Precise Graphic Images of Minimal Surfaces Using Computer

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Suppose a line segment is a part of a boundary of a minimal surface. Then the surface and its image by the line symmetry forms a smooth minimal surface. This property is called the reflection principle of minimal surfaces. Also, if a minimal surface contains a line segment, then there is a symmetric neighborhood.Now assume that a spatial poligon is given, that is, a cycle of a finite number of edges, and it is not contained in any plane. It often bounds a minnial surface (the Plateau problem). By the reflection principle, such a minimal surface can be extended infinitely. When the given poligon is very special, the extended minimal surface is embedded and becomes a triply periodic minimal surface. In such a case, we say a spatial poligon generates a triply periodic minimal surface.Such a minimal surface contains many lines. Each line gives a symmetry of the whole surface. Then, the group generated by all such line symmetries becomes a crystalographic group. The crystallographic groups are classified completely, and we can list up the such groups generated by line symmetries. And then, we found 35 systems of lines that generate crystallographic group and can be contained in minimal surfaces. Finally, we can count 21 spatial poligons that generate triply periodic minimal surfaces. About one half of the poligons are known, but we have listed all the spatial poligons of this property.