The Stellation of the Icosahedron

In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Only five solids meet these criteria, and each is named after its number of faces.

Named after the ancient Greek philosopher, Plato, who theorised in his dialogue, the Timaeus (360BC), that the classical elements were made of these regular solids. It was thought that they represented the five basic elements of the world; earth, air, fire, water, and the universe.

 

If it not required for polyhedra to be convex, there are four more regular solids. In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. These can be created through the stellation of the regular convex dodecahedron and icosahedron, and differ from these forms due to their regular pentagrammic faces or vertex figures.

Together, the Platonic solids and the Kepler-Poinsot polyhedra form the set of 9 regular polyhedra. It was Augustin Cauchy who first proved that no other polyhedra can exist with identical regular faces and identical regular vertices in 1859. The small and great stellated dodecahedra, sometimes called the Kepler polyhedra, were first recognised as regular by Johannes Kepler in 1619. In 1809, Louis Poinsot rediscovered Kepler’s figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more regular stars, the great icosahedron and great dodecahedron.

Stellation is the process of constructing polyhedra by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect (Wenninger 1989). Starting with an original figure, the process extends specific elements such as its edges, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original.

The stellation diagram below shows a face plane of an icosahedron (the inner triangle), giving the lines of intersection with the other face planes. The lines define the edges of the various stellations.

A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound. The interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, and as the stellation process continues then more of these cells will be enclosed. For a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells.

A set of cells forming a closed layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types.

In the book ‘The Fifry-Nine Icosahedra (1938), H.S.M. Coxeter, P. Du Val, H.T. Flather and J.F. Petrie illustrated and catalogued the complete set of the stellation of the icosahedron – in accordance to a set of rules put forward by J.C.P. Miller. These stellations are demonstrated in the following pages.

The Infinity Tree

Scorched, dry, barren, the desert wilderness sits silent, still, soundless, lifeless. Only the pale whispers of the delicate desert fauna can be heard. Whispering, softly a whisper becomes a murmur, gently a murmur becomes a sound before the faintest of echoes can be heard between; it begins. At first a drop, motionless.

Weeks pass, slowly, a drop becomes a trickle, trickling, creeping, a journey begins, weaving, the trickle meanders. With time it grows, a small stream flows, filling the cracks of the playa. New life left in its wake, the desert is alive once more, the sounds of joy approaching. A sapling emerges from the stream, twisting and winding it grows high above the ground, encircling itself, entwining itself.

A figure of our memories, a myriad of our existence, its infinitude transcends time and like the stream to the tree we breathe life into the playa. A rest for our minds, a shelter for our bodies, a place not only to remember, a place to never forget. As our lives become more complex, diluted by materialistic culture, we find ourselves absorbed in the external forces placed upon us.It can sometimes be easy in a world of fake ideals and counterfeit culture to forget what is really real to us, what is truly important.

Above the consumerist sphere exists a timeless space, our memories, our joy, our loved ones, an eternal, unbreakable, everlasting chain. The Infinity Tree is symbol of what can never be lost, what will always be found, and what makes us who we are.

The Infinity Tree is inspired by Daniel Piker’s “Rheotomic Surface” research.

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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

I N T I

Philosophical Statement:

Inti: The Incan Sun God, his face portrayed as a gold disk from which rays and flames extended. Inti is the Sun and controls all that implies: warmth, light and sunshine. During the festival of Inti Ramyi, held during the Summer Solstice, Inti is celebrated with much drinking, singing and dancing – special statues are made of wood are burned at the end of the festival. This sculpture is an extended physical manifestation of this; decadent ritualism and a spiritual experience.

Inti incorporates 288 petals are self-assembled into 12 concentric rings, with each petal representing the hours of the day and each ring every month of the year. These are held together using mirror polished circular brackets, designed to catch the light and reflect circles of sunlight around the structure interior. Inti’s focus is the sunrise; as the sun rises on the playa, Inti is designed to catch the light at this precise moment and funnel through the piece, enveloping and bathing the burners inside with it’s warmth and spirit.

Silk Cave

The Silk Cave is an art installation proposal for Burning Man festival that allows participants to play and relax in the desert. The project was derived from the forms created by a model that explored the build up of latex on string. Using the forms that had been created to form spaces that can be inhabited and explored.

The concept is to create a space that allows people to play and interact within the fabric structure. The spaces can be used for shade, relaxing, and climbing, creating a fun and interactive art installation.

The main inspiration for the Silk Cave was from Ernesto Neto’s sculpture shown above.

One of the ten principles of Burning Man is participation and within this structure everyone is encouraged to play and therefore everyone is encouraged to participate. In addition some of the spaces will mean that people will have to support one another to climb in and out of the structure encouraging a strong sense of community.

There can be various colour arrangements for the Silk Cave as shown below:

Equipotential / Streamlines / Vector Field – early experiments

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Early experiments with DS10 this term have involved the use of the Flow-L plug-in for grasshopper to generate streamlines through fields containing charged point locations. The points act as negative magnetic forces directing the streamlines away through space resulting in an interesting visual mix of converging and diverging directional bias for the streamlines. For the purpose of the visualisation the lines are then simply extruded to form an interesting sculptural form that seemingly combines visual elements of both structure and randomness.

Other experiments have used the alternative function provided within the Flow-L plug-in components to generate  equipotential lines and pseudo-random patterns in 2D. The components allow the creation of complex patterns influenced by point charges of varying intensity by jittering the values for the point charges.

 

 

Further investigations will follow by which these early experiments are translated into 3D physical models and structural systems.

Parade: Instalaltion.

Critical Practice is a public arts group based at the University of the Arts, London. They operate under the proposition that developing aesthetic and programmatic space is a radial rather than lineal process and created the installation Parade to explore the effectiveness of their process in the public square. Made from 4300 black milk crates tied together with zip ties the structure’s components were minimized in order to focus on special relationships during the design and assembly process. It was constructed on the Rootstein Hopkins Parade Ground at Chelsea Collage of Arts and Design during the third week of May, 2010.

The temporary installation was designed by Polish architects Ola Wasilkowska and Michał Piasecki who developed it as an exploration of changes and context of public space and how user-built structures could evolve. The seed of the design came from a series of algorithms manipulated to provide structural soundness. As participants began the assembly process they were encouraged to add to the design layout by manipulating space and adding “furniture” or human scale seating and platforms designed into the overall structure. This had the effect of the structure spreading out. The building process had layers of predetermined design and spontaneous space creation which then became indistinguishable.

The installation is intended to explore the role, intent, and process of communal places- how people interact by creating space and engage in group design.

Magnetic Tetrahedra

This animation shows a model made from modular magnetic tetrahedra. Each tetrahedron has a side length of 50mm, and contains four spherical neodymium magnets.

The tetrahedra build up according to rules that stem from their dihedral angle [angle between two faces]. The dihedral angle of a tetrahedron given by θ=arccos(1/3) [approx 70.5288°]. This means that five tetrahedra placed face to face around a single axis fall approximately 7.2° short of a full 360°. Because of this, the tetrahedra do not fill space, and instead form sections of helical structures called Boerdijk–Coxeter Helices [Named ‘Tetrahelices’ by Buckminster Fuller].

The magnets in the tetrahedra ensure that when placed by hand, they lock together face to face to form structures that completely follow these rules. When pushed just within range of the magnets of other tetrahedra, they exhibit self organising properties, but due to the power of the magnets, occasionally stick edge to edge or vertex to vertex instead of face to face.