:
Simplified versions of the diamonds and the dots —
I prefer Jung. For those who prefer LéviStrauss —
First edition, Cornell University Press, 1970.
A related tale — "A Meaning, Like."
From the Web this morning —
A different 35year wait:
A monograph of August 1976 —
Thirtyfive years later, in a post of August 2011, "Coordinated Steps" —
"SEE HEAR READ" — Walt Disney Productions
Some other diamondmine productions —
"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."
— T. S. Eliot, Four Quartets
See also a recurrent image
from this journal —
See also W. Tecumseh Fitch in this journal.
From the publisher (click to enlarge) —
The above publication date, 01 September 2015, suggests a review
of posts now tagged A Mirror Darkly.
From a web page quoted here on the
Feast of St. Louis, 2003 —
Case 9 of Hekiganroku:
A monk asked Joshu,
Joshu said, Setcho's Verse:
Its intention concealed,
Setcho (9801052), 
The epigraph to Lefebvre's
The Production of Space (1974, translated in 1991) —
(Adapted from a prose poem, "La Higuera ,"
in ¿Águila o Sol? (1951).)
From posts tagged Design Deadline —
A quotation from Lefebvre:
"… an epochmaking event so generally ignored
that we have to be reminded of it at every moment.
The fact is that around 1910 a certain space was shattered…
the space… of classical perspective and geometry…."— Page 25 of The Production of Space
(Blackwell Publishing, 1991)
This suggests, for those who prefer Harvard's past glories
to its current state, a different Raum from the Zeit 1910.
In January 1910 Annals of Mathematics , then edited at Harvard,
published George M. Conwell's "The 3space PG (3, 2) and Its Group."
This paper, while perhaps neither epochmaking nor shattering, has
a certain beauty. For some background, see this journal on February 24, 2009.
Text —
"A field is perhaps the simplest algebraic structure we can invent."
— Hermann Weyl, 1952
Context —
See also yesterday's Personalized Book Search.
Full text of Symmetry – Internet Archive — https://archive.org/details/Symmetry_482
A field is perhaps the simplest algebraic 143 structure 
From a Log24 search for Mathematics+Nutshell —
From Monday morning's post Advanced Study —
"Mathematical research currently relies on
a complex system of mutual trust
based on reputations."
— The late Vladimir Voevodsky,
Institute for Advanced Study, Princeton,
The Institute Letter , Summer 2014, p. 8
Related news from today's online New York Times —
A heading from the above screenshot: "SHOW US YOUR WALL."
This suggests a review of a concept from Galois geometry —
(On the wall — a Galoisgeometry inscape .)
Click to enlarge —
The quote from Hermann Weyl on which the above search is based
is from a search within this journal for Springer + Knight.
Thanks to a Harvard math major for the following V. I. Arnold quote
in a weblog post yesterday titled "Abstraction and Generality"—
"… the author has attempted to adhere to the principle of
minimal generality, according to which every idea should first
be clearly understood in the simplest situation;*
only then can the method developed be extended to
more complicated cases.
— Vladimir I. Arnold, Lectures on Partial Differential Equations
(Russian edition 1997; English translation 2004),
Preface to the second Russian edition
Thanks also to the math major for his closing post today.
* For instance… Natalie Angier's New Year's meditation
on a Buddha Field—
"… the multiverse as envisioned in Tibetan Buddhism,
'a vast system of 1059 [sic ; corrected to 10^59 on Jan. 3]
universes, that together are called a Buddha Field,' said
Jonathan C. Gold, who studies Buddhist philosophy at
Princeton."
— versus a search in this journal for "Japanese character" that yields…
Sacerdotal Jargon
From the website
Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:
Paper: clfalg/good9601
From: David M. Goodmanson
Address: 2725 68th Avenue S.E., Mercer Island, Washington 98040
Title: A graphical representation of the Dirac Algebra
Abstract: The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….
Published: Am. J. Phys. 64, 870880 (1996)
The following is a picture of K_{6}, the complete graph on six points. It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.
From
"The Relations between Poetry and Painting,"
by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And 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 to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
Notes toward a Supreme Fact
In "Notes toward a Supreme Fiction," Wallace Stevens lists criteria for a work of the imagination:
For a work that seems to satisfy these criteria, see the movable images at my diamond theory website. Central to these images is the interplay of rational sides and irrational diagonals in square subimages.
"Logos and logic, crystal hypothesis,
Incipit and a form to speak the word
And every latent double in the word…."— "Notes toward a Supreme Fiction," Section 1, Canto VIII
Recall that "logos" in Greek means "ratio," as well as (human or divine) "word." Thus when I read the following words of Simone Weil today, I thought of Stevens.
"The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi , was the first splendor in mathematics."
— Simone Weil, Oeuvres Choisies , éd. Quarto, Gallimard, 1999, p. 100
In the conclusion of Section 3, Canto X, of "Notes," Stevens says
"They will get it straight one day at the Sorbonne.
We shall return at twilight from the lecture
Pleased that the irrational is rational…."
This is the logoi alogoi of Simone Weil.
Plato's 

From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:
"What is Omega? It's just the diamondhard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…"
Charles H. Bennett has written about Omega as a cabalistic number.
Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamondhard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, viceversa, the square has twice the area of the diamond).
From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:
"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.
Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."
From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:
SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….
From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:
Galileo on Plato's diamond:
"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"
Roger Bacon on Plato's diamond:
"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"
It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.
Today's birthday: James Joseph Sylvester
"Mathematics is the music of reason." — J. J. Sylvester
Sylvester, a nineteenthcentury mathematician, coined the phrase "synthematic totals" to describe some structures based on 6element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. See also the abstract of a December 7, 2000, talk, Mathematics and the Art of M. C. Escher, in which Curtis notes that graphic designs can "often convey a mathematical idea more eloquently than pages of symbolism."
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