NUMERICAL MODELS OF THE FIFTY-NINE ICOSAHEDRA

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1 NUMERICAL MODELS OF THE FIFTY-NINE ICOSAHEDRA JEFF MARSH Johannes Kepler s first published work, The Secret of the Universe: On the Marvelous Proportion of the Celestial Spheres, and on the true and particular causes of the number, size, and periodic motions of the heavens, established by means of the five regular Geometric solids [4], attempts to explain the sizes of the orbits of the known planets in terms of geometric properties of polyhedra. For example, the ratio of the orbital radii of Venus and the Earth was supposed to be related to the ratio of the radii of the in-sphere and the circum-sphere of an icosahedron. Needless to say, Kepler s planetary theory has not stood the test of time, however many of his geometric discoveries have. Kepler was the first to describe stellations of polyhedra, which he called echini (hedgehogs). In particular, he discovered two stellations of the dodecahedron, the echinus major icosaëdricus and the echinus minor icosaëdricus (the greater and lesser icosahedral hedgehogs) now known as the great stellated dodecahedron and the small stellated dodecahedron [3, p. 170]. In this paper we describe how a computer was used to create numerical models of the icosahedron s hedgehogs: greater, lesser, and in-between. 1. The icosahedron and its symmetries The regular icosahedron, one of the five Platonic solids, has 12 vertices, 20 faces (all of which are equilateral triangles), and 30 edges. A line through two opposite vertices defines an axis of 5-fold rotational symmetry, that is, a rotation of 2π/5 radians about this axis leaves the icosahedron invariant. Similarly, a line through the centers of two opposite faces defines an axis of 3-fold rotational symmetry, and a line through the centers of two opposite edges defines an axis of 2-fold rotational symmetry. There are six 5-fold axes, ten 3-fold axes, and fifteen 2-fold axes. The set of all distinct rotations that leave the icosahedron invariant form a group with 60 elements, the icosahedral symmetry group. If we label the faces of the icosahedron with the integers 1 through 20, we may then represent each element of the icosahedral group as a permutation of these integers. As we shall see, such a representation is particularly useful for the work described here. Date: 2008 December 5. Submitted in partial satisfaction of the requirements of the course Mathematics 521: Higher Geometry. 1

2 2 JEFF MARSH We define a reflexible polyhedron (or polygon) as one that is congruent to its mirror image. A chiral polyhedron, in contrast, is not congruent to its mirror image and exists in two forms, laevo and dextro. The icosahedron is a reflexible polyhedron. 2. Stellating the icosahedron The term stellation refers to a process of building a new polyhedron by extending the faces of an existing polyhedron past their edges until they intersect. Stellations of the icosahedron are particularly complicated: each face plane intersects 18 of the others creating the pattern shown in Figure 1, which we refer to as the stellation diagram. Each of the 18 lines in this diagram represents the intersection of two face planes, and each vertex the intersection of three (or more) face planes. The polygons we refer to as parts and the union of congruent parts we refer to as regions. There are 11 reflexive regions in the diagram, which Coxeter [2] labels with the boldface integers 0, 1, 2, 3, 4, 7, 8, 11, 12, 13, and 14. In addition there are 8 regions which occur in chiral pairs, labeled with the roman and italic integers 5, 6, 9, 10, 5, 6, 9, and 10. Each part in the diagram is a candidate for a face of a stellated icosahedron, but in order to choose the parts we need to define exactly which combinations are considered distinct and significant. Coxeter [2, p. 15] proposed the following five rules: (1) The faces must lie in the twenty planes containing the faces of the original icosahedron. (2) All parts composing the faces must be the same in each plane, although they may be disconnected. (3) The union of parts included in any one plane must have 3-fold rotational symmetry, with or without reflection. This ensures icosahedral symmetry for the whole solid. (4) The parts included in any plane must all be accessible in the completed solid, that is, they must be on the outside. (5) We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure. But we allow the combination of an mirror-image pair having no common part. Applying these rules to the stellation diagram produces 32 reflexible polyhedra (one of which is the original icosahedron) and an additional 27 chiral forms. These are the fifty-nine icosahedra. 3. Computing the vertices To construct numerical models of the icosahedra we will need the Cartesian coordinates of all the vertices in the stellation diagram for each face plane.

3 FIFTY-NINE ICOSAHEDRA 3 Figure 1. The stellation diagram. The 18 straight lines correspond to where the plane of any one face meets the planes of 18 other faces. The inner equilateral triangle corresponds to a face of the original icosahedron. The Cartesian coordinates of the vertices V i (i = ) of an icosahedron of edge length 2 and centered at the origin are given by (0, ±τ, ±1), (±1, 0, ±τ), (±τ, ±1, 0) where τ = ( 5 + 1)/2 is the so-called golden ratio [1, p. 52]. The orientation is such that the Cartesian axes coincide with three of the icosahedron s axes of 2-fold symmetry. To obtain the coordinates for an icosahedron of arbitrary edge length e we simply multiply by e/2. We define the plane of each face F i (i = ) of the icosahedron using a vector n i such that n i is perpendicular to F i and has norm equal to R 2, the radius of the in-sphere, which is given by R 2 = 3 6 τ 2 e

4 4 JEFF MARSH [1, p. 292]. Although these vectors can be computed directly from the vertex coordinates, it is simpler to use the published coordinates of the icosahedron s reciprocal polyhedron, which is a dodecahedron: (0, ±τ 1, ±τ), (±τ, 0, ±τ 1 ), (±τ 1, ±τ, 0), (±1, ±1, ±1) [1, p. 53]. Each vertex of the reciprocal dodecahedron corresponds to an n i, and scaling by a factor of τ 2 e/6 gives the desired norm. We will need an equation for the plane P i containing the face F i. Any point x in a plane satisfies the equation n (x x 0 ) = 0 where n is perpendicular to the plane and x 0 is a point in the plane. By definition n i is perpendicular to P i and also represents a point in P i. Noting that n i n i = R 2 2, we obtain n i x = R 2 2 as the equation of the plane P i. We now require a method to compute the point of intersection of the three planes P i, P j, and P k, which we denote by p ijk. Since this point lies in all three planes we must have which we write as where A = n i p ijk = R 2 2, n j p ijk = R 2 2, n k p ijk = R 2 2 (n i ) x (n i ) y (n i ) z (n j ) x (n j ) y (n j ) z (n k ) x (n k ) y (n k ) z Ap = b (1), p = This linear system is easily solved for p. (p ijk ) x (p ijk ) y (p ijk ) z, b = R Numerical models of the Stellations We used the commercial numerical computing environment matlab to create a program to model and display all fifty-nine icosahedra. The 20 faces of the original icosahedron were more-or-less arbitrarily labeled by the integers 1 through 20, and the coordinates of the face vectors n i were defined. Equation 1 was then used to compute all possible vertex points p ijk. Of course, if any two of these indices are equal or correspond to opposite faces the resulting system is singular and has no solution, so care was taken to exclude these cases. Also it should be pointed out that many of these computations were redundant. For example p 1,3,7 and p 1,3,9 are the same point, since face planes F 1, F 3, F 7, and F 9 all intersect in the same point. These redundant computations were not eliminated as the effort required to do so would have been excessive but the computational overhead was insignificant. Each vertex point p ijk is indexed by the three integers i, j, and k, but it was convenient to index this same point with a single integer m, which was arbitrarily assigned and stored in a mapping array.

5 FIFTY-NINE ICOSAHEDRA 5 A region catalog was defined, which is a numerical structure that lists, for each face plane, all the regions in the stellation diagram, the parts which comprise that region, and the vertices which define the parts. Coxeter s convention of labeling the regions with a combination of boldface, roman, and italic integers was vexing, as matlab (or any computer programming language we are aware of) makes no distinction between the typefaces of integers. For this reason, we indexed regions 5, 6, 9, and 10 as 15, 16, 19, and 20. Furthermore, matlab does not allow an index of 0, so region 0 was indexed as 21. The region catalog for face plane 1 was tediously populated by hand, using a paper printout of the stellation diagram and a cardboard model of an icosahedron. Populating the region catalog for face planes 2 through 20, however, was easy: since every vertex was specified by the indices of the three face planes that define it, we simply applied the appropriate rotation in the icosahedral symmetry group, as represented by a permutation of the faces. The hard part was defining the rotations, which we did by hand (again using the cardboard model) simply because we did not have the time to figure out a better way to do it. With the region catalog complete, it was straightforward to create structures to display each of the icosahedra. A given icosahedron is defined as a union of regions; using the region catalog this is translated into a list of parts, each of which is defined by a list of vertices. This is exactly what is required to create a structure called a patch, which is passed to a routine that draws the icosahedron so defined. This routine does all the difficult tasks such as orthographic projection of the 3-dimensional solid on a 2-dimensional view plane, as well as hidden surface removal. 5. Results A gallery of stellations was produced displaying all 32 reflexible icosahedra (labeled 1 to 32), and the dextro forms of the 27 chiral icosahedra (labeled 33 to 59). These labels are the same as in Coxeter [2, p ]. The face planes are colored one of five hues (red, orange, yellow, green, and blue) in four tints, thus each plane receives a unique color. These colors are assigned to the planes such that for icosahedron 47 (which can be thought of as a compound of five tetrahedra) each component tetrahedron has a single hue. There is a large variation in the sizes of the icosahedra, so each illustration is scaled to fit in the view plane. Some highlights and brief comments follow: 5.1. The regular icosahedron. The original icosahedron, here labeled icosahedron number 1, is not without its charms. From the coordinates of the vertices presented above, it can be seen that the vertices may be grouped into three sets of four, each of which describe a golden rectangle. Furthermore, these rectangles are mutually perpendicular.

6 6 JEFF MARSH 5.2. The triakisicosahedron. Icosahedron number 2 can be thought of as the first stellation, and has been given a name, the triakisicosahedron. It is very clear that the three faces surrounding each face of the original icosahedron have been extended until they form a little pyramid sitting on each face. This is a very pretty and not too spiky polyhedron Second stellation. Icosahedron number 3 can be thought of as the second stellation. It is also a compound of five octahedra, but it is difficult to see this without the proper coloring. Although reflexive, it almost appears chiral The great icosahedron. Number 7 is also known as the great icosahedron and is one of the four regular (Kepler-Poinsot) star polyhedra. It can be thought of as a polyhedron whose faces are regular pentagrams, which are allowed to intersect one another. M. J. Wenninger calls it most beautiful and attractive [5, p. 63] but we find it rather ugly The complete stellation. Number 8 is the complete stellation, that is, it encloses all other stellations and thus can be thought of as the final stellation. It is spiky in the extreme, with the spikes clustering in groups of five Icosahedron 9. This stellation is a bit of a surprise, as it consists of pyramid-like shapes connected only at vertices. Several of the other icosahedra share this property and seem to contain voids or appear hollow. It would be difficult to construct physical models of these polyhedra for this reason. Number 34 is similar in that it is connected only at the vertices, but in addition is delightfully chiral Icosahedron 47. This beautiful chiral polyhedron is a compound of five tetrahedra, made obvious by the coloring scheme. References 1. H. S. M. Coxeter, Regular polytopes, third ed., Dover Publications, New York, US, H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie, The fifty-nine icosahedra, third ed., Tarquin Publications, St. Albans, UK, P. R. Cromwell, Polyhedra, Cambridge University Press, Cambridge, UK, J. Kepler, Mysterium cosmographicum, second ed., Press of Erasmus Kempfer, Frankfurt, 1621, (translated by A. M. Duncan, Abaris Books, New York, US, 1981). 5. M. J. Wenninger, Polyhedron models, Cambridge University Press, Cambridge, UK, 1971.

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19 Icosahedron Model Showing face and vertex numberings, and Cartesian axes y+y z z x +x z x -x z y -y Jeff Marsh, 2008 Fall