A "Fractal Crystal"
This "fractal crystal" is constructed by starting with a first generation cube and placing a half-scale cube on the center of each face. The second-generation cubes have the same orientation as the first-generation cube. Third-generation cubes again scaled by half are placed on each unoccupied face of a second-generation cube. This process is continued ad infinitum to form the fractal crystal. The growth of the crystal occurs more rapidly along normals (perpendicular directions) to the faces of the starting cube, leading to an overall envelope (convex hull) that is the Archimedean dual of the cube. The structure has been created as a VRML object through 11 generations (below). A physical model approximately 20 cm across was fabricated by Hank Kaczmarski and Nicholas Duchnowski (who also created the VRML description), at the Beckman Institute of the University of Illinois at Urbana-Champaign, using a zCorp color printer, with a different color assigned to each generation of cube.
Click on the images below for larger versions.
The 3-D structure was inspired by a 2-D arrangement of squares (below, left) discovered by Robert Fathauer in August of 1999. There are three planar slices of the 3-D object that yield this arrangement of squares. The 3-D structure was discovered by Robert in September of 2000. (The structure is so simple it's likely that others thought of it first. Without drawing it through at least six generations, though, it's difficult to see that the facets will form Sierpinski triangles. It's this feature that makes it such a fasinating visual object.) A drawing was made at that time through six generations using FreeHand, a 2-D drawing program (see below, right). The fact that the faces are Sierpinski triangles is evident in this figure, though it doesn't jump out at the viewer. Nothing more was done with it until 2007, when Hank Kaczmarski contacted Robert. They worked together to create the 3-D model shown above.
To view this object in vrml through six generations of cubes, click here. In the vrml version, your mouse can be used to control the orientation.
An animation illustrating the construction of this object has been created by Greg Trounson for YouTube.
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