Hexagonal Bipyramids, Triangular Prismids, and their Fractals

Hideki Tsuiki Kyoto University Yoshida-Nihonmatsu Kyoto 606-8501, Japan E-mail: [email protected]

Abstract

The Sierpinski tetrahedron is a fractal object in the three-dimensional space, but it is two-dimensional with respect to fractal dimensions. Correspondingly, it has square projections in three orthogonal directions. We study its generalizations and present two two-dimensional fractal objects 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 prismid. These two polyhedrons form a tiling of the three- dimensional space, which is a Voronoi tessellation of the three-dimensional space with respect to the union of two cubic lattices.

Sierpinski Tetrahedron and its Projections

Three-dimensional objects are attractive in that they change their appearances according to the way they are looked at. When it has a fractal structure, the shadow images are very attractive in that, in many cases, they also have fractal structures which change smoothly as the object is rotated three-dimensionally. For example, Sierpinski tetrahedron is a popular fractal three-dimensional object, which 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 kn copies of itself with /1 k scale. Therefore, the Sierpinski tetrahedron is two dimensional with respects to the similarity dimension. Correspondingly, a Sierpinski tetrahedron has a two-dimensional solid square shadow image when it is 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 4n regular tetrahedrons. The artistic object in Figure 2 is an application of this property. It has the shape of the n-th level approximated Sierpinski tetrahedron, and two square pictures extended with the rate 3:1 are cut into 4n pieces and pasted on two faces of the 4n tetrahedrons. We can view one picture from an edge, and the other one from the opposite side.

(a) (b) Edge view (c) Vertex view Figure 1: Some projection images of a Sierpinski Tetrahedron. Figure 2: Fractal University KYOTO (Polyurethane tetrahedron pieces connected by wires).

Generalization of the Sierpinski Tetrahedron

Now, we consider other fractal three-dimensional objects with the same properties as the Sierpinski tetrahedron. That is, (1) it is the union of k2 copies of itself with 1/k scale, and thus the similarity dimension is two, (2) it has square projections from three orthogonal directions, just as a cube has, (3) the similarity functions do not include rotational parts, and therefore the k2 copies have the same direction as itself.

The theory of fractal geometry says that such a fractal object is determined only by the similarity functions, which are again determined by their centers (i.e. the fixed points) for each k, because they do not include rotational part and the scale is fixed to 1/k. For the case of the Sierpinski tetrahedron, the centers are the four vertices of the tetrahedron. The theory also says that, starting with any three- dimensional figure containing the fractal, successive application of the IFS (i.e., the union of the similarity functions) will decrease its size and converge to the fractal. For a Sierpinski tetrahedron, we normally start with a tetrahedron to obtain a series of its approximations. For our purpose, condition (2) says that there is a cube with the same three projection images, and cubic approximation will make the structure of such a fractal object very clear. Figure 3 shows the first three cubic approximations of the Sierpinski tetrahedron.

…

A A1 A2 A3

Figure 3: Cubic approximations of the Sierpinski tetrahedron.

Notice that the first approximation A1 consists of four cubes which are selected from the 8 cubes obtained by dividing the cube A into ×× 222 so that they are not overlapping when viewed from all the three surface-directions of A. Since A2, A3,… are obtained by repeating this procedure, it ensures that each level of the approximation has three square projections. This fact also ensures, through a little topological argument, that its limit, i.e., the Sierpinski tetrahedron, also has this property. We can generalize this procedure to every k. That is, dividing the cube A into ×× kkk small cubes and selecting × kk of them so that they are not overlapping when viewed from the three surface directions. Each selection of such × kk cubes determines the similarity functions and thus a fractal object with three square projections, and a little mathematical argument ensures that all fractals with the properties (1) to (3) are obtained in this way. When k = 2, A1 is the only possible configuration if we identify those congruent through the rotation of the cube. When k = 3, we have two (non-congruent) ways of selecting 9 cubes from the 27 cubes, which are F and G in Figure 4. In the following two sections, we study those fractals generated by F and G.

F:

Lower Middle Upper

G:

Figure 4: Two non-overlapping configurations of nine small cubes out of 27.

The Hexagonal Bipyramid Fractal

In the lower part of Figure 5 are computer graphics images of the fractal object generated by F viewed from many directions. As (c) shows, it has hexagonal symmetry. Because of this symmetry, it has square projections (d) not in three but in six directions. It also has two kinds of dendrites (a, e), two kinds of snowflakes (c, g), and two kinds of tiling patterns (b, f), which we will investigate in later. Note that all the projection images are fractal, i.e., the union of its copies of small sizes, because the similarity functions do not rotate the object. As we normally start not with a cube but with a regular tetrahedron to obtain a Sierpinski tetrahedron, it is more natural to start with the convex-hull of the fractal object. The convex-hull of this fractal is simply obtained by connecting the centers of the similarity functions except for the middle one, which are two vertices and middle points of six edges of the cube. As a result, we have a hexagonal bipyramid as in the top of Figure 6. This dodecahedron is the intersection of two cubes which share a diagonal. Each face is an isosceles triangle whose height is 3/2 of the base. This hexagonal bipyramid is itself a polyhedron which has six square projections. Since this fractal is based on a hexagonal bipyramid, we will call it the hexagonal bipyramid fractal. Level one and level two approximation models of the hexagonal bipyramid fractal are made with papers and piano wires as in Figure 5. This level two approximation model is colored in nine colors so that every nine rows and nine columns and nine blocks contain all the nine colors in all the 12 square appearances. We will call it a SUDOKU coloring. It has 30 SUDOKU colorings, and this is the most symmetric one, with the same coloring order on the nine rings.

The Triangular Prismid Fractal

Figure 6 is about the fractal object generated by the configuration G. On the bottom, there are figures of this fractal from some directions. As property (2) says, this fractal has square projection images in three orthogonal directions (Figure (d)). As (e) shows, this fractal is not connected. It is composed of slices which form the Cantor set. Figures (g) are fractal tiling image. There is another tiling image not in this list. The convex-hull of this object is obtained by taking the convex-full of the centers of the similarity maps, which are (1) three vertices, (2) three midpoints of edges, and (3) three center-points of faces. Here, Hexagonal Bipyramid.

1st level approximation of the Hexagonal Bipyramid Fractal.

2nd level approximation with a SUDOKU coloring.

(a) Dendrite1 (3) (b) Tiling1 (c) Snowflake1 (1) (d) Square (6)

(e) Dendrite2 (3) (f) Tiling2 (6) (g) Snowflake2 (6)

Figure 5: The hexagonal bipyramid fractal and its finite approximation models. In the parenthesis are the numbers of directions in which we have such projection images (opposite directions counted once).

(a) (b) (c)

(d) Square (3) (e) Tiling2 (3) (f) Triangle (1) (g) Cantor Set (3) Figure 6: The triangular prismid fractal viewed from many directions. the points in (3) are the middle points of the edges of the triangle generated from (1), and thus can be omitted in taking the convex-full. Therefore, it has the form in Figure 6(a). This octahedron has the square shape when viewed from three orthogonal directions. The two pictures on the top left are taken from the opposite directions. Figure 6(b) shows another explanation of this octahedron, that is, it is obtained from a triangular prism by truncating three vertices of one triangular face. Such a truncated prism is the intersection of a prism and a pyramid of an appropriate size. Therefore, we will call this octahedron a triangular prismid. Figure 9(c) shows yet another property of this octahedron. That is, the six vertices are on the positive and the negative parts of the three axes of coordinates and the distance of the vertices from the origin on the negative parts is twice those on the positive parts. One can see these properties of a triangular prismid from the pictures on the right. The k ≥ 4 Case

When k is equal to 4, we have 36 different fractal objects, including the Sierpinski tetrahedron. Only one of them is connected except for the Sierpinski tetrahedron. When k is equal to 5, we have 3482 fractal objects. There is a Java applet which displays all the fractals up to k=5 with rotation [2]. As the number k increases, the number of two-dimensional fractal objects obtained in this way increases explosively. Among them, there is a series of fractals. Recall the configuration of the small cubes for the prismid fractal in Figure 4(G). The lower level is a diagonal sequence of cubes, and the middle level is its shift to the left, with the overflowed one appearing from the right. This construction of a configuration applies to every k, as the following figure shows.

…

Figure 7: A series of configurations whose fractals converge to two sheets of triangles.

Correspondingly, we have a sequence of fractals with three square projections. This sequence converges with respect to the Hausdorff metric, and as the limit, we obtain a simple object in Figure 8 which consists of two triangular sheets. Though this limit is not a self-similar fractal, when it is viewed from three directions, we have square images. As an application, the author has made a simple object. It has two triangular tiles which are tessellating as the triangular tiling of the plane. One can see the three square connected images of the two tiles.

Note: the author is preparing one made of acrylic resin with a picture by Escher. Though it is not in time for this submission, he expects that it will be in time for the final version.

Figure 8: A toy which displays three connected images of the triangular tessellation.

Tiling Patterns

We explain the tiling pattern Figure 5(b). It appears when the hexagonal bipyramid is projected from a vertex of the cube containing the bipyramid fractal. Figure 9 shows the similarity functions of a bipyramid fractal projected from this vertex, and they are the similarity functions of this fractal figure. The big hexagon in Figure 9(a) is the projection image of the cube from this direction and the thick-lined hexagon is 3/1 of this big hexagon. One can see that the similarity functions map the thick-lined hexagon to the 9 gray hexagons of size 1/3. Note that the 9 gray hexagons are taken from the hexagonal tiling and therefore they are not overlapping and they in all have the same area as the original hexagon. In addition, this gray figure is a tiling pattern, and when we extend the thick-lined hexagon to the hexagonal tiling and apply this procedure to all the tiles, we obtain a tiling with this gray figure. Therefore, its repetition does not cause overlapping and we obtain a series of tiling patterns with the same area. One can see that the limit is also a tiling pattern and also has the same area, that is, one third of the projection image of the cube.

(a) (b) (c)

Figure 9: Tiling pattern of Figure 5(b).

Other tiling patterns like Figure 1(c) and Figure 6(g) have the same kind of explanation. Tiling pattern Figure 5(f) is a projection from an edge of the cube, and it also has a similar explanation.

A 3D Tiling with Hexagonal Bipyramids and Triangular Prismids

The two polyhedrons introduced as the convex full of the fractals for the case k=3 have a strong connection. That is, there is a tiling of the three-dimensional space with hexagonal bipyramids and triangular prismids as in Figure 11(a). The ratio of prismids to bipyramids used for this tessellation is 4 to 1. In this section, we give mathematical backgrounds of this tiling. We consider, in addition to the configurations in Figure 4, the following configuration which also produces a prismid fractal with a different direction.

H:

Figure10: Another configuration which generates a prismid fractal.

Note that the three configurations F, G, and H are not overlapping. It means that the 1st level approximations of these three fractals are not overlapping. Therefore, we have ×× 333 cubes which contain 9 bipyramids, 9 prismids, and 9 prismids with a different direction. The problem is the shapes of the voids left in the large cube. It has 6 voids inside, around the corners of the small cubes. Figure 11(b) is the configuration of the lower level. Note that polyhedrons in adjoining cubes have the same polygon (or line) on their shared face. Therefore, each void is an octahedron surrounded by polyhedrons in the surrounding 8 cubes. One can see that this octahedron is just a prismid, because three of the vertices exist on other corners of cubes and the other three exist on middle points of edges of cubes. Three prismids are (b) (c)

Bipyramids have dark color, H-Prismids are (a) (a) bright, and G-prismids medium. Figure 11: Tessellation with bipyramids and prismids. Prismids have four colors. placed in the voids in Figure 11(c). This procedure can be applied to a tiling of the 3-dimensional space with large cubes, and as the result, we obtain a tiling of the 3D space with bipyramids and prismids. This tiling can also be obtained as the Voronoi tessellation in the following way. Let X be the cubic lattice of the centers of the small cubes containing bipyramids and prismids, described above. We put the origin at the center of a bipyramid and give the coordinate system as in Figure 12. y

x

Figure 12: The relation between a cubic 3D lattice and equilateral triangle lattices.

Then, bipyramids are located on planes x + y + z = 3k for k integers, and prismids coming from the configuration G and H are located on planes x + y + z = 3k+2 and x + y + z = 3k + 1, respectively. On these three kinds of planes, the points of the cubic lattice form equilateral triangle lattices. Let Y be the lattice obtained by rotating X around the axis x = y = z by 60 degrees. The intersection of the two lattices X and Y is the union of the triangle lattices corresponding to x + y + z = 3k, that is, the centers of the bipyramids. Since our 3D-tiling has order 6 (60 degrees) rotation symmetry around this axis, we also have bipyramids and prismids on the lattice Y. One can see that bipyramids are mapped to bipyramids and prismids are mapped to void spaces by this rotation. In this way, we have the following.

Theorem: This tiling with bipyramids and prismids is the Voronoi tessellation corresponding to the union of a cubic lattice and its 60-degree rotation around a diagonal of one of the cubes.

References: [1] Falconer, K.J. (1990) Fractal Geometry – Mathematical Foundations and Applications, John Wiley [2] http://www.i.h.kyoto-u.ac.jp/~tsuiki/index-e.html