See as well this journal on the above FlixLatino date: Dec. 3, 2015.
Friday, October 25, 2019
Thursday, September 26, 2024
Puzzle Art
From the post Belgian Puzzle Art —
Related reading . . .
— "The Devil, unlike the angels, was at home in the world of phenomena.
He knew how to combine pure concepts with empirical intuitions …
which is the basic principle of linguistic creation."
(Noah Jonathan Jacobs, Naming-Day in Eden, Macmillan, 1958 …
In Macmillan 1969 revised edition, page 21.)
The figure of 25 parts discussed in
"On Linguistic Creation"–
— "Such is the square dance of Numbers."
(Jacques Derrida, Dissemination, 1972)
— "It all adds up."
(Saul Bellow, book title, 1994)
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Thursday, April 4, 2024
Wednesday, October 25, 2023
House Call
"When you build your house
Then call me home"
— Fleetwood Mac, "Sara"
“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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Wednesday, October 11, 2023
Void Game
Fans of the phrase "God-shaped hole" may have some opinions
about what should fill the inner 3×3 void of the above 5×5 array.
Update of 3:53 pm ET — The White Paper —
The Source —
The Atlantic . . . Technology:
Washington and Beijing have been locked in a conflict
over AI development. Now a new battle line is being drawn.
By Karen Hao. October 11, 2023, 9:13 AM ET
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Tuesday, June 7, 2022
A Square for the Circle
The new URL matrix.bingo forwards to
http://m759.net/wordpress/?s=5×5 .
“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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Saturday, March 26, 2022
Box Geometry: Space, Group, Art (Work in Progress)
| Name Tag | .Space | .Group | .Art |
|---|---|---|---|
| Box4 |
2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space) |
S4 = AGL(2,2) (Box4.group) |
(Box4.art) |
| Box6 |
3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set. |
S6 | |
| Box8 | 2x2x2 cube or 4×2 (4-row, 2-column) array. | S8 or A8 or AGL(3,2) of order 1344, or GL(3,2) of order 168 | |
| Box9 | The 3×3 square. | AGL(2,3) or GL(2,3) | |
| Box12 | The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M12. | Symmetries of the cube or elements of the group M12 | |
| Box13 | The 13 symmetry axes of the cube. | Symmetries of the cube. | |
| Box15 |
The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field. |
Collineations of PG(3,2) | |
| Box16 |
The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field. |
AGL(4,2), the affine group of |
|
| Box20 | The configuration representing Desargues's theorem. | ||
| Box21 | The 21 points and 21 lines of PG(2,4). | ||
| Box24 | The 24 points of the Steiner system S(5, 8, 24). | ||
| Box25 | A 5×5 array representing PG(2,5). | ||
| Box27 |
The 3-dimensional Galois affine space over the 3-element Galois field GF(3). |
||
| Box28 | The 28 bitangents of a plane quartic curve. | ||
| Box32 |
Pair of 4×4 arrays representing orthogonal Latin squares. |
Used to represent elements of AGL(4,2) |
|
| Box35 |
A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2) |
PGL(3,2), order 20,160 | |
| Box36 | Eurler's 36-officer problem. | ||
| Box45 | The 45 Pascal points of the Pascal configuration. | ||
| Box48 | The 48 elements of the group AGL(2,3). | AGL(2,3). | |
| Box56 |
The 56 three-sets within an 8-set or |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
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Thursday, December 3, 2020
Bunker Bingo
See as well 5×5, The Matrix of Abraham, and Deutsche Schule Montevideo .
“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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Tuesday, August 25, 2020
Geometric Pedigree
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Friday, December 28, 2018
Blackline Master
From a Log24 post of September 4, 2018, "Identity Crisis" —
From the 2011 Spanish film "Verbo" — (Click to enlarge) —
From a Blackline Master —
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Friday, September 7, 2018
A Square for Sims
The American Mathematical Society on Wednesday, September 5,
reported a death from October 23 last year —
See also Higman-Sims and 5×5 in this journal.
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Tuesday, September 4, 2018
Monday, June 25, 2018
The Trials of Device
"A blank underlies the trials of device."
"Designing with just a blank piece of paper is very quiet."
Related material —
An image posted at 12 AM ET December 25, 2014:
The image stands for the
phrase "five by five,"
meaning "loud and clear."
Other posts featuring the above 5×5 square with some added structure:
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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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Wednesday, October 18, 2017
Dürer for St. Luke’s Day
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 .
Two-part decomposition of base-4 array
as two (non-Latin) 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 .
Four-part decomposition of base-2 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.
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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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Sunday, December 25, 2016
Monday, July 25, 2016
The Retinoid Self
"… which grounds the self" . . .
Popular Mechanics online today —
"Verizon exec Marni Walden seemed to
indicate Mayer's future may still be up in the air."
See also 5×5 in this journal —
"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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Tuesday, February 2, 2016
Depth Haiku
From yesterday's 2 PM post —
From "Inception" —
Paraphrase of remarks by "Inception" director Christopher Nolan
at Princeton on June 1, 2015 —
"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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Monday, January 25, 2016
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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Monday, December 29, 2014
Dodecahedron Model of PG(2,5)
Recent posts tagged Sagan Dodecahedron
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.
For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:
For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.
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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
Thursday, December 18, 2014
Platonic Analogy
(Five by Five continued)
As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged Galois-Plane Models.
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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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Tuesday, November 25, 2014
Euclidean-Galois Interplay
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
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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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Tuesday, September 23, 2014
Matrix
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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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Tuesday, January 25, 2011
Another Reappearing Number
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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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Tuesday, June 24, 2008
Tuesday June 24, 2008
Random Walk with
X's and O's
X's and O's
Part I: Random Walk
Part II: X's
3/22:
"Shakespeare, Rilke, Joyce,
Beckett and Levi-Strauss are
instances of authors for whom
chiasmus and chiastic thinking
Beckett and Levi-Strauss are
instances of authors for whom
chiasmus and chiastic thinking
are of central importance,
for whom chiasmus is a
generator of meaning,
tool of discovery and
philosophical template."
for whom chiasmus is a
generator of meaning,
tool of discovery and
philosophical template."
— Chiasmus in the
Drama of Life
Drama of Life
Part III: O's —
A Cartoon Graveyard
in honor of the late
Gene Persson †
Today's Garfield —
See also
Midsummer Eve's Dream:
"The meeting is closed
with the lord's‡ prayer
and refreshments are served."
† Producer of plays and musicals
including Album and
The Ruling Class
‡ Lower case in honor of
Peter O'Toole, star of
the film version of
The Ruling Class.
(This film, together with
O'Toole's My Favorite Year,
may be regarded as epitomizing
Hollywood's Jesus for Jews.)
Those who prefer
less randomness
in their religion
may consult O'Toole's
more famous film work
involving Islam,
as well as
the following structure
discussed here on
the date of Persson's death:
"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah."
Comments Off on Tuesday June 24, 2008
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
Comments Off on Friday June 6, 2008
Sunday, February 19, 2006
Sunday February 19, 2006
But seriously…
Raiders of the Lost Matrix
(continued)
(continued)
The Matrix:
Click on pictures for details.
In memory of George T. Davis,
who died on February 4,
a Hollywood ending:
“Santa Claus rides alone.”
— Clint Eastwood
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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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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
Sunday, April 27, 2003
Sunday April 27, 2003
ART WARS:
Graphical Password
From a summary of “The Design and Analysis of Graphical Passwords“:
“Results from cognitive science show that people can remember pictures much better than words….
The 5×5 grid creates a good balance between security and memorability.”
— Ian Jermyn, New York University; Alain Mayer, Fabian Monrose, Michael K. Reiter, Bell Labs, Lucent Technologies; Aviel Rubin, AT&T Labs — Research
Illustration — Warren Beatty as
a graphical password:
“Town & Country,”
released April 27, 2001
Those who prefer the simplicity of a 3×3 grid are referred to my entry of Jan. 9, 2003, Balanchine’s Birthday. For material related to the “Town & Country” theme and to Balanchine, see Leadbelly Under the Volcano (Jan. 27, 2003). (“Sometimes I live in the country, sometimes I live in town…” – Huddie Ledbetter). Those with more sophisticated tastes may prefer the work of Stephen Ledbetter on Gershwin’s piano preludes or, in view of Warren Beatty’s architectural work in “Town & Country,” the work of Stephen R. Ledbetter on window architecture.
As noted in Balanchine’s Birthday, Apollo (of the Balanchine ballet) has been associated by an architect with the 3×3, or “ninefold” grid. The reader who wishes a deeper meditation on the number nine, related to the “Town & Country” theme and more suited to the fact that April is Poetry Month, is referred to my note of April 27 two years ago, Nine Gates to the Temple of Poetry.
Intermediate between the simplicity of the 3×3 square and the (apparent) complexity of the 5×5 square, the 4×4 square offers an introduction to geometrical concepts that appears deceptively simple, but is in reality fiendishly complex. See Geometry for Jews. The moral of this megilla?
32 + 42 = 52.
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