Precise Graphic Images of Minimal Surfaces Using Computer

使用计算机获得最小表面的精确图形图像

• 批准号: 09640127
• 负责人: KAWASAKI Tetsuro
• 金额: $ 1.6万
• 依托单位: Gakushuin University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640127/
• 关键词: minimal surface; crystallographic group; triply periodic minimal surface; precise graphic image

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.

假设一条直线段是极小曲面边界的一部分。然后，曲面与其图像通过线对称形成一个光滑的极小曲面。这一性质称为极小曲面的反射原理。同时，如果极小曲面包含直线段，则存在对称邻域。现在假设给定了一个空间多边形，即有限个边的循环，并且它不包含在任何平面中。它通常以一个极小曲面为边界(高原问题)。根据反射原理，这样的极小曲面可以无限扩张。当给定的极子非常特殊时，扩展的极小曲面被嵌入，成为一个三周期极小曲面。在这种情况下，我们说一个空间多边形会生成一个三周期的极小曲面，这样的极小曲面包含许多直线。每条线都表示整个曲面的对称性。然后，由所有这些线对称性生成的群成为结晶学群。对晶体基团进行了完整的分类，我们可以列出由线对称产生的这些基团。然后，我们发现了35个生成结晶群的线系，它们可以包含在极小曲面中。最后，我们可以计算出生成三周期极小曲面的21个空间多边形。大约一半的多边形是已知的，但我们已经列出了该属性的所有空间多边形。