This post was suggested by a review in the Jan. 2022
Notices of the American Mathematical Society :
My own sympathies are with Veblen.
Saturday, December 18, 2021
Desargues, Galois, Veblen, and Young
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Saturday, July 8, 2017
Desargues and Galois in Japan
Related material now available online —
A less business-oriented sort of virtual reality —
For example, "A very important configuration is obtained by
taking the plane section of a complete space five-point."
(Veblen and Young, 1910, p. 39)—
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Sunday, December 18, 2016
Two Models of the Small Desargues Configuration
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Thursday, August 11, 2016
The Large Desargues Configuration
(Continued from April 2013 and later)
This is what I called "the large Desargues configuration"
in posts of April 2013 and later.
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Tuesday, August 6, 2013
Desargues via Galois
The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.
Update of Aug. 7, 2013: See also an expanded PDF version.
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Friday, April 19, 2013
The Large Desargues Configuration
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6-volume
Principles of Geometry .
Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
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Monday, April 1, 2013
Desargues via Rosenhain
Background: Rosenhain and Göpel Tetrads in PG(3,2)
|
Introduction: The Large Desargues Configuration Added by Steven H. Cullinane on Friday, April 19, 2013 Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point The converse, from the same book:
"If two triangles are perspective from a line
Desargues' theorem according to Wikipedia
"Two triangles are in perspective axially [i.e., from a line]
A figure often used to illustrate the theorem,
A discussion of the "if and only if" version of the theorem
This large Desargues configuration involves a third triangle,
Point-line incidence in this larger configuration is,
The third triangle, within the larger configuration,
|
A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desargues-theorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original Veblen-Young figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.
In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):
|
Volume I of Baker's Principles has a cover closely related to the Rosenhain tetrads in PG(3,2) |
Volume IV of Baker's Principles has a cover closely related to the Göpel tetrads in PG(3,2) |
|
Foundations (click to enlarge)
|
Higher Geometry (click to enlarge)
|
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Sunday, December 24, 2023
Infolded Perspectives
Infolded:
“In the garden of Adding,
Live Even and Odd….”
— The Midrash Jazz Quartet in
City of God , by E. L. Doctorow
Perspectives:
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Friday, July 21, 2023
Monday, April 10, 2023
Space
(Perspective Not as Symbolic Form)
From a post of June 8, 2014 —
See August 6, 2013 — Desargues via Galois.
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Friday, November 25, 2022
“Open the pod bay doors, Hal.”
"Both Hal and the students were great interlocutors
with lots of ideas and differing perspectives."
—Leah Dickerman, MoMA, May 10, 2010, at
https://www.princeton.edu/news/2010/05/10/
class-snapshot-origins-abstraction
Later . . .
See as well Desargues + Galois.
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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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Tuesday, March 15, 2022
The Rosenhain Symmetry
See other posts now so tagged.
Hudson's Rosenhain tetrads, as 20 of the 35 projective lines in PG(3,2),
illustrate Desargues's theorem as a symmetry within 10 pairs of squares
under rotation about their main diagonals:
See also "The Square Model of Fano's 1892 Finite 3-Space."
The remaining 15 lines of PG(3,2), Hudson's Göpel tetrads, have their
own symmetries . . . as the Cremona-Richmond configuration.
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Thursday, April 2, 2020
Friday, February 21, 2020
Thursday, August 29, 2019
As Well
For some backstory, see
http://m759.net/wordpress/?s=”I+Ching”+48+well .
See as well “elegantly packaged” in this journal.
“Well” in written Chinese is the hashtag symbol,
i.e., the framework of a 3×3 array.
My own favorite 3×3 array is the ABC subsquare
at lower right in the figure below —
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Sunday, June 16, 2019
Tuesday, October 23, 2018
Sunday, September 9, 2018
Plan 9 Continues.
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
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Friday, September 29, 2017
Principles Before Personalities*
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Friday, August 11, 2017
Symmetry’s Lifeboat
A post suggested by the word tzimtzum (see Wednesday)
or tsimtsum (see this morning) —
Lifeboat from the Tsimtsum in Life of Pi —
Another sort of tsimtsum, contracting infinite space to a finite space —
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Tuesday, July 11, 2017
A Date at the Death Cafe
The New York TImes reports this evening that
"Jon Underwood, Founder of Death Cafe Movement,"
died suddenly at 44 on June 27.
This journal on that date linked to a post titled "The Mystic Hexastigm."
A related remark on the complete 6-point from Sunday, April 28, 2013 —
(See, in Veblen and Young's 1910 Vol. I, exercise 11,
page 53: "A plane section of a 6-point in space can
be considered as 3 triangles perspective in pairs
from 3 collinear points with corresponding sides
meeting in 3 collinear points." This is the large
Desargues configuration. See Classical Geometry
in Light of Galois Geometry.)
This post was suggested, in part, by the philosophical ruminations
of Rosalind Krauss in her 2011 book Under Blue Cup . See
Sunday's post Perspective and Its Transections . (Any resemblance
to Freud's title Civilization and Its Discontents is purely coincidental.)
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Sunday, July 9, 2017
Perspective and Its Transections
The title phrase is from Rosalind Krauss (Under Blue Cup , 2011) —
Another way of looking at the title phrase —
"A very important configuration is obtained by
taking the plane section of a complete space five-point."
(Veblen and Young, 1910, p. 39) —
For some context, see Desargues + Galois in this journal.
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Tuesday, May 2, 2017
Image Albums
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
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Friday, April 28, 2017
A Generation Lost in Space
The title is from Don McLean's classic "American Pie."
A Finite Projective Space —
A Non-Finite Projective Space —
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Thursday, April 27, 2017
Partner, Anchor, Decompose
See also a figure from 2 AM ET April 26 …
" Partner, anchor, decompose. That's not math.
That's the plot to 'Silence of the Lambs.' "
— Greg Gutfeld, September 2014
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Wednesday, April 12, 2017
Expanding the Spielraum
"Cézanne ignores the laws of classical perspective . . . ."
— Voorhies, James. “Paul Cézanne (1839–1906).”
In Heilbrunn Timeline of Art History . New York:
The Metropolitan Museum of Art, 2000–. (October 2004)
Some others do not.
This is what I called "the large Desargues configuration"
in posts of April 2013 and later.
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Monday, January 2, 2017
John Berger Has Died
Screenshot of 11:07 PM ET tonight —
A sample of his work —
An antidote to Berger's remarks —
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Monday, December 19, 2016
Tetrahedral Cayley-Salmon Model
The figure below is one approach to the exercise
posted here on December 10, 2016.
Some background from earlier posts —
Click the image below to enlarge it.
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Sunday, December 11, 2016
Complexity to Simplicity via Hudson and Rosenhain*
*The Hudson of the title is the author of Kummer's Quartic Surface (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.
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Saturday, December 10, 2016
Folk Etymology
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the small-configuration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
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Saturday, October 15, 2016
A Marxist Perspective
The previous post, Nobel Perspective, suggests a review of
the following passage pictured here on August 27, 2013.
Click image for a better view of the original.
There are, of course, more sophisticated approaches
to the place of perspective in the history of art.
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Sunday, August 28, 2016
Wednesday, August 24, 2016
Core Statements
"That in which space itself is contained" — Wallace Stevens
An image by Steven H. Cullinane from April 1, 2013:
The large Desargues configuration of Euclidean 3-space can be
mapped canonically to the 4×4 square of Galois geometry —
On an Auckland University of Technology thesis by Kate Cullinane —
The thesis reportedly won an Art Directors Club award on April 5, 2013.
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Tuesday, May 24, 2016
Rosenhain and Göpel Revisited
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
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Tuesday, December 1, 2015
Pascal’s Finite Geometry
See a search for "large Desargues configuration" in this journal.
The 6 Jan. 2015 preprint "Danzer's Configuration Revisited,"
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the
interesting context of Pascal's Hexagrammum Mysticum .
They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .
This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.
The duality was, as they note, previously described in 1898.
Related material on six-set geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 1-2, pp 161-176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160
Related material on six-set geometry from a more recent source —
Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage
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Friday, August 14, 2015
Schoolgirl Problem
But first, a word from our sponsa* …
Sir Laurence Olivier in "Term of Trial" (1962),
a film starring Sarah Miles as a schoolgirl —
* Bride in Latin. See also "bride's chair,"
a phrase from mathematical pedagogy.
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Sunday, May 31, 2015
Wednesday, April 22, 2015
Purely Aesthetic
G. H. Hardy in A Mathematician's Apology —
|
What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid’s or Pythagoras’s? I will not risk more than a few disjointed remarks. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way. |
Related material:
-
A post at noon on Sunday, April 19, 2015, with a link,
"Ageometretos medeis eisito," to an image search
for "large Desargues configuration" that includes …
-
A review, dated April 20, 2015, of "After Hours at the
Harvard Art Museums" in The Harvard Crimson -
An article, also dated April 20, 2015, in Harvard Magazine
titled "Harvard Installs 'Triangle Constellation'"
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Sunday, April 19, 2015
Preoccupied
"The Cardinal seemed a little preoccupied today."
See also a post found via a search in
this journal for "April 19 ".
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Thursday, September 25, 2014
Mystery
"Welcome to America." — Harrison Ford in "The Devil's Own"
On readings at Mass on Sunday, Sept. 21, 2014 —
"Isaiah 55:8-9: 'For my thoughts are not your thoughts,
neither are your ways my ways, says the Lord.
For as the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts.'
The Gospel reading… was a perfect complement to
the passage from Isaiah…."
The America piece quoting Isaiah was titled "The Mystery of God."
The author "currently works at Xavier College Preparatory
in Palm Desert, CA, where he teaches theology…."
Related material: This journal that Sunday morning:
See also "The Mystery of God, Part II" —
Other secular stand-ins for "the thing one doesn't know"—
The mysteries of the late Joseph D. McNamara.
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Sunday, September 21, 2014
Uncommon Noncore
This post was suggested by Greg Gutfeld’s Sept. 4 remarks on Common Core math.
Problem: What is 9 + 6 ?
Here are two approaches suggested by illustrations of Desargues’s theorem.
Solution 1:
9 + 6 = 10 + 5,
as in Common Core (or, more simply, as in common sense), and
10 + 5 = 5 + 10 = 15 as in Veblen and Young:
Solution 2:
In the figure below,
9 + 6 = no. of V’s + no. of A’s + no. of C’s =
no. of nonempty squares = 16 – 1 = 15.
(Illustration from Feb. 10, 2014.)
The silly educationists’ “partner, anchor, decompose” jargon
discussed by Gutfeld was their attempt to explain “9 + 6 = 10 + 5.”
As he said of the jargon, “That’s not math, that’s the plot from ‘Silence of the Lambs.'”
Or from Richard, Frank, and Marcus in last night’s “Intruders”
(BBC America, 10 PM).
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Sunday, July 20, 2014
Sunday School
Paradigms of Geometry:
Continuous and Discrete
The discovery of the incommensurability of a square’s
side with its diagonal contrasted a well-known discrete
length (the side) with a new continuous length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous configuration at
left is embodied in the discrete unit cells of the square at right.
See Desargues via Galois (August 6, 2013).
Comments Off on Sunday School
Sunday, July 6, 2014
Sunday School
The date of a Vanity Fair post on Hugh Jackman—
Sunday, June 8, 2014— suggests a review of the following
quotation from this journal on that date —
Zen ideal —
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Sticks and Stones
The title is from this morning's previous post.
From a theater review in that post—
… "all flying edges and angles, a perpetually moving and hungry soul"
… "a formidably centered presence, the still counterpoint"
A more abstract perspective:
See also Desargues via Galois (August 6, 2013).
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Sunday, June 8, 2014
Vide
"The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof."
— Gian-Carlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The "simplest case" link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
Comments Off on Vide
Tuesday, May 20, 2014
Play
From a recreational-mathematics weblog yesterday:
"This appears to be the arts section of the post,
so I’ll leave Martin Probert’s page on
The Survival, Origin and Mathematics of String Figures
here. I’ll be back to pick it up at the end. Maybe it’d like
to play with Steven H. Cullinane’s pages on the
Finite Geometry of the Square and Cube."
I doubt they would play well together.
Perhaps the offensive linking of the purely recreational topic
of string figures to my own work was suggested by the
string figures' resemblance to figures of projective geometry.
A pairing I prefer: Desargues and Galois —
For further details, see posts on Desargues and Galois.
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Thursday, March 20, 2014
Classical Galois
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Monday, February 10, 2014
Mystery Box III: Inside, Outside
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Comments Off on Mystery Box III: Inside, Outside
Tuesday, August 27, 2013
Perspective
For Fans of Bad Movies*
This post was suggested by my viewing last night
the 1995 horror film Species , and by news that
Scarlett Johansson will be starring in a similar
production at the Venice Film Festival, which
opens tomorrow.
The new Johansson film, Under the Skin ,
is based on a novel by one Michel Faber.
Faber on books that have influenced him—
"Most influential has possibly been John Berger's Ways of Seeing —
not a novel at all (although Berger has written fiction) but a book of
art criticism. The influence of these wonderfully perceptive and
thought-provoking essays peeps out everywhere in my own work."
An excerpt from the Berger book—
Click image for a better view of the original.
Related material: Johansson in this journal, Sunday's NY Times
teaser for a piece on Saturday Night Live, and a more serious
approach to the geometry of perspective.
* And of Ben Kingsley, who starred both in Species and in
a previous film by the director of Under the Skin .
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Saturday, August 17, 2013
Up-to-Date Geometry
The following excerpt from a January 20, 2013, preprint shows that
a Galois-geometry version of the large Desargues 154203 configuration,
although based on the nineteenth-century work of Galois* and of Fano,**
may at times have twenty-first-century applications.
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 154203 configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428-435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106-132.
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Sunday, July 28, 2013
Sermon
(Simplicity continued)
"Understanding a metaphor is like understanding a geometrical
truth. Features of various geometrical figures or of various contexts
are pulled into revealing alignment with one another by the
demonstration or the metaphor.
What is 'revealed' is not that the alignment is possible; rather,
that the alignment is possible reveals the presence of already-
existing shapes or correspondences that lay unnoticed. To 'see' a
proof or 'get' a metaphor is to experience the significance of the
correspondence for what the thing, concept, or figure is ."
— Jan Zwicky, Wisdom & Metaphor , page 36 (left)
Zwicky illustrates this with Plato's diamond figure
from the Meno on the facing page— her page 36 (right).
A more sophisticated geometrical figure—
Galois-geometry key to
Desargues' theorem:
| D | E | F | |
| S' | P | Q | R |
| S | P' | Q' | R' |
| O | P1 | Q1 | R1 |
For an explanation, see
Classical Geometry in Light of Galois Geometry.
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Wednesday, July 24, 2013
The Broken Tablet
This post was suggested by a search for the
Derridean phrase "necessary possibility"* that
led to web pages on a conference at Harvard
on Friday and Saturday, March 26**-27, 2010,
on Derrida and Religion .
The conference featured a talk titled
"The Poetics of the Broken Tablet."
I prefer the poetics of projective geometry.
An illustration— The restoration of the full
15-point "large" Desargues configuration in
place of the diminished 10-point Desargues
configuration that is usually discussed.
Click on the image for further details.
* See a discussion of this phrase in
the context of Brazilian religion.
** See also my own philosophical reflections
on Friday, March 26, 2010:
"You Can't Make This Stuff Up."
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Sunday, May 19, 2013
Sermon
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the Conway-Sloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
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Tuesday, April 30, 2013
Projective Analysis
A Nested Sequence of Complete N-points and Their Sections
The complete space 6-point
(6 points in general position in space,
5 lines on each point, and 15 lines, 2 points on each)
has as a section
the large Desargues configuration
(15 points, 4 lines on each, and 20 lines, 3 points on each).
(Veblen and Young, Vol. 1, exercise 11, p. 53)
The large Desargues configuration may in turn be viewed as
the complete space 5-point
(5 points, 4 lines on each, and 10 lines, 2 points on each)
together with its section
the Desargues configuration
(10 points, 3 lines on each, and 10 lines, 3 points on each).
(Veblen and Young, Vol. I, pages 40-42)
The Desargues configuration may in turn be viewed as
the complete space 4-point (tetrahedron)
(4 points, 3 lines on each, and 6 lines, 2 points on each)
together with its section
the complete (plane) 4-side (complete quadrilateral)
(6 points, 2 lines on each, and 4 lines, 3 points on each).
The complete quadrilateral may in turn be viewed as
the complete 3-point (triangle)
(3 points, 2 lines on each, and 3 lines, 2 points on each)
together with its section
the three-point line
(3 points, 1 line on each, and 1 line, 3 points on the line).
The three-point line may in turn be viewed as
the complete 2-point
(2 points, 1 line on each, and 1 line with 2 points on the line)
together with its section
the complete 1-point
(1 point and 0 lines).
Update of May 1: For related material, see the exercises at the end of Ch. II
in Veblen and Young's Projective Geometry, Vol. I (Ginn, 1910). For instance:
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Sunday, April 28, 2013
The Octad Generator
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M24 —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
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Sunday, April 21, 2013
The Grandmother Ship
Tina Jordan at EW.com yesterday:
"E.L. Konigsburg— the author of one of my favorite
childhood books, the brilliantly quirky mystery
From The Mixed-Up Files of Mrs. Basil E. Frankweiler—
died April 19 at the age of 83."
From other mixed-up files:
Detail:
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Abstraction
(Continued from December 31st, 2012)
"Principles before personalities." — AA saying
Art Principles
Part I:
Part II:
Baker's 1922 Principles of Geometry—
Art Personalities
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Saturday, April 13, 2013
Veblen and Young in Light of Galois
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
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Wednesday, April 10, 2013
Caution: Slow Art
"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."
— Geoff Dyer, review of Point Omega
Related material:
A phrase of critic Robert Hughes,
"slow art," in this journal.
A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:
The above 3×3 grid is embedded in a
somewhat more sophisticated example
of conceptual art from April 1, 2013:
Update of April 12, 2013
The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a three-triangle version of Desargues's theorem.
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
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Wednesday, April 3, 2013
Museum Piece
Roberta Smith in 2011 on the American Folk Art Museum (see previous post):
"It could be argued that we need a museum of folk art
the way we need a museum of modern art,
to shine a very strong, undiluted light on
a very important achievement."
Some other aesthetic remarks:
"We have had a gutful of fast art and fast food.
What we need more of is slow art: art that holds time
as a vase holds water: art that grows out of modes
of perception and whose skill and doggedness
make you think and feel; art that isn't merely sensational,
that doesn't get its message across in 10 seconds,
that isn't falsely iconic, that hooks onto something
deep-running in our natures. In a word, art that is
the very opposite of mass media. For no spiritually
authentic art can beat mass media at their own game."
— Robert Hughes, speech of June 2, 2004,
quoted here June 15, 2007.
Perhaps, as well as museums of modern art and of folk art,
we need a Museum of Slow Art.
One possible exhibit, from this journal Monday:
The diagram on the left is from 1922. The 20 small squares at right
that each have 4 subsquares darkened were discussed, in a different
context, in 1905. They were re-illustrated, in a new context
(Galois geometry), in 1986. The "key" square, and the combined
illustration, is from April 1, 2013. For deeper background, see
Classical Geometry in Light of Galois Geometry.
Those who prefer faster art may consult Ten Years After.
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Tuesday, April 2, 2013
Rota in a Nutshell
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
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Baker on Configurations
The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as
Classical Geometry in Light of Galois Geometry.
Some background:
See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212-218)—
On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem
and Vol. II, Note II (pp. 219-236)—
On the Hexagrammum Mysticum of Pascal.
Monday's elucidation of Baker's Desargues-theorem figure
treats the figure as a 154203 configuration (15 points,
4 lines on each, and 20 lines, 3 points on each).
Such a treatment is by no means new. See Baker's notes
referred to above, and
"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.
What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).
See also Cremona's kernel, or nocciolo :
Baker on Cremona's approach to Pascal—
"forming, in Cremona's phrase, the nocciolo of the whole."
A related nocciolo :
Click on the nocciolo for some
geometric background.
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Thursday, September 8, 2011
Starring the Diamond
"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)
Continued from Tuesday, Sept. 6—
The Diamond Star
The above is a version of a figure from Configurations and Squares.
Yesterday's post related the the Pappus configuration to this figure.
Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.
Some excerpts from Coxeter—
The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).
The relabeled configuration is used in a discussion of Pappus—
(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)
Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—
The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts—
"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."
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Saturday, January 1, 2011
For a Dead Philosopher —
Dagger Definitions
Part I —
Part II —
Update of Jan. 2, 2011 —
Singer goes on to say that "A finite projective plane, PG (2, pn ), defined in this way is Pascalian and Desarguesian ; it exists for every prime p and positive integer n , and there is only one such PG (2, pn ) for a given p and n …."
His definitions therefore deliberately exclude non -Desarguesian finite projective planes, which were known to exist at the time he wrote.
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Thursday, January 26, 2006
Thursday January 26, 2006
In honor of Paul Newman’s age today, 81:
On Beauty
Elaine Scarry, On Beauty (pdf), page 21:
“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”
The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures. Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points. Less obvious are some related projective structures. Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with
See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:
Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.
For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:
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Wednesday, September 15, 2004
Wednesday September 15, 2004
Translation Plane
for Rosh Hashanah
Figure A
|
From the website of Priv.-Doz. Dr. H. Klein, The Translation Plane of Order Nine There are exactly four projective planes of order nine, and one of these planes is a non-Desarguesian translation plane. Theorem. Up to isomorphism, there exists exactly one non-Desarguesian translation plane of order 9. This translation plane is defined by a spreadset in a 2-dimensional vector space over the field GF(3), consisting of the following matrices.
  As it turns out, the coordinatizing quasifield is a nearfield. Moreover the non-Desarguesian translation plane of order 9 has Lenz-Barlotti type IVa.3. |
Two versions of the defining spreadset for this plane are shown in Figure A. In the left part of Fig. A, the matrices of Dr. Klein are altered by the use of “2” instead of
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