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Precise Graphic Images of Minimal Surfaces Using Computer

Research Project

Project/Area Number 09640127
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionGakushuin University

Principal Investigator

KAWASAKI Tetsuro  Gakushiin U., Dept.of Mathematics, Associate Professor, 理学部, 助教授 (90107061)

Co-Investigator(Kenkyū-buntansha) MIZUTANI Akira  Gakushuin U., Dept.of Mathematics, Professor, 理学部, 教授 (80011716)
FUJIWARA Daisuke  Gakushuin U., Dept.of Mathematics, Professor, 理学部, 教授 (10011561)
KURODA Shigetoshi  Gakushuin U., Dept.of Mathematics, Professor, 理学部, 教授 (20011463)
KATASE Kiyoshi  Gakushuin U., Dept.of Mathematics, Professor, 理学部, 教授 (70080489)
IITAKA Shigeru  Gakushuin U., Dept.of Mathematics, Professor, 理学部, 教授 (20011588)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 1998: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1997: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsminimal surface / crystallographic group / triply periodic minimal surface / precise graphic image
Research Abstract

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.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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