Click the above image for some context.
See also "accomplished in steps."
Friday, April 8, 2016
Prominent Modernist Foreground
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Saturday, April 7, 2012
Foreground
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Saturday, December 30, 2023
December 2023, Day XXX
It was in connexion with his discovery of the solution of the problem _To move a given weight by a given force_ that Archimedes uttered the famous saying, "Give me a place to stand on, and I can move the earth" ([Greek: dos moi pou sto kai kino ten gen], or in his broad Doric, as one version has it, [Greek: pa bo kai kino tan gan]). The Project Gutenberg EBook of Archimedes, by Thomas Little Heath
Related material . . .
"The southwest furthers."
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Monday, December 18, 2023
AI Class
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Bringing Up Bing : A Tutorial for Bing on Monday, December 18, 2023 Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?
Response: Searching for: Generating answers for you… The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2. The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants1,2. The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2. The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements1,2. This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets1,2. The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1. It preserves all of the symmetries and maximal subgroups of the Mathieu group M241. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2. In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2. Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2. Learn more Reply to Bing's Response — Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements. Response to the Above Reply — Generating answers for you… Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!
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Saturday, December 9, 2023
Donde Crece la Palma
In the foreground of the above image
from today's online New York Times
is the pediment of Harvard's McKean Gate.
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Monday, November 1, 2021
Plata para los muertos
The PLATA on the sign at right means "silver." The car in the foreground
is turning left onto Jardín Juárez, a street named for the plaza it adjoins
in Cuernavaca, Morelos, Mexico.
An image suggested by Stacy Martin this morning —
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Monday, October 11, 2021
Silverview
The title is that of a novel by John le Carré — apparently his last —
reviewed this morning in the online New York Times by Joseph Finder:
" The great Graham Greene didn’t quite take his own
spy novels seriously, labeling them 'entertainments,'
but le Carré revamped the genre to fit his considerable
ambitions. 'Out of the secret world I once knew,' he wrote,
'I have tried to make a theater for the larger worlds we inhabit.' "
The title suggests an image related to another novel …
Under the Volcano , by Malcolm Lowry (1947) —
The PLATA on the sign at right means "silver." The car in the foreground
is turning left onto Jardín Juárez, a street named for the plaza it adjoins
in Cuernavaca, Morelos, Mexico.
As for "to make a theater" . . .
My own modest efforts along those lines include a Log24 post
from le Carré's date of death:
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Sunday, May 7, 2017
Art Space
Detail of an image in the previous post —
This suggests a review of a post on a work of art by fashion photographer
Peter Lindbergh, made when he was younger and known as "Sultan."
The balls in the foreground relate Sultan's work to my own.
Linguistic backstory —
The art space where the pieces by Talman and by Lindbergh
were displayed is Museum Tinguely in Basel.
As the previous post notes, the etymology of "glamour" (as in
fashion photography) has been linked to "grammar" (as in
George Steiner's Grammars of Creation ). A sculpture by
Tinguely (fancifully representing Heidegger) adorns one edition
of Grammars .
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Tuesday, January 10, 2012
Defining Form
(Continued from Epiphany and from yesterday.)
Detail from the current American Mathematical Society homepage—
Further detail, with a comparison to Dürer’s magic square—
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The three interpenetrating planes in the foreground of Donmoyer‘s picture
provide a clue to the structure of the the magic square array behind them.
Group the 16 elements of Donmoyer’s array into four 4-sets corresponding to the
four rows of Dürer’s square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.
Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—
Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.
Just as the decomposition theorem’s interpenetrating lines explain the structure
of a 4×4 square , the foreground’s interpenetrating planes explain the structure
of a 2x2x2 cube .
For an application to theology, recall that interpenetration is a technical term
in that field, and see the following post from last year—
| Saturday, June 25, 2011
— m759 @ 12:00 PM “… the formula ‘Three Hypostases in one Ousia ‘
Ousia
|
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Tuesday, April 19, 2011
Romancing the Omega
Today's news from Oslo suggests a review—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
Some related philosophical remarks—
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Thursday, July 15, 2010
Brightness at Noon, continued
"What exactly was Point Omega?"
This is Robert Wright in Nonzero: The Logic of Human Destiny.
Wright is discussing not the novel Point Omega by Don DeLillo,
but rather a (related) concept of the Jesuit philosopher Pierre Teilhard de Chardin.
My own idiosyncratic version of a personal "point omega"—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
For some other sorts of points, see the drawings
on the wall and Geometry Simplified—
The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affine-space squares.
For related darkness at noon, see Derrida on différance
as a version of Plato's khôra—
The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institution— Deconstruction in a Nutshell—
For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.
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Wednesday, June 30, 2010
Field Dream
In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday—
"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
"The Chinese painter Wu Tao-tzu was famous because he could paint nature in a unique realistic way that was able to deceive all who viewed the picture. At the end of his life he painted his last work and invited all his friends and admirers to its presentation. They saw a wonderful landscape with a romantic path, starting in the foreground between flowers and moving through meadows to high mountains in the background, where it disappeared in an evening fog. He explained that this picture summed up all his life’s work and at the end of his short talk he jumped into the painting and onto the path, walked to the background and disappeared forever."
— Jürgen Teichmann. Teichmann notes that "the German poet Hermann Hesse tells a variation of this anecdote, according to his own personal view, as found in his 'Kurzgefasster Lebenslauf,' 1925."
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Friday, May 27, 2005
Friday May 27, 2005
Drama of the Diagonal,
Part Deux
Part Deux
Wednesday’s entry The Turning discussed a work by Roger Cooke. Cooke presents a
“fanciful story (based on Plato’s dialogue Meno).”
The History of Mathematics is the title of the Cooke book.
Associated Press thought for today:
“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
— Henry Kissinger (whose birthday is today)
For Henry Kissinger on his birthday:
a link to Geometry for Jews.
a link to Geometry for Jews.
This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares (ps),
a discussion of a 4×4 array
of square linear designs.
Cipra says that
“If you like, there are three symmetry groups lurking within the LeWitt puzzle: the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16. The first group is the most obvious. The third, once you see it, is also obvious.”
* Jean-Paul Sartre,
Being and Nothingness,
Philosophical Library, 1956
[reference by Cipra]
For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.
Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:
Logos Alogos II:
Horizon
This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–
From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:
| 1. | on Page 51: |
| “He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being. Man is ‘a being of distances.'” | |
| 2. | on Page 154: |
| “… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities. Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….” | |
| 3. | on Page 155: |
| “… end of the desires. But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the glass-in-the-midst-of-the-world.” | |
| 4. | on Page 158: |
| “… it is in time that my possibilities appear on the horizon of the world which they make mine. If, then, human reality is itself apprehended as temporal….” | |
| 5. | on Page 180: |
| “… else time is an illusion and chronology disguises a strictly logical order of deducibility. If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….” | |
| 6. | on Page 186: |
| “… It appears on the horizon to announce to me what I am from the standpoint of what I shall be.” | |
| 7. | on Page 332: |
| “… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon. It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….” | |
| 8. | on Page 359: |
| “… eyes as objects which manifest the look. The Other can not even be the object aimed at emptily at the horizon of my being for the Other.” | |
| 9. | on Page 392: |
| “… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.” | |
| 10. | on Page 502: |
| “… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.” |
| 11. | on Page 506: |
| “… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?” | |
| 12. | on Page 661: |
| “I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.” | |
| 13. | on Page 754: |
| “Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself. The pen and the pipe, the clothing, the desk, the house– are myself. The totality of my possessions reflects the totality of my being. I am what I have. It is I myself which I touch in this cup, in this trinket. This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.” |
|
Illustration of the
last horizon remark: |
For more on the horizon, being, and nothingness, see
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