Does it Look Square? Hexagonal Bipyramids, Triangular Antiprismoids, and their Fractals

The Sierpinski tetrahedron is two-dimensional with respect to fractal dimensions though it is realized in threedimensional space, and it has square projections in three orthogonal directions. We study its generalizations and present two-dimensional fractals with many square projections. One is generated from a hexagonal bipyramid which has square projections not in three but in six directions. Another one is generated from an octahedron which we call a triangular antiprismoid. These two polyhedra form a tiling of three-dimensional space, which is a Voronoi tessellation of three-dimensional space with respect to the union of two cubic lattices. We also present other fractals and show a simple object with three square projections obtained as the limit of such fractals. Sierpinski Tetrahedron and its Projections Three-dimensional objects are interesting in that they change their appearances according to the way they are looked at and their shadow images change smoothly as they are rotated three-dimensionally. For some fractals in three-dimensional space, their shadow images are complicated fractal figures in many directions but sometimes they become simple shapes like the square, and they are really attractive. For example, Sierpinski’s tetrahedron is a popular fractal realized in three-dimensional space. It is self-similar in that it is equal to the union of four half-sized copies of itself. In fractal geometry, we say that an object has similarity dimension n when it is equal to the union of k copies of itself with scale. Therefore, the Sierpinski tetrahedron is two dimensional with respects to the similarity dimension. We refer the reader to [1] and [2] for the theory of fractals. Figure 1 shows some of the shadow images of the Sierpinski tetrahedron. As Figure 1(b) shows, it has a solid square shadow image when projected from an edge. Since it has three pairs of edges opposite to each other, a Sierpinski tetrahedron has square shadow images when it is projected from three directions which are orthogonal to each other. Here, we count opposite directions once. It is true both for the mathematically-defined pure Sierpinski tetrahedron and for its n-th level approximation, which is composed of 4 regular tetrahedrons. k / 1 (a) (b) Edge view (c) Vertex view Figure 1: Some projection images of the Sierpinski Tetrahedron. Figure 2: Fractal University KYOTO (Polyurethane tetrahedron pieces connected by wires). The sculpture in Figure 2 is an application of this property. It has the shape of the 3rd level approximation of the Sierpinski tetrahedron, and two square pictures extended with the rate 3 : 1 are cut into 4 pieces and the pieces are pasted on two faces of the 4 tetrahedrons. We can view one picture from an edge, and the other one from the opposite side. Generalization of the Sierpinski Tetrahedron Now, we consider other fractals in three-dimensional space with the same properties as the Sierpinski tetrahedron. For each k≥2, we consider self-similar fractals which satisfy the followings, (1) it is the union of k copies of itself with 1/k scale, (2) it has square projections from three orthogonal directions, just as a cube has, (3) the similarity functions do not include rotational parts. From property (1), the fractal has the similarity dimension two, and from property (3), the k copies have the same orientation as itself. Property (3) also ensures that the projection images are also selfsimilar, because each projection image is the union of the projection images of its small copies which are similar to the projection image of the whole because they have the same orientation. The theory of fractal geometry says that a fractal is determined only by the similarity functions, and for our case, they are again determined by their centers (i.e. the fixed points), because they do not include rotational parts and the scale is fixed to 1/k. For the case of the Sierpinski tetrahedron, k is equal to 2 and the centers are the four vertices of the tetrahedron. The theory also says that, starting with any three-dimensional object containing the fractal, successive application of the IFS (i.e., the union of the similarity functions) will decrease its size and converges to the fractal. For the Sierpinski tetrahedron, we normally start with a tetrahedron to obtain a series of its approximations. However, condition (2) says that there is a cube with the same three projection images and this cubic approximation will make the structure of such a fractal very clear. Figure 3 shows the first three cubic approximations of the Sierpinski tetrahedron. Figure 3: Cubic approximations of the Sierpinski tetrahedron. A1 A2 A3 A0 = A ... Notice that the first approximation A1 consists of four cubes which are selected from the 8 cubes obtained by dividing the cube A into and the four cubes are not overlapping when viewed from all the 2 2 2 × × three surface-directions of A. Since A2, A3,... are obtained by repeating this procedure, this fact ensures that each level of the approximation has three square projections. The Sierpinski tetrahedron is obtained as the intersection ∩ of these approximations, and it also has the same square projection for the following reason. Suppose that p is a point of this square and Bi Ai its preimage by the projection from Ai. Each Bi is a closed set, and Bi ⊃ Bj when i < j. Therefore, there intersection C=∩ is a nonempty set contained in the fractal and every point in C is projected to the point p. 1A i i ∞ = ⊂ 1B i i ∞ = We can generalize this procedure to every k. That is, dividing the cube A into small cubes and selecting of them so that they are not overlapping when viewed from the three surface directions. Each selection of such cubes determines the similarity functions and thus a fractal object with three square projections. On the other hand, when a fractal satisfies properties (1) to (3), there exists a cube determined by the three projections in (2), and there are cubes obtained by applying the similarity maps to this cube. Since their union contains the fractal, it must also have the square projection in three directions from (2). Therefore, every fractal with the properties (1) to (3) is obtained in this way. k k k × × 2 k 2 k