# Antiparallelograms

Tim Hoffmann, Jürgen Richter-Gebert

### Description

The Discrete Darboux transform of a polygon (as detailed here [1]) is given by a parallelogram folded along its diagonal: Given an edge from $F$ to $F_1$ of the polygon and an initial point $F_2$ there is a unique point $F_{12}$ that has distance $\Vert F_1-F \Vert$ from $F_2$ and distance $\Vert F_2 - F\Vert$ from $F_1$ and that does not make $F,F_1,F_{12},F_2$ a parallelogram. One can think of this as an evolution equation mapping initial Cauchy data $F,f_1,F_2$ to a unique point $F_{12}$. This applet shows that this equation is 3d-consistent and thus integrable. If one picks a fourth initial point $F_3$ three folded parallelograms can be formed giving unique points $F_{12},F_{13}$, and $F_{23}$. Thinking of them as the first three sides of a cube three more can then be formed giving rise to three more points $F_{123},F_{231}, F_{312}$. It is a theorem that these three points always coincide and the construction always gives a combinatorial 3d-cube with all sides being folded parallelograms.

### References

• Tim Hoffmann.
Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow.
Discrete Differential Geometry, 2008(38):95–115, 2008.

#### Prof. Dr. Tim Hoffmann   +

Projects: A02
University: TU München, Department of Mathematics, 02.06.021
Address: Boltzmannstr. 3, 85748 Garching, GERMANY
Tel: +49 89 28918384
E-Mail: tim.hoffmann[at]ma.tum.de
Website: https://geo.ma.tum.de/de/personen/tim-hoffmann.html

#### Prof. Dr. Dr. Jürgen Richter-Gebert   +

University: TU München, Zentrum Mathematik, M10, 02.06.054
Address: Boltzmannstr. 3, 85748 Garching bei München, GERMANY
Tel: +49 89 28918354
E-Mail: richter[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/RichterGebert