This is a family of discrete minimal surfaces given by an explicit formulation for the position of its vertices. It is a discrete variant of the smooth TT surface described by Karcher [1] which is a minimal surface bounded by two equilateral triangles in parallel planes and in twisted positions relative to each other.

A discrete minimal surface is a triangulated piecewise linear simplicial surface. It is critical for its surface area with respect to variations of the position of the inner vertices as defined in [2].

This surface is constructed using a surface patch consisting of five vertices `v`

and four triangles _{i}, i=0..4`T`

with _{i}=(v_{0}, v_{i}, v_{i+1}), i=1..4`v`

. The explicit coordinates of the vertices are up to a scaling factor given by:_{5}:=v_{0}

```
``````
v
```_{0} = (0, 3-1/sqrt(h^{2}/12+1/3), h/(4 sqrt(h^{2}/12+1/3))),

v_{1} = (0, 0, 0),

v_{2} = (sqrt(3), 3, 0),

v_{3} = (0, 2, h),

v_{4} = (-sqrt(3), 3, 0)

with a height parameter `h>0`

. The shown models have height h=5.

A calculation of the area gradient at the inner vertex `v`

shows that this surface patch is a minimal surface._{0}

In a second step, reflect the whole geometry across the plane `{(x,y,h/2) | x,y in R}`

and rotate the reflected geometry around the z axis by 60 degrees. The result is a second surface which joins the original surface at the vertices `v`

and _{2}`v`

. Repeat the reflection and rotation until 6 surface patches are obtained. They arrange to a discrete minimal surface which is bounded by two twisted triangles as the smooth TT surface is (figure 1)._{3}

The second figure shows how the discrete TT surface can be extended by rotation about the outer edges to a triply periodic discrete minimal surface.

In the special case `h=2 sqrt(2)`

, the TT surface gets additional symmetries. Pairs of adjacent triangles become planar quadrilaterals and the geometry becomes a discrete variant of the Schwarz D surface. A fundamental part of this triply periodic surface is shown in figure 3. By swapping each planar edge in the quadrilaterals, one gets a discrete minimal surface, which was constructed by Rossman [5] and called a discrete variant of the superman surface.

Some likewise constructed discrete minimal surfaces are given in [5] (Rossman), e.g. discrete versions of the Schwarz CLP surface or H surface from Schoen. Polthier and Rossman [4] presented an explicit description of a family of discrete catenoids and helicoids.

Model produced with: JavaView 3.80

Keywords | TT surface; Minimal surface; Discrete surface | |

MSC-2000 Classification | 53-04 (53-XX, 68Uxx, 68Rxx, 65Kxx, 65M50) | |

Zentralblatt No. | 05264903 |

- H. Karcher:
*The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions*, Manuscripta Math.**64**(1989), 291--357, . - U. Pinkall and K. Polthier:
*Computing Discrete Minimal Surfaces and Their Conjugates*, Experimental Mathematics**2**(1993), 15--36, . - K. Polthier, K. Hildebrandt, E. Preuss and U. Reitebuch:
*JavaView Homepage*, http://www.javaview.de/. - K. Polthier and W. Rossman:
*Index of Discrete Constant Mean Curvature Surfaces*, J. Reine und Angew. Math. (Crelle Journal)**549**(2002), 47--77, . - W. Rossman:
*Infinite periodic discrete minimal surfaces without selfintersections*, eprint arXiv:math/0410314 (10/2004), .

- Master File: TTsurface_Master.jvx
- Master File: TTsurface_Extended_Master.jvx
- Master File: Schwarz_Master.jvx
- Applet File: TTsurface_Master.jvx
- Applet File: TTsurface_Extended_Master.jvx
- Applet File: Schwarz_Master.jvx
- Preview: TTsurface_Preview.gif
- Preview: TTsurface_Extended_Preview.gif
- Preview: Schwarz_Preview.gif

Submitted: Fri Apr 8 14:09:40 CEST 2005.

Revised: Thu Jun 2 13:05:42 CEST 2005, Wed Jun 8 13:12:35 MET DST 2005.

Accepted: Mon Jul 4 13:49:28 MET DST 2005.

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