The previous post suggests a flashback to June 8, 2014 —
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In the simplest case of a projective space Vide Classical Geometry in Light of Galois Geometry.
* The two types of lines named are derived from
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Friday, April 3, 2026
Diagon Alley
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.
Sunday, December 18, 2016
Two Models of the Small Desargues Configuration
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.
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.
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.
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
Wednesday, April 22, 2015
Purely Aesthetic
G. H. Hardy in A Mathematician's Apology —
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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:
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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 …
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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'"
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 ".
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:
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:
