A double tower of icosahedra is aligned along each edge of a larger enveloping icosahedron. A tower stacks gradually diminishing icosahedra on the face of an icosahedron. The rate of reduction is 1/2, and each tower could theoretically have an infinite number of icosahedra converging to a vertex of the enveloping icosahedron. Each icosahedron in the assembly could be replaced with a fractal icosahedron to form an infinite fractal structure.
This Demonstration can serve as a reminder of certain geometrical features—for instance, that the icosahedron has 56 edges, corresponding to the number of faces of the rhombic triacontahedron, and that the arrangement of 16 edges corresponds to the faces of the cube. The assembly is a good illustration of the selfÃ¢ÂÂsimilarity property of fractals. It also shows a geometrical example of how an infinite set of volumes can have a finite boundary. Related schoolroom exercises could include the calculation of the height of a tower, the volume of a tower, and the proportions of icosahedra.
Wolfram Demonstrations Project
Published: April 27 2007