Minimal surfaces with self-intersections along straight lines. I. Derivation and properties

Area of Science:

• Differential Geometry
• Topology
• Materials Science

Background:

• Three-periodic minimal surfaces (TPMS) are fundamental in various scientific fields.
• Understanding their generation and properties is crucial for applications in materials science and architecture.
• Existing methods for deriving TPMS can be complex and limited.

Purpose of the Study:

• To introduce a novel procedure for deriving a specific class of three-periodic minimal surfaces.
• To establish a direct link between the generating skew polygon and the surface's intrinsic properties.
• To provide a systematic method for analyzing these surfaces and their associated spatial subunits.

Main Methods:

• Development of a new derivation procedure for TPMS based on disc-like-spanned skew polygons.
• Analysis of geometric and topological properties directly from the generating polygon.
• Illustration of the procedure and deduced properties with concrete examples.

Main Results:

• A new method for constructing specific self-intersecting TPMS is presented.
• Key surface properties, including symmetry groups, orientability, and self-intersection patterns, are directly linked to the generating polygon.
• The symmetry, periodicity, and Euler characteristics of the demarcated spatial subunits can be determined from the polygon.

Conclusions:

• The proposed method offers a simplified and insightful approach to understanding and generating these special TPMS.
• The direct correlation between the generating polygon and surface properties facilitates detailed analysis and design.
• This work provides a valuable tool for researchers and designers working with periodic minimal surfaces.