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Classical Geometry in Light of Galois Geometry

The following journal notes discuss the relationship of three frontispiece figures from Baker's Principles of Geometry (six volumes, the first four in 1922-1925) to the Galois projective 3-space PG(3,2).

The Rosenhain and Göpel tetrads within PG(3,2) are shown to underlie these figures. The Göpel tetrads underlie the Cremona-Richmond figure that is the frontispiece of Baker's Volume IV (Higher Geometry).  The Rosenhain tetrads underlie the figure illustrating Desargues's theorem that is the frontispiece of Baker's Volume I (Foundations).

The Göpel tetrads also underlie Baker's frontispiece illustrating Pascal's Hexagrammum Mysticum in Volume II (Plane Geometry). A Galois-geometry key to the mystic hexagram is taken from a May 26, 1986 note on 2-subsets of a 6-set and PG(3,2).

The 8-sets (and the associated Galois geometry PG(3,2)) in the Miracle Octad Generator (MOG) of R. T. Curtis turn out be closely related to the Hexagrammum via the 354 configuration of Danzer discussed in a recent paper.

The Journal Notes:

Sunday, March 31, 2013 (Easter)

For Pascal

By Steven H. Cullinane at 2:28 PM

A related image search, for Cremona synthemes, is omitted here,
but one result from the search is retained...

Note particularly the following image:

This is from Inscapes.

Sunday, March 31, 2013

For Baker

By Steven H. Cullinane at 8:00 PM

Baker, Principles of Geometry, Vol. IV  (1925), Title:

Baker, Principles of Geometry, Vol. IV  (1925), Frontispiece:

Baker's Vol. IV frontispiece shows "The Figure of fifteen lines 
and fifteen points, in space of four dimensions."

Another such figure in a vector space of four dimensions 
over the two-element Galois field  GF(2):

(Some background grid parts were blanked by an image resizing process.)

Here the "lines" are actually planes  in the vector 4-space over GF(2),
but as planes through the origin  in that space, they are projective  lines .

For some background, see today's previous post and Inscapes.

Update of 9:15 PM March 31

The following figure relates the above finite-geometry 
inscape  incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.

Monday, April 1, 2013

Desargues via Rosenhain

By Steven H. Cullinane at 6:00 PM

Background: Rosenhain and Göpel Tetrads in PG(3,2)

Introduction (an update added on April 19, 2013):

The Large Desargues Configuration

By Steven H. Cullinane 

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.

IMAGE- The proof of the converse of Desargues' theorem involves a third triangle.

A connection discovered today (April 1, 2013)—

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:

IMAGE- Desargues' theorem with three triangles, and Galois-geometry version

(End of  April 12, 2013 update)

Update of April 13, 2013

Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:

IMAGE- Veblen and Young, 1910 Desargues illustration, with 2013 Galois-geometry version

See also the original Veblen-Young figure in context.

(End of April 13, 2013 update.)

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) 
(click to enlarge)

Higher Geometry
(click to enlarge)

Tuesday, April 2, 2013

Baker on Configurations

By Steven H. Cullinane at 11:11 AM 

The geometry posts of Sunday and Monday have been
placed in 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 15420configuration (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."

IMAGE- Definition of 'nocciolo' as 'kernel'

A related nocciolo :

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Click on the nocciolo  for some
geometric background.

Saturday, April 6, 2013

Pascal via Curtis

By Steven H. Cullinane at 9:17 AM 

Click image for some background.

IMAGE- The Miracle Octad Generator (MOG) of R.T. Curtis

Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)

The 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell in each is disregarded)
the thirty-five 3-subsets of a 7-set.

Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the mystic hexagram of Pascal.

On Danzer's 354 Configuration:

IMAGE- Branko Grünbaum on Danzer's configuration

"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines 
(represented by 4-sets) that contain the 3-set."

– Branko Grünbaum, "Musings on an Example of Danzer's," 
European Journal of Combinatorics , 29 (2008), 
pp. 1910–1918 (online March 11, 2008)

"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."

– Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited,", Jan. 6, 2013

For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see

Classical Geometry in Light of Galois Geometry.

Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).

Sunday, April 7, 2013

Pascal Inscape

By Steven H. Cullinane at 1:00 PM 

A Galois-geometry figure as a key to the mystic hexagram of Pascal

Background:  Inscapes and The 2-subsets of a 6-set are the points of a PG(3,2).

Tuesday, April 23, 2013

The Six-Set

By Steven H. Cullinane at 3:00 AM

The configurations recently discussed in
"Classical Geometry in Light of Galois Geometry"
are not unrelated to the 27 "Solomon's Seal Lines
extensively studied in the 19th century.

See, in particular—

IMAGE- Archibald Henderson on six-set geometry (1915)

The following figures supply the connection of Henderson's six-set
to the Galois geometry previously discussed in "Classical Geometry…"—

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Update of August 16,  2013:

Click above image for some background from 1986.

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

Page created April 1, 2013.