Simple 3-D Visualization of Some Common Mathematical Minimal Surfaces Using MATLAB

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Simple 3-D Visualization of Some Common Mathematical Minimal Surfaces using MATLAB

Nasiha Muna1 and Albert E. Patterson2

1Department of Physics, Chemistry, and Mathematics, Alabama Agricultural and Mechanical (A&M) University, Normal, Alabama, 35762, USA. Email: [email protected] 2Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801, USA. Email: [email protected]

The MATLAB code for this work is available at: https://github.com/pttrsnv2/Minimal_ Surfaces_Visualization_Code. This report and its code are published under a CC-BY 4.0 International license, so it can be used and modiﬁed with proper attribution to the authors.

Abstract This report presents a simple approach for visualizing some common mathematical minimal surfaces using MATLAB tools. The studied minimal surfaces were the gyroid, lidinoid, Schwarz P surface, Schwarz D surface, Scherk tower, helicoid surface, catenoid surface, and Mobius strip. For each, the mathematic deﬁnition is given along with a simple matlab code for it and an example 3-D surface plot generated from the given code.

1 Introduction The selected minimal surfaces to be discussed in this report were the gyroid, lidinoid, Schwarz P surface, Schwarz D surface, Scherk tower, helicoid surface, catenoid surface, and Mobius strip. These were selected, as they represent a suitable subset of the available minimal surfaces and serve as good cases to demonstrate the simple approach proposed in this report. Some very complex surfaces, such as Costa surfaces, are not covered here and will be discussed in future works. Most minimal surfaces are sometimes relatively difficult to plot in three dimensions, as they are all typically defined by implicit or parametric functions (sometimes in the complex domain). Fortunately, appropriate built-in MATLAB functions can be used to plot each of the surfaces relative to its defining equation. The gyroid and lidinoid functions are triply periodic minimal surfaces of genus 3 embedded in R3 that possess no planar symmetry curves nor straight lines. They are the distinctive embedded surfaces of the associate families of the Schwarz P and H surfaces [1–3]. The gyroid function is typically defined as

fgy = sin(x) cos(y) + sin(y) cos(z) + sin(z) cos(x) = 0 (1) while the lidinoid function is most often expressed as:

1 flid = (sin(2x) cos(y) sin(z) + sin(2y) cos(z) sin(x) + sin(2z) cos(x) sin(y)) 2 (2) 1 − (cos(2x) cos(2y) + cos(2y) cos(2z) + cos(2z)cos(2x)) + 0.15 = 0 2

The Schwarz minimal surfaces span many diﬀerent kinds of forms, but the two most commonly- seen are the Schwarz Primitive (“Schwarz P”) and Schwarz Diamond (“Schwarz D”) surfaces [3–5]. The Schwarz P surface is the triply periodic minimal surface built from a skew hexagon inscribed in the edges of a regular octahedron. The Schwarz P surface encompasses sixfold junctions and is typically expressed as:

1 Minimal Surface Visualization with MATLAB Muna and Patterson

fsp = cos(x) + cos(y) + cos(z) (3)

The Schwarz D surface is similar to the Schwarz P except that it contains six skew quadrilaterals from a skew hexagon inscribed in the edges of a cube. It has two intertwined corresponding mazes, each holding the ﬁgure of a version of the diamond arrangement; unlike Schwarz P, Schwarz D had fourfold junctions. Schwarz D is most often expressed as:

fsd = sin(x) sin(y) sin(z) + sin(x) cos(y) cos(z)+ (4) cos(x) sin(y) cos(z) + cos(x) cos(y) sin(z) = 0

The Scherk surface is a 1-parameter singly-periodic, complete, properly-embedded surface in R3 and can be modified in shape by defining new handles for it [6,7]. Several forms exist, but the most common is the Scherk tower, which can be defined as:

fsc = sinh(x) sinh(y) − sin(z) (5)

The helicoid surface is simply a non-algebraic surface that coils around an axis and is one of the “parametric” minimal surfaces discussed in this report. When used as a minimal surface, the edge of the surface is bounded by a circular helix and is therefore the only ruled minimal surface besides the plane. The helicoid is very commonly encountered in nature, with examples including seashells, horns, and polymer structures [8–10]. A good parametric equation to describe the surface is:

x = u cos(t) y = u sin(t) (6) 2 z = t 3 where u and t are the parameters speciﬁed by the user. The catenoid surface is another parametric minimal surface, which is an inﬁnite, boundless, and locally isometric to the helicoid surface previously discussed. The equations are mathematically similar to the helicoid, with removal of the helical boundary and jointing of the ends [3, 11]. This involves the use of hyperbolic sines and cosines in the equation, which can be expressed as:

x = cos(α) sinh(v) sin(u) + sin(α) cosh(v) cos(u) y = − cos(α) sinh(v) cos(u) + sin(α) cosh(v) sin(u) (7) z = u cos(α) + v sin(α) where u and v are the parameters speciﬁed by the user and α = π/2 is the value for a catenoid (α = 0 corresponds to a helicoid) [12, 13]. Finally, the Mobius strip was examined. This is one of the most commonly-seen and known minimal surface in mathematics and the basic surface related to many other mathematical structures, including knots and Klein bottles [14–16]. A common parametric expression for the Mobius strip is:

v u x = cos(u) 1 + cos 2 2 v u y = sin(u) 1 + cos (8) 2 2 v u z = sin 2 2 where u and v are the parameters speciﬁed by the user.

Technical Report 2 Minimal Surface Visualization with MATLAB Muna and Patterson

In each of the following subsections, the basic MATLAB code and a visualization of the 3-D surface are given. The first five (gyroid, lidinoid, Schwarz P, Schwarz D, and Scherk) are defined above by 3-D implicit equations, requiring the use of the fimplicit3 tool to plot. The others (helicoid, catenoid, and Mobius strip) are defined according to parametric equations and are therefore easier to plot using a standard fsurf built-in function. In all cases, the basic equation definition is given, followed by limits on the values of the variables or parameters, the mesh density setting, and the face color for the plot. All of these settings are infinitely-adjustable within the domain of the defining functions. It is recommended that MATLAB version 2017a or higher be used.

2 Minimal Surface Cases

2.1 Gyroid

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2.2 Lidinoid

Figure 1: Lidinoid

Figure 2: Lidinoid

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2.3 Schwarz P Surface

Figure 3: Schwarz P

Figure 4: Schwarz P

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2.4 Schwarz D Surface

Figure 5: Schwarz D

Figure 6: Schwarz D

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2.5 Scherk Surface

Figure 7: Scherk

Figure 8: Scherk

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2.6 Helicoid Surface

Figure 9: Helicoid

Figure 10: Helicoid

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2.7 Catenoid Surface

Figure 11: Catenoid

Figure 12: Catenoid

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2.8 Mobius Strip

Figure 13: Mobius

Figure 14: Mobius

3 Summary and Conclusions

Minimal surfaces are very important in mathematics, art, and design. Various forms of minimal surfaces can be identiﬁed; visualizing minimal surfaces is important because it allows users to view and handle physical objects representing various natural and mathematical phenomena. By studying minimal surfaces, structures that are otherwise diﬃcult to represent mathematically can be graphically displayed and understood.

References

[1] A. Weyhaupt, “Deformations of the gyroid and lidinoid minimal surfaces,” Paciﬁc Journal of Mathe- matics, vol. 235, no. 1, pp. 137–171, 2008. [2] S. N. Krivoshapko and V. N. Ivanov, “Minimal surfaces,” in Encyclopedia of Analytical Surfaces, pp. 427–457, Springer International Publishing, 2015.

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[3] H. Karcher and K. Polthier, “Construction of triply periodic minimal surfaces,” Philosophical Trans- actions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 354, no. 1715, pp. 2077–2104, 1996. [4] H. Karcher, “The triply periodic minimal surfaces of alan schoen and their constant mean curvature companions,” Manuscripta Mathematica, vol. 64, no. 3, pp. 291–357, 1989. [5] D. M. Anderson, H. T. Davis, J. C. C. Nitsche, and L. E. Scriven, “Periodic surfaces of prescribed mean curvature,” in Springer Proceedings in Physics, pp. 130–130, Springer Berlin Heidelberg, 1987. [6] A. J. Y. Hancco, G. A. Lobos, and V. R. Batista, “Explicit minimal scherk saddle towers of arbitrary even genera in r3,” Publicacions Matemàtiques, vol. 58, pp. 445–468, 2014. [7] J. Perez and M. Traizet, “The classiﬁcation of singly periodic minimal surfaces with genus zero and scherk-type ends,” Transactions of the American Mathematical Society, vol. 359, pp. 965–990, 2007. [8] C. Mamaloukas, “On determination of developable and minimal surfaces,” International Journal of Pure and Applied Mathematics, vol. 39, pp. 193–201, 2007. [9] W. Meeks and H. Rosenberg, “The uniqueness of the helicoid,” Annals of Mathematics, vol. 161, no. 2, pp. 727–758, 2005. [10] R. Ghafouri and R. Bruinsma, “Helicoid to spiral ribbon transition,” Physical Review Letters, vol. 94, no. 13, 2005. [11] A. Boudaoud, P. Patrcio, and M. B. Amar, “The helicoid versus the catenoid: Geometrically induced bifurcations,” Physical Review Letters, vol. 83, no. 19, pp. 3836–3839, 1999. [12] A. Ogawa, “Helicatenoid,” The Mathematica Journal, vol. 2, no. 2, 1992. [13] H. Rosenberg, “Deformations of complete minimal surfaces,” Transactions of the American Mathe- matical Society, vol. 295, no. 2, pp. 475–475, 1986. [14] G. Pradisi and A. Sagnotti, “Open string orbifolds,” Physics Letters B, vol. 216, no. 1-2, pp. 59–67, 1989. [15] E. L. Starostin and G. H. M. van der Heijden, “The shape of a möbius strip,” Nature Materials, vol. 6, no. 8, pp. 563–567, 2007. [16] V. Petresin and L.-P. Robert, “The double möbius strip studies,” Nexus Network Journal, vol. 4, no. 2, pp. 54–64, 2002.

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