Finite Geometry Notes   | Home | Site Map | Author |

Rosenhain and Göpel Tetrads in PG(3,2)

by Steven H. Cullinane on March 17, 2013

Hudson's 1905 classic Kummer's Quartic Surface
discusses the 80 Rosenhain odd tetrads and the 60 Göpel even tetrads.

These 140 tetrads occur naturally in the Galois projective 3-space PG(3,2).

IMAGE- Rosenhain and Göpel Tetrads in Hudson

 The Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of PG(3,2):IMAGE- Rosenhain and Göpel tetrads in PG(3,2)
As in the proof of the diamond theorem, each 4x4 array above has a set of three line diagrams. Arrays containing the Rosenhain odd tetrads (turquoise or white) have either 3 or 1 unidirectional line diagrams (all-horizontal or all-vertical lines). Arrays containing the Göpel even tetrads (yellow or gold) have either 2 or 0 unidirectional line diagrams.

The Göpel tetrads appear in my "Inscapes" notes and in "Picturing the Smallest Projective 3-Space" (April 1986). See Inscapes and Notes on Groups and Geometry, 1978-1986.

Göpel tetrads in an inscape, April 1986

More recently, they have appeared in "Birational Automorphisms of Quartic Hessian Surfaces," by Igor Dolgachev and JongHae Keum, in Transactions of the American Mathematical Society, Volume 354, Number 8, pages 3031-3057 (published online on April 3, 2002).

Dolgachev and Keum represent the Göpel tetrads as occuring in a set of 16 points that they identify with the empty set plus the fifteen 2-element subsets of a 6-set (p. 3037).  For an earlier version of this representation, see "The 2-subsets of a 6-set are the points of a PG(3,2)" (May 1986). That note also presents the Göpel tetrads as they relate to the Miracle Octad Generator of R. T. Curtis and to its underlying Moore correspondence.

Update: August 3, 2014

The Diamond-Theorem Correlation

Update: August 6-7 and 15, 2014

Symplectic Structure*

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ’plaited together’ or ‘woven.’
This is wonderfully apt….”

The above symplectic  structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

* The title phrase is a deliberate abuse of language.
For some background, see a definition of "symplectic structure"
as the relevant symplectic form. See also the phrase in Wikipedia's
"Symplectic vector space" article.

** See Steven H. Cullinane, Inscapes III, 1986

Update: August 15, 2014

Related readings:

Gonzalez-Dorrego on symplectic structure in PG(3,2)

Dolgachev and Keum, 2002, on symplectic structure

Six-set geometry, illustrated by Steven H. Cullinane

Symplectric structure in PG(3,2), 1986, by Steven H. Cullinane

Updates: March 26, 2013, August 3, 2014, August 6-7, 2014, August 15, 2014