Raymond Aschheim
As one of a few Hypersculptors, i'm using mathematics, computer science, and rapid prototyping technologies, to produce sculptures of high dimensional objects. To exhaust the fact that they includes an other dimension, I make them electomagetically levitate and freely floating in our space, without any physical link to the earth. My hyper sculptures illustrates my NKS-E8 Theory of Everything, It says that the void is an hypercrystal made of a trivalent network. From Set theory, topology gives a data structure, which gives a geometry. Then data defaults creates forces and matter following Lisi's E8 model. I exposed in intersculpt biennale since 2003, and in Grand Palais's 2009 Art en Capital, Bridges 2010 Pecs, 'TOE' exhibition at Centre Art Bastille (Grenoble 2011). I'm trying to express what the universe is on its fundamental level, searching for the Truth, and luckily, finding Beauty. Follow me on twitter @24cl
Trivalent Honeycomb is a 3D slice of a 4D honeycomb. The 4D honeycomb is the dual 2-complex of the icosatetrachoric honeycomb, a 4D space-filling (regular tesselation)by 24-cell. It can also be seen as the projection of the unimodular E8 lattice, or as the laminated lattice lambda4=D4. It is sliced as a 3D space tesselation by keplerian rhombic dodecahedron, and an internal structure of triple binary tree is added to make the graph trivalent. This sculpture is not regular because the 3D coordinates emerge from the graph topology instead of the lattice geometry. This is a representation of only the graph and we see the 3D space emerging. Minor topological modifications of the internal structure can encode the extended standard model, as E8 projection to D4. Each dodecahedron encodes 3 particles (and may holds a proton or a neutron) along 5 consecutive time slices, as 24 trits (valued in {-1,0,1}), one for each octahedron (choosing one of it axis) of the sliced 24-cell.
Isabella Graph is a graph resulting from the application of one unique rule to a graph, repeatedly, from an original simple tetrahedron. This is a stunning example of the emergence of complexity from a simple rule in a network substitution system. To apply a rule to a trivalent graph, we have to color each half-edge, so that any node has three colors, like the baryons in physics. The rule is to replace an edge made of 2 demi-edges with one same fixed color (instead of color we can also use trit, the extension of bit to the values -1, 0 and 1), by a square made from four edges (thus by adding 2 nodes and 3 edges to the graph). This is described in the "Network as Manifolds" short paper by Isabella Thiesen made with the help of Stephen Wolfram and Todd Rowland, available here: http://www.wolframscience.com/summerschool/resources/IsabellaThiesen.pdf
Kepler discovered the rhombic dodecahedron, a non-archimedean dual
of the archimedean cuboctahedron.
The honeycomb slice is a lattice of keplerian rhombic dodecahedra,
rendered as a trivalent graph.
On it float a platonic ball including all platonic polyhedra (with
same length), and an archimedean ball including all archimedean
polyhedra.
Platon associated its five polyhedra to the five elements, as the
components of the universe. Kepler found a less symmetric but more
beautiful polyhedron, as it is able to tile the space. It is in
fact the projection of the sixth element which lives in 4
dimensions and gives it shape and symmetry to a regular tiling of
the spacetime. Loop Quantum Gravity shows that the dual 2-complex
(the skeleton) of this tiling is enough to define space, time,
energy and matter, in term of information coded in the topology of
this graph.