Carlo Sequin
Looking for 3D analogies to interesting 2D patterns is a fertile
approach to the development of abstract geometrical 3D sculptures. In
2001, this technique has led me to “Hilbert Cube 512” -- the third
generation of a Hilbert curve in 3D space.
It started with a closed Hamiltonian circuit on the edges of a cube,
and each vertex in this path was then replaced with a scaled-down
version of an open Hamiltonian path on the cube.
In this year’s submission, I am inspired by yet another intriguing 2D
space filling curve: the Gosper curve.
I study how it can lead to a variety of 3D sculptures that preserve
some basic characteristics of the 2D curve, and which might also lead
to a recursive filling of the whole 3D space.
One presentation of the Gosper curve is on a collection of hexagonal tiles. With every recursive generation, a single hexagon is replaced with a cluster of seven scaled-down hexagons. To take this construction into 3D, the first idea is to replace a single sphere with a scaled-down cluster of 13 densely packed spheres. To obtain a proper tiling, the clusters have to be packed in such a way that all the sphere centers fall onto a common regular lattice; in this case this is the face-centered cubic (FCC) lattice. However, the 13-sphere clumps do not completely tile the FCC lattice – one of every 14 lattice-sites is left void. Still, two generations yield an interesting sculpture with 13x13 atoms, but with a somewhat uneven "surface."
Another approach starts with a rhomboid lattice, which can be obtained by affinely stretching a cubic lattice by a factor of 2.0 in the direction of one of the space diagonals. The smallest unit that seamlessly tiles this lattice is a rhomboid composed of 8 atoms. This is just large enough to capture the key angles of 60 and 120 degrees present in the Gosper curve. A larger rhomboid tile composed of 27 lattice sites offers a richer variation in local path behavior. The presented sculpture design is based on an 8-segment rhomboid frame, in which the vertices have been populated with eight suitably connected, rhomboid clusters consisting of 27-segment 3D Gosper curves. It has a smoother surface than the 13-clump Gosper Ball.