**… 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 M_{24} —

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.