Tuesday, July 9, 2024
Saturday, September 24, 2022
Nine and Sixteen: In Search of Common Ground
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Saturday, February 16, 2019
Bunker Bingo Continues.
From a post of October 16, 2017, "Halloween Meditation" —
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Friday, May 25, 2018
Grid Design
Click the grid for the tag 5×5 in this journal.
A related book —
See also the previous post, Bucharest Semiotics.
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Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
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
Base-5:
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 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
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
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Thursday, June 4, 2015
Foundations
Princeton University's president on June 2, 2015 —
“Dream Audaciously,” Eisgruber Urges Graduates
Related news —
Film director Christopher Nolan at Princeton on June 1, 2015:
“In these graduation speeches, generally, you have the
speaker say something along the lines of, ‘You need to
chase your dreams,’ ” Nolan said. “But I’m not going to
say that because I don’t believe it. I don’t want you to
chase your dreams. I want you to chase your realities.
And I want to say: Don’t chase your realities at the
expense of your dreams, but as the foundation of your
dreams.”
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
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Sunday, December 28, 2014
A Christmas Carol
See also Sagan Dodecahedron, which includes
an image posted at 12 AM ET December 25, 2014:
The image stands for the
phrase "five by five,"
meaning "loud and clear."
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Thursday, December 25, 2014
Wednesday, December 17, 2014
Launched from Cuber
Continued from Nobel Note (Jan. 29, 2014).
From Tradition in Action , "The Missal Crisis of '62,"
remarks on the revision of the Catholic missal in that year—
"Neither can the claim that none of these changes
is heretical in content be used as an argument
in favor of its use, for neither is the employment of
hula girls, fireworks, and mariachis strictly speaking
heretical in itself, but they belong to that class of novel
and profane things that do not belong in the Mass."
— Fr. Patrick Perez, posted Sept. 11, 2007
See also this journal on November 22, 2014…
… and on Bruce Springsteen's birthday this year —
Tuesday, September 23, 2014
|
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Saturday, October 25, 2014
Foundation Square
In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…
-
"The Hoffman-Singleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and -
the Web pages "Hoffman-Singleton Graph" and "Higman-Sims Graph"
of A. E. Brouwer.
Hafner's abstract:
We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)
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Sunday, January 26, 2014
A Midnight Exorcism
The summoning of the spirit of Bertrand Russell
yesterday by Peter J. Cameron at his weblog
suggests a review of this weblog’s posts of
Christmas Eve, December 24-25, 2013.
(Recall that Robert D. Carmichael, who, in a book
linked to at midnight last Christmas Eve discusses
some “magic” mathematical structures,
reportedly was trained as a Presbyterian minister.
See also The Presbyterian Exorcist.)
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Wednesday, December 25, 2013
Sunday, December 2, 2012
Knight’s Labyrinth
A magic— indeed, diabolic— square:
For the construction, see a book
by W. W. Rouse Ball, founding president
of a Cambridge University magic society.
For some related religious remarks,
see Raiders of the Lost Matrix.
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Thursday, October 14, 2010
Diamond Theory and Magic Squares
"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical
ultraomnipotence."
— e. e. cummings, 1944
For one such specimen, see The Matrix of Abraham—
a 5×5 square that is hypermagical… indeed, diabolical.
Related material on the algebra and geometry underlying some smaller structures
that have also, unfortunately, become associated with the word "magic"—
- Finite Geometry of the Square and Cube
- Clifford Pickover on a 4×4 square
-
Christopher J. Henrich on the geometry of 4×4 magic squares
(without any mention of [1] above or related work dating back to 1976)
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Happy birthday, e. e.
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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
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Friday, June 6, 2008
Friday June 6, 2008
The Dance of Chance
"Harvard seniors have
every right to demand a
Harvard-calibre speaker."
— Adam Goldenberg in
The Harvard Crimson
"Look down now, Cotton Mather"
— Wallace Stevens,
Harvard College
Class of 1901
For Thursday, June 5, 2008,
commencement day for Harvard's
Class of 2008, here are the
Pennsylvania Lottery numbers:
Mid-day 025
Evening 761
Thanks to the late
Harvard professor
Willard Van Orman Quine,
the mid-day number 025
suggests the name
"Isaac Newton."
(For the logic of this suggestion,
see On Linguistic Creation
and Raiders of the Lost Matrix.)
Thanks to Google search, the
name of Newton, combined with
Thursday's evening number 761,
suggests the following essay:
PHILOSOPHY OF SCIENCE:
|
What can a non-scientist add?
Perhaps the Log24 entries for
the date of Koshland's death:
The Philosopher's Stone
and The Rock.
Or perhaps the following
observations:
On the figure of 25 parts
discussed in
"On Linguistic Creation"–
"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah. "
— Clifford Pickover
"At the still point,
there the dance is."
— T. S. Eliot,
Harvard College
Class of 1910
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Saturday, February 4, 2006
Saturday February 4, 2006
Raiders of
the Lost Matrix
the Lost Matrix
(continued)
|
The Archaeologist
with a Thousand Faces "From often humble beginnings, and often with a childhood fascination for antiquity, the archaeologist leaves familiar surroundings to undergo exacting professional training under a series of mentors and when armed, at last, with the intellectual weapons of the profession, sets off for unfamiliar or exotic realms, braving opposition and danger to solve an ancient mystery. The lives of… real-life archaeologists… have lent themselves to this style of retelling… as have such fictional heroes as John Cullinane (Michener 1965) and Indiana Jones." — From "Promised Lands and Chosen Peoples: The Politics and Poetics of Archaeological Narrative," by Neil Asher Silberman, pp. 249-262 in Nationalism, Politics, and the Practice of Archaeology, edited by Philip L. Kohl and Clare Fawcett, Cambridge University Press, paperback, published Feb. 8, 1996. |
From Today in History,
by the Associated Press:
Thought for Today:
"Character consists of what you do
on the third and fourth tries."
— James Michener,
American author (1907-1997),
attributed by
Simpson's Contemporary Quotations
to Chesapeake, Random House, 78.
The Matrix:
First try:
On Linguistic Creation
June 25, 1999
Second try:
Art Wars: Picasso's Birthday,
Oct. 25, 2002
Third try:
Matrix of the Death God,
May 25, 2003
Fourth try:
Happy Birthday,
July 26, 2004
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Monday, July 26, 2004
Monday July 26, 2004
Happy Birthday
to Kate Beckinsale
(star of Cold Comfort Farm)
and Kevin Spacey
(star of The Usual Suspects).
From a novel,
The Footprints of God,
published August 12, 2003 —
A tour guide describes
stations of the cross in Jerusalem:
"Ibrahim pointed down the cobbled street to a half circle of bricks set in the street. 'There is where Jesus began to carry the cross. Down the street is the Chapel of Flagellation, where the Roman soldiers whipped Jesus, set on him a crown of thorns, and said, "Hail, King of the Jews!" Then Pilate led him to the crowd and cried, "Ecce homo! Behold the man!" '
Ibrahim delivered this information with the excitement of a man reading bingo numbers in a nursing home."
In keeping with this spirit of religious fervor and with the spirit of Carl Jung, expositor of the religious significance of the mandala,
Behold —
The Mandala of Abraham
For the religious significance of this mandala,
see an entry of May 25, 2003:
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Tuesday, July 1, 2003
Tuesday July 1, 2003
Jew’s on First
This entry is dedicated to those worshippers of Allah who have at one time or another cried
“Itbah al-Yahud!” … Kill the Jew!
(See June 29 entries).
|
Dead at 78 Comedian Buddy Hackett died on Tuesday, July First, 2003, according to the New York Times. According to Bloomberg.com, he died Sunday or Monday. |
Associated Press
Buddy Hackett, |
| Whatever. We may imagine he has now walked, leading a parade of many other stand-up saints, into a bar. |
|
|
MIDRASH From my May 25 entry, Matrix of the Death God: R. M. Abraham’s Diversions and Pastimes, published by Constable and Company, London, in 1933, has the following magic square:
The Matrix of Abraham A summary of the religious import of the above from Princeton University Press: “Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah. Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.” — Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72 Other appearances of this religious icon on the Web include:
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In the Picasso’s Birthday version, 22 of the 25 magic square cells are correlated with pictures on the “Class of ’91” cover of Rolling Stone magazine. Number 7 is Rod† Stewart. In accordance with the theological rhyme “Seven is heaven, eight is a gate,” our site music for today is “Forever Young,” a tune made famous by Stewart.
† Roderick, actually — the name of the hero in “Madwoman of Chaillot”
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Sunday, May 25, 2003
Sunday May 25, 2003
— ART WARS —
Mental Health Month, Day 25:
Matrix of the Death God
Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham. (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.) For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.
See, too, The Matrix of Abraham, illustrated below. This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.
The Matrix of Abraham
A summary of the religious import of the above from Princeton University Press:
“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah. Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”
— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72
Other appearances of this religious icon on the Web:
A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).
This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.
Friday, October 25, 2002
Friday October 25, 2002
ART WARS:
Picasso's Birthday
From an art quotes website:
Dore Ashton's Picasso on Art —
"We all know that Art is not truth.
Art is a lie that makes us realize truth,
at least the truth that is given us
to understand." — Pablo Picasso
From "Xanadu" —
"You have to believe we are magic."
— Olivia Newton-John
|
The Muse |
Soul Kiss |
A is for Art |
"A work of art has an author and yet,
when it is perfect, it has something
which is essentially anonymous about it."
— Simone Weil, Gravity and Grace
when it is perfect, it has something
which is essentially anonymous about it."
— Simone Weil, Gravity and Grace
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