"Sharpie, we have condensed six dimensions into four,
then we either work by analogy into six, or we have to use math
that apparently nobody but Jake and my cousin Ed understands.
Unless you can think of some way to project six dimensions into three–
you seem to be smart at such projections."
I closed my eyes and thought hard. "Zebbie, I don't think it can be done.
Maybe Escher could have done it."
Related art —
(For some backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
Detail of the above screen (click to enlarge) —
See also this journal on the above date — June 10, 2021.
From this journal on May 6, 2009 —
A related picture of images that "reappear metamorphosed
in the coordinate system of the high region" —
(For the backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.
In this field, "1 + 1 = 0," as noted here Wednesday.
See also other posts now tagged Interstice.
"An awful lot of important dualities in four and fewer dimensions
follow from this six-dimensional theory and its properties."
— Edward Witten, interviewed by Natalie Wolchover,
in Quanta Magazine on November 28, 2017
See also Six Dimensions in this journal.
A comment today on yesterday's New York Times philosophy column "The Stone"
notes that "Augustine… incorporated Greek ideas of perfection into Christianity."
Yesterday's post here for the Feast of St. Augustine discussed the 2×2×2 cube.
Today's Augustine comment in the Times reflects (through a glass darkly)
a Log24 post from Augustine's Day, 2006, that discusses the larger 4×4×4 cube.
For related material, those who prefer narrative to philosophy may consult
Charles Williams's 1931 novel Many Dimensions . Those who prefer mathematics
to either may consult an interpretation in which Many = Six.
Click image for some background.
“My pursuits are a joke
in that the universe is a joke.
One has to reflect
the universe faithfully.”
— John Frederick Michell
Feb. 9, 1933 –
April 24, 2009
— Robert A. Heinlein,
The Number of the Beast
For Marisa Tomei
(born Dec. 4, 1964) —
on the day that
Bob Seger turns 64 —
A Joke:
Points All Her Own
Points All Her Own,
Part I:
(For the backstory, see
the Log24 entries and links
on Marisa Tomei’s birthday
last year.)
Points All Her Own,
Part II:
(For the backstory, see
Galois Geometry:
The Simplest Examples.)
Points All Her Own,
Part III:
(For the backstory, see
Geometry of the I Ching
and the history of
Chinese philosophy.)
In simpler terms:
Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
to hear about our religion
… that we made up."
|
From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:
… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer… A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination. |
Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."
As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.
Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.
"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."
|
A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:
For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:
The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...
The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,
Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.
The rock is the habitation of the whole,
Its strength and measure, that which is near,
point A
In a perspective that begins again
At B: the origin of the mango's rind.
(Collected Poems, 528)
|
Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:
B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'…. Its subject is its speaker's sense of nothingness and his need to be cured of it."
More positively…
Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space (or the corresponding
5-dimensional projective space)
over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."
Cara:
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.
"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."
— "Was Modernism Born
in Toddler Toolboxes?"
by Trip Gabriel, New York Times,
April 10, 1997
Figure 1 —
Concept from 1819:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.
(Footnote 3)
Figure 3 —
The Third Gift, 1906:
Figure 4 —
Solomon's Cube,
1981 and 1983:
Figure 5 —
Design Cube, 2006:
The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).
(To see how the display works,
try the Kaleidoscope Puzzle first.)
"An acute study of the links
between word and fact"
— Nina daVinci Nichols
| Virginia | /391062427/item.html? | 2/22/2008 7:37 PM |
Johnny Cash:
"And behold,
a white horse."
Chess Knight
(in German, Springer)
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
(vonbayern.de/german/anna.html)
zu Springer kommt?"
Background of "Frau vBayern" from thePeerage.com:
Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg
F, #64640, b. 15 March 1978Last Edited=20 Oct 2005
Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.
The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:
From Log24 on
St. Bridget's Day, 2007:
The quotation
"Science is a Faustian bargain"
and the following figure–
Change
From a short story by
the above Princess:
"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."
To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in
"For Princeton's Class of 2007"—
Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–
Related material:
Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),
and
(Click on image for details.)
“… we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections.”
I closed my eyes and thought hard. “Zebbie, I don’t think it can be done. Maybe Escher could have done it.”
The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2-element field).
(To see how the display works,
try the Kaleidoscope Puzzle first.)
Compare and contrast:
Solomon’s Cube, the five
Log24 entries ending on 3/14,
and the
American Mathematical Society
on Mathematical Imagery.
Related material:
A more extensive excerpt from
The Number of the Beast, and
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