scholarly journals Minimal Surfaces with Only Horizontal Symmetries

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Márcio Fabiano da Silva ◽  
Guillermo Antonio Lobos ◽  
Valério Ramos Batista
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Reflection Principle ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
Horizontal Type ◽  
Straight Lines ◽  
Schwarz Reflection Principle

The Schwarz reflection principle states that a minimal surface S in ℝ3 is invariant under reflections in the plane of its principal geodesics and also invariant under 180°-rotations about its straight lines. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type.

Description of a 3-periodic minimal surface family with trigonal symmetry

1994 ◽  
Vol 209 (1) ◽  
Author(s):  
A. Fogden
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Systematic Analysis ◽  
Trigonal Symmetry ◽  
Triply Periodic Minimal Surfaces ◽  
Periodic Minimal Surface ◽  
Triply Periodic ◽  
The Family

AbstractA systematic analysis of a family of triply periodic minimal surfaces of genus seven and trigonal symmetry is given. The family is found to contain five such surfaces free from self-intersections, three of which are previously unknown. Exact parametrisations of all surfaces are provided using the Weierstrass representation.

Periodic minimal surfaces embedded in ℝ3 derived from the singly periodic Scherk minimal surface

2018 ◽  
Vol 22 (01) ◽  
pp. 1850075
Author(s):  
Filippo Morabito
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Parameter Family ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic

We construct three kinds of periodic minimal surfaces embedded in [Formula: see text] We show the existence of a [Formula: see text]-parameter family of minimal surfaces invariant under the action of a translation by [Formula: see text] which seen from a distance look like [Formula: see text] equidistant parallel planes intersecting orthogonally [Formula: see text] equidistant parallel planes, [Formula: see text] [Formula: see text] We also consider the case where the surfaces are asymptotic to [Formula: see text] equidistant parallel planes intersecting orthogonally infinitely many equidistant parallel planes. In this case, the minimal surfaces are doubly periodic, precisely they are invariant under the action of two orthogonal translations. Last we construct triply periodic minimal surfaces which are invariant under the action of three orthogonal translations in the case of two stacks of infinitely many equidistant parallel planes which intersect orthogonally.

scholarly journals Porous polylactic acid scaffolds for bone regeneration: A study of additively manufactured triply periodic minimal surfaces and their osteogenic potential

2020 ◽  
Vol 11 ◽  
pp. 204173142095654
Author(s):  
Anna Diez-Escudero ◽  
Hugo Harlin ◽  
Per Isaksson ◽  
Cecilia Persson
Keyword(s):  
Polylactic Acid ◽  
Cancellous Bone ◽  
Minimal Surfaces ◽  
Pore Distribution ◽  
Osteogenic Potential ◽  
Osteoblastic Cell ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
3D Printed ◽  
Grade Material

Three different triply periodic minimal surfaces (TPMS) with three levels of porosity within those of cancellous bone were investigated as potential bone scaffolds. TPMS have emerged as potential designs to resemble the complex mechanical and mass transport properties of bone. Diamond, Schwarz, and Gyroid structures were 3D printed in polylactic acid, a resorbable medical grade material. The 3D printed structures were investigated for printing feasibility, and assessed by morphometric studies. Mechanical properties and permeability investigations resulted in similar values to cancellous bone. The morphometric analyses showed three different patterns of pore distribution: mono-, bi-, and multimodal pores. Subsequently, biological activity investigated with pre-osteoblastic cell lines showed no signs of cytotoxicity, and the scaffolds supported cell proliferation up to 3 weeks. Cell differentiation investigated by alkaline phosphatase showed an improvement for higher porosities and multimodal pore distributions, suggesting a higher dependency on pore distribution and size than the level of interconnectivity.

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