## Triply Periodic Minimal Surfaces

A minimal surface is the surface of minimal area between any given boundaries. In nature such shapes result from an equilibrium of homogeneous tension, e.g. in a soap film.

Minimal surfaces have a constant mean curvature of zero, i.e. the sum of the principal curvatures at each point is zero. Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense of repeating themselves in three dimensions, in other words being triply periodic.

Many triply periodic minimal surfaces are known. The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student Neovius in 1883. In 1970 Alan Schoen, a then NASA scientist, described 12 more TPMS, and in 1989 H. Karcher proved their existence.

My research into grid structures with the goal of simplifying fabrication through repetitive elements prompted an exploration of TPMS. The highly symmetrical and optimised physical properties of a TPMS, in particular the Gyroid surface, inspired my studio proposal, Minimal Matters.

The gyroid is an infinitely connected periodic minimal surface discovered by Schoen in 1970. It has three-fold rotational symmetry but no embedded straight lines or mirror symmetries.

The boundary of the surface patch is based on the six faces of a cube. Eight of the surface patch forms the cubic unit cell of a Gyroid.

For every patch formed by the six edges, only three of them is connected with the surrounding patches.

Note that the cube faces are not symmetry planes. There is a C3 symmetry axis along the cube diagonal from the upper right corner when repeating the cubic unit cell.

Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by a short equation:

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

Using Grasshopper and the â€˜Iso Surfaceâ€™ component of Millipede, many TPMS can be generated by finding the result of itâ€™s implicit equation.

Standard F(x,y,z) functions of minimal surfaces are defined to determine the shapes within a bounding box. The resulting points form a mesh that describes the geometry.

1. A cube of points are constructed via a domain and fed into a function. Inputs of standard minimal surfaces are used as the equation.
2. The resulting function values are plugged into Millipedeâ€™s Isosurface component.
3. The bounding box sets up the restrictions for the geometry.
4. Xres, Yres, Zres [Integer]: The resolution of the three dimensional grid.
5. Isovalue: The â€˜IsoValueâ€™ input generates the surface in shells, with zero being the outermost shell, and moving inward.
6. Merge: If true the resulting mesh will have its coinciding vertices fused and will look smoother (continuous, not faceted)

The above diagrams show Triply Periodic Minimal surfaces generated from their implicit mathematical equations. The functions are plotted with a domain of negative and positive Pi. By adjusting the domain to 0.5, the surface patch can be generated.

Many TPMS can best be understood and constructed in terms of fundamental regions (or surface patches) bounded by mirror symmetry planes. For example, the fundamental region formed in the kaleidoscopic cell of a Schwarz P surface is a quadrilateral in a tetrahedron, which 1 /48 of a cube (shown below left). Four of which create the surface patch. The right image shows a cubic unit cell, comprising eight of the surface patch.

Schoen’s batwing surface has the quadrilateral tetrahedron (1/48 of a cube) as it’s kaleidoscopic cell, with a C2 symmetry axis. As shown in the evolution diagram below, the appearance of two fundamental regions is the source of the name ‘batwing’. Twelve of the fundamental regions form the cubic unit cell; however this is still only 1/8 of the complete minimal surface lattice cell.

## Growth From The Ger

#### Introduction

‘Growth From The Ger’ seeks to analyse the vernacular structure of the traditional nomad home and use parametric thinking to create a deployable structure that can grow by modular.

‘Ger’ meaning ‘home’ is a Mongolian word which describes the portable dwelling. Commonly known as a ‘yurt’, a Turkish word, the yurt offered a sustainable lifestyle for the nomadic tribes of the steppes of Central Asia. It allowed nomads to migrate seasonally, catering to their livestock, water access and in relation to the status of wars/conflicts. An ancient structure, it has developed in material and joinery, however the concept prominently remaining the same.

#### Inspiration

Growing up in London, I fell in love with the transportable home when I first visited Mongolia at the age of 17. The symmetrical framework and circulating walls create a calm and peaceful environment. In the winter it keeps the cold out and in the summer keeps the heat out. The traditional understanding of placement and ways of living within it, which seems similar to a place of worship, builds upon the concept of respect towards life and its offerings.

Understanding the beauty of the lifestyle, I also understand the struggles that come with it and with these in mind, I wanted to explore ways of solving it whilst keeping the positives of the lifestyle it offers.

Pros: Deployable, transportable, timber, vernacular, can be assembled and dissembled by one family, can vary in size/easily scaleable depending on user, low maintenance, sustainable, autonomous.

Cons: Difficult to sustain singularly, not water proof, no privacy, no separation of space, low ceiling height, can’t attach gers together, low levels of security.

#### Lattice Analysis and Testing

To understand the possibilities of the lattice wall, I created a 1:20 plywood model using 1mm fishing wire as the joinery. This created various circular spirals and curves. The loose fit of the wire within the holes of timber pieces allowed such curves to happen and created an expanding body. The expansion and flexible joinery allows it to cover a wider space in relation to the amount of material used.

I created the same latticework at 1:2 scale to see if the same curvature was created.

Locking the curve to create a habitable space. I did this by changing the types of joints in different parts of the structure.

To create a smoother and more beautiful curve I change the baton to a dowel and densify the structure.

To lock the lattice curve in expansion I extrude legs that meet the ground and tie together.

#### Manufacturing and assembly

The model made from sheet plywood cost approximately Â£30 and took one working day to make for one person. However, a more sustainable material and process needed to be considered as the process of making plywood contradicted this.

This model can be made by one person with the use of a wood workshop. The timber pieces were bought at 18mm x 95mm x 4200mm, 13 pieces of these were enough to make three modules, roughly costing Â£170 in total. Each module takes approximately 5 hours to construct, this involves the tying of the measured length twine joints. The structure is lightweight and each module is easily transportable by one person.