## Secondary polytope of a cyclic 8-polytope with 12 vertices

Electronic Geometry Model No. 2000.09.032#### Author

Julian Pfeifle#### Description

Secondary polytope of an 8-dimensional polytope with 12 vertices which is
combintorially equivalent to the cyclic polytope.

The polytope displayed is **NOT the secondary polytope of
C**_{8}(12) as realized on the *standard* moment curve
(t,t^{2},...,t^{8}). Instead, you see the secondary
polytope of the 8-dimensional *Carathéodory cyclic polytope* with 12
vertices, that is, it is realized on the *trigonometric* moment curve
(sin t, cos t, sin 2t, cos 2t, ..., cos 4t). The secondary of the usual
cyclic polytope is too flat to be viewed well. The two types of cyclic
polytopes are combinatorially equivalent, but not geometrically. Therefore,
the secondary polytopes differ.

The standard realization of the
standard C_{8}(12) has **42(!)** non-regular triangulations, all
at flip distance 1 from regular ones. They are marked with red vertices. The
realization of the Carathéodory cyclic polytope shown has only 38
non-regular triangulations, all of them also at flip distance 1.

The
files `cyclic*` resp. `car_cyclic*` contain the information
about the standard resp. Carathéodory cyclic polytope.

Model produced with: polymake 1.4, TOPCOM 0.9.0

**Keywords**
| | secondary polytope; triangulation |

**MSC-2000 Classification**
| | 52B11 (52B55) |

**Zentralblatt No.**
| | 01682981 |

#### References

- I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky: Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser (1994).

#### Files

#### Submission information

Submitted: Sun Sep 10 07:10:24 CET 2000.

Accepted: Mon Nov 20 17:06:57 CET 2000.

#### Author's Address

Julian Pfeifle
Technische Universität Berlin

Fachbereich Mathematik, MA 7-1

Straße des 17. Juni 136

10623 Berlin, Germany

pfeifle@math.tu-berlin.de

http://www.math.tu-berlin.de/~pfeifle