Triply periodic minimal surfaces _topology

Frequently occuring in nature, minimal surfaces are defined as surfaces with zero mean curvature.  These surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.

The thin membrane that spans the wire boundary is a minimal surface of all possible surfaces that span the boundary, it is the one with minimal energy. One way to think of this “minimal energy” is that to imagine the surface as an elastic rubber membrane: the minimal shape is the one that in which the rubber membrane is the most relaxed.

 

A minimal surface parametrized as x=(u,v,h(u,v)) therefore satisfies Lagrange`s equation

(1+h(v)^2)*h(uu)-2*h(u)*h(v)*h(uv)+(1+h(u)^2)*h(vv)=0

(Gray 1997, p.399)

This year`s research focuses on triply periodic minimal surfaces (TPMS). A TPMS is a type of minimal surface which is invariant under a rank-3 lattice of translations. In other words, a TPMS is a surfaces which, through mirroring and rotating in 3D space, can form an infinite labyrinth. TPMS are of particular relevance in natural sciences, having been observed in observed as biological membranes, as block copolymers, equipotential surfaces in crystals, etc.

From a mathematical standpoint, a TPMS is the most interesting type of surface, as all connected RPMS have genus >=3, and in every lattice there exist orientable embedded TPMS of every genus >=3. Embedded TPMS are orientable and divide space into disjoint sub-volumes. If they are congruent the surface is said to be a balance surface.

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.

The first part of my research focuses on understanding TPMS geometry using a generation method that uses a marching cubes algorithm to find the results of the implicit equtions describing each particular type of TMPS. The resulting points form a mesh that describes the geometry.

Schwartz_P surface

   

Neovius surface

 

Gyroid surface

 

Generated from mathematical equations, these diagrams show the plotting of functions with different domains. Above, the diagrams on the left illustrate the process of forming a closed TMPS, starting from a domain of 0.5, which generates an elementary cell, which is mirrored and rotate 7 times to form a closed TPMS. A closed TMPS can also be approximated by changing the domain of the function to 1.

The diagrams below show some examples generating a TMPS from a function with a domain of 2. The views are front, top and axonometric.

FRD surface

dd = 8 * Math.Cos(px) * Math.Cos(py) * Math.Cos(pz) + Math.Cos(2 * px) * Math.Cos(2 * py) * Math.Cos(2 * pz) – Math.Cos(2 * px) * Math.Cos(2 * py) – Math.Cos(2 * py) * Math.Cos(2 * pz) – Math.Cos(2 * pz) * Math.Cos(2 * px)

D Prime surface

dd = 0.5 * (Math.Sin(px) * Math.Sin(py) * Math.Sin(pz) + Math.Cos(px) * Math.Cos(py) * Math.Cos(pz)) – 0.5 * (Math.Cos(2 * px) * Math.Cos(2 * py) + Math.Cos(2 * py) * Math.Cos(2 * pz) + Math.Cos(2 * pz) * Math.Cos(2 * px)) – 0.2

FRD Prime surface

dd = 4 * Math.Cos(px) * Math.Cos(py) * Math.Cos(pz) – Math.Cos(2 * px) * Math.Cos(2 * py) – Math.Cos(2 * pz) * Math.Cos(2 * py) – Math.Cos(2 * px) * Math.Cos(2 * pz)

Double Gyroid surface

dd = 2.75 * (Math.Sin(2 * px) * Math.Sin(pz) * Math.Cos(py) + Math.Sin(2 * py) * Math.Sin(px) * Math.Cos(pz) + Math.Sin(2 * pz) * Math.Sin(py) * Math.Cos(px)) – 1 * (Math.Cos(2 * px) * Math.Cos(2 * py) + Math.Cos(2 * py) * Math.Cos(2 * pz) + Math.Cos(2 * pz) * Math.Cos(2 * px))

Gyroid surface

dd = Math.Cos(px) * Math.Sin(py) + Math.Cos(py) * Math.Sin(pz) + Math.Cos(pz) * Math.Sin(px)

This method of approximating a TPMS is high versatile, useful in understanding the geometry, offsetting the surfaces and changing the bounding box of the lattice in which the surface is generated. In other words, trimming the surface and isolating parts of the surface. However, the resulting topology is unsuitable for fabrication purposes, as the generated mesh is unclean, being composed of irregular polygons consisting of triangulations, quads and hexagons.

The following diagrams show the mesh topology for a Gyroid surface, offset studies and trimming studies.

 

  

For fabrication purposes, my proposed method for computationally simulating a TPMS is derived from discrete differential geometry, relying on the use of Kangaroo Physics, a Grasshopper plugin for modeling tensile membranes. Bearing in mind that a TPMS has 6 edge conditions, a planar hexagonal mesh is placed within the space defined by a certain TPMS`s edge conditions. The edge conditions are interpreted as Nurbs curves. Constructed from 6 predefined faces, the initial planar hexagonal mesh, together with the curves defining the surface boundaries are split into the same number of subdivisions. The subdivision algorithm used on the mesh is WeaveBird`s triangular subdivision. The points resulted from the curve division are ordered so that they match the subdivided mesh`s edges, or its naked vertices. The naked vertices are then moved in the corresponding points on the curve, resulting in a new mesh describing a triply periodic surface, but not a minimal one. From this point, Kangaroo Physics is used to find the minimal surface for the given mesh parameters, resulting in a TPMS.

Sequential diagram showing the generation of a Schwartz_P surfaces using the above method.

A Gyroid surface approximated with the above method

  

This approach towards approximating a TPMS leads to a study in the change of boundary conditions, gaining control over the geometry. The examples below present various gyroid distorsions generated by changing the boundary conditions.

  

  

Being able to control the boundary conditions defining a gyroid, or any TPMS, opens up to form optimization through genetic algorithms. Here, various curvatures for the edge conditions have been tested with regards to solar gain, using Galapagos for Grasshopper.

                

                 

The following examples show some patterns generated by different topologies of the starting mesh.

 

 

 

 

 

 

 

 

 

 

 

 

 

Temple of Autarky

This installation hovers around the idea of a society that is not familiar with the concept of hierarchy, and is instead based on the autonomy of each individual. The built form aims to sintesize such a concept and provoke critical thinking towards ideas of self organisation.

To achieve this, Temple of Autarky is a design driven by a Cellular Automata Algorithm, an algorithm which challenges the conventions of modern science. In short, a Cellular Automata is a collection of cells arranged on  grid which evolve through a series of time steps based on rules guided by the conditions of the neighbouring cells. In other words, it is a collection of cells that constantly evolves and organises itself, without any external input, but just by following a set of predefined rules. Some theories state that our whole Universe might have started from a single cell that evolved following a dozen rules. That cell would be the initial configuration of the system, or the First Mover.

In the case of the installation, the First Mover is a collection of cells plotted along the word “autarky”, a word that describes ideals of autonomy, self sufficiency and self organisation, ideals which I believe need be echoed.

The temple stands as a physical manifestation of such concepts, and at the same time challenges our perception of aesthetics. Do we find beauty only in systems/objects we have engineered to be beautiful ? Can we find beauty in something which we had no input upon ?

 

 

System development – Cellular Automata

A cellular automaton is a collection of (coloured) cells arranged on a grid. The cells evolve on the grid through a number of time steps, according to a set of rules based on the states of the neighboring cells. The rules can be applied iteratively for as many steps as desired. Such a model was first considered in the 1950s by von Neumann, who used it to build his “universal constructor”. Further studies were conducted in the 1980s by S. Wolfram, whose extensive research culminated in the publication of the book “A new kind of science”, which provides an exhaustive collection of results concerning cellular automata.

 

The fundamental parameter concerning a cellular automaton is the grid on which it is computated. A CA can be computed on a 1D line, a 2D or a 3D grid which can both vary in terms of shapes. CAs can be computated on grids consisting of squares, triangles, hexagons, etc. Another parameter is the number k, representing the colours or states a cell can have. K=2 (binary CA) is the simplest choice, and also the one I have been using in my experiments. In the case of a binary automaton, the number 0 is usually assigned to the colour white and 1 to the colour black. In my experiments the number 0 refers to a cell being dead, and 1 refers to a cells state being alive. An alive cell generates a point in spaces, whereas a dead one generates a void. Governing the evolution of the CA is also the set of rules applied. For 2D cellular automata, the one I am using for my experiments, there is a total of 255 possible rules depending on the states of the neighboring cells of each cell. For my form finding experiments each iteration of a 2D CA has been memorized by the computer and stored in 3D spaces. The result was a collection of points generated by a CA controlled by its initial configuration ( or the initial state of each cell in the grid ), the evolving rule and the number of iterations.

 

The rules governing the evolution of a CA are vast and produce interesting results, varying from ordered CAs which die after few iterations to chaotic patterns. Upon experimenting with a few rules I have decided to research rule 30 in more detail, also known as the Game of Life rule. Rule 30 has been discovered by John Conway in the 1970s and popularized in Martin Gardner`s Scientific American columns. The game of Life is a binary (k=2) totalistic cellular automaton with a Moore neighbourhood of range r=1. The evolving rule states that a dead cell can come to life if surrounded by 3 alive neighbours, and an alive cell survives if surrounded by 2 or 3 alive neighbours. Such a simple rule can produce very interesting results when computated in 3D space.

 

For my experiments I have been using the Rabbit plugin by Morphocode, using their sample CA definition as a starting point.

Cellular automata starting from the same configuration and following different rule. First rule is Conway`s Game of Life

Game of Life CA evolution – Initial configuration=Pentonimo Puzzle – Time=150

Game of Life CA evolution – Initial configuration=Queen Bee Shuttle – time=150

Game of Life CA evolution – Initial configuration=Diehard – Time=150

For this experiment the same evolution rule was applied, but the CA grew in both directions

The same CA definition explored in vertical growth was explored in a circular growth

Following the circular growth experiments various curves and rotation angles were explored for the growth pattern

Proximity experiments using the points generated by the CA

The lines generated by the proximity experiments were used to generate structural frames

Experiments in building the frames generated by the CA

Technology, politics and visionaries

The beginning of the 20^th century, and more specifically, the interwar, witnessed many changes, especially technology wise, which, along with the economical climate of the time and the emerging social ideas, favored new political regimes and carved the way for visionaries to imagine new cities and new worlds.

The spirit of the machine age was becoming dominant, industry was beginning to shape the economy and advancements in the car industry (new engines and new tires were being developed at that time) started changing the way people lived.

At the same time the radio was taking shape, with the first broadcasting station being established in the US in 1920 and quickly spreading. In 1928 the radio beacon was invented, and by 1930 the radio was mainstream, providing people with cheap entertainment (the US were going to the Great Depression, radio was cheap and fun) and political powers with a great tool for propaganda. The TV was also invented in this period, with the electrical TV being discovered in 1927.

Politically, dictatorship was beginning to take shape.

Mussolini came to power in 1922, promoting a cult for personality and laying down the principles of the doctrine of fascism. Propaganda was one of his main tools, and the radio was a very good medium for doing this. He presented his ideas of idealism by imposing ideas of collective and hierarchy.

Shortly after, Germany was become Nazi Germany, with Hitler rising to power. Again, this was a regime were the power was centralized in the hands of the dictator, who, through propaganda, burning of books and controlling the radio, was controlling public opinion and the arts society. Needles to say, Hitler`s ideal society was one based on race, and homogeneity.

Russia was also seeing changes. Stalin rose to power and set the goal for a communist society. He promoted authoritarianism, a centralized state and collectivization. He saw the opportunity of the machine age and carved an industry based economy, reshaping the way Russian society was organized, both at a social level, and at a physical one, by promoting urbanization (villages were turned into cities).

Within this context, visionaries began responding.

First moving image produce by Baird`s “televisor”, 1926

Baird`s televisor equipment

Snippet from “The Jazz Singer”, the first talking movie by Warner Bros

We

Yevgeny Zamyatin

We is a Russian dystopian novel first published in 1924 in New York.

Set in the future, the novel depicts an urbanized setting constructed entirely by glass, which allows the secret political police to supervise the public with ease. Life is organized in such a way as to promote maximum production in a system were the power is centralized in the hands of one person, The Benefactor. Principles of egalitarianism are promoted, the people not having names but numbers, and all wearing identical clothing. The only form of entertainment for the society is the marching in forms, while listening to the State Anthem.

However, the novel is a criticism of an organized dystopia, tackling the theme of the rebellion of the human primitive spirit against a rationalized, machined world. This is apparent from the plot, which is centered around the love story between the two main characters, who play with the idea of a revolution.

In his satire, Zamytian had in mind the Soviet Union, which at that time was a single party dictatorship. Future conditions depicted in the novel might also have been informed by Mussolini`s incipient fascist order. Even thought at that time life in the U.S.S.R. wasn`t exactly as depicted in the novel, Zamytian tackled the inevitable outcome of modern totalitarianism.

A center piece in sci fi literature, We has influenced future works, such as George Orwell`s 1984, which depicts a very similar scenario.

Zamytian`s imaginary world

Brave New World

Aldous Huxley

Brave New World is a novel published in 1932 and a milestone in modern Sci Fi. The novel anticipates changes in society through developments in reproductive technology, psychological manipulation, classical conditioning and sleep-learning.

Similar to We, the society depicted in this novel is a manipulated one, but, in this instance, it is so by the use of chemically controlled substances and hypnotic persuasion, rather than brute force.

Huxley used his novels as a means to express widely held opinions of that time., probably the most notable one being the fear of the loss of identity in a fast paced world. He feared that no one would want to read a book and that society would be given so much information that it would be reduced to passivity and egotism.

A trip which Huxley made to New York gave the novel much of its essence. Huxley was outraged by the youth culture, by the sexual promiscuity and by the commercial cheeriness he had witnessed. In his novel, he talks about “feelies”, which seem to be a response to “talkie” motion pictures (talking television was invented by Warner Bros at that time) and the sex-hormone chewing gum, which draws parallels to the ubiquitous chewing gum, which was a symbol of American youth at that time.

Brave New World

Radiant City

Le Corbusier

In response to the same political and technological context, Corbusier proposed his plans for Ville Radieuse, or the Radiant City.

The Radiant City was Corbusier`s ideal for a utopia which would respond to the world`s rapid development of that time.

Centered around rapid urbanization (specifically present in Russia at that time), advancements in transportation and industry, Corbusier`s ideas depicted high rise housing blocks, free circulation and abundant green spaces. Corbusier also believed that only a dictatorial government would be equipped to inaugurate the “age of harmony”, following the opposing values of benevolent imperialism and community control from European and English perspectives respectively.

Model of Ville Radieuse

Aerial View

Broadacre City

Frank Lloyd Wright

Broadacre City is a concept for suburban living presented By Frank Lloyd Wright in his 1932 book The Dissapearing City. Is stood as a planning statement, as well as a socio-political scheme by which each American family would be allocated an Acre of land and a new community would be built based on this. Wright depicts a community were all transport would be done by automobile and the pedestrian can exist safely only within the allocated one acre. This proposal was again a decentralized one, with the homestead considered the conceptual center.

Plan of Broadacre City

3.7m x 3.7m model of one part of Broadacre City, exhibited by Frank Lloyd Wright

Aerial View

Car sketches for Broadacre City, by Frank Lloyd Wright