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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 1

Platonic Stars 2

3 ALEXANDRA FRITZ AND HERWIG HAUSER 4

6 ut of beauty, I repeat again that we saw her there such as u and v from above. Their role will become clear 33 5 7 BB shining in company with the celestial forms; and when we introduce some invariant theory. 34 8 coming to earth we find her here too, shining in The general task is to construct an algebraic surface, that 35 Author Proof 9 clearness through the clearest aperture of sense. For sight is is, the zero set X = V(f) of a polynomial f R x; y; z , with 36 12 ½ Š 10 the most piercing of our bodily senses; though not by that is prescribed symmetriesPROOF and singularities. By ‘‘prescribed 37 11 wisdom seen; her loveliness would have been transporting if symmetries’’ we mean that we insist the surface should be 38 12 there had been a visible image of her, and the other ideas, if invariant under the action of some finite subgroup of the real 39 13 they had visible counterparts, would be equally lovely. But orthogonal group O R . Most of the time we will consider 40 3ð Þ 14 this is the privilege of beauty, that being the loveliest she is the symmetry group of some Platonic solid S R3. 41  15 also the most palpable to sight The symmetry group of a set A R3 is the subgroup of 42 16  Plato, Phaedrus the orthogonal group O3 R , formed by all matrices that 43 17 transport the set intoð itself,Þ that is, Sym A M 44 O R ; M a A for all a A O R . (Oftenð Þ¼f the sym-2 45 3ð Þ ð Þ2 2 g 3ð Þ 18 EXAMPLE 1 The picture above shows the zero set of the metry group is defined as a subgroup of SO3 instead of O3. 46 19 following equation, The subgroup of O3 we consider here is referred to as the 47 full symmetry group.) 48 5 f u; v 1 u 3 cu3 cv; with c 0; 1 A Platonic solid is a convex polyhedron whose faces are 49 ð Þ¼ð À Þ À 27 þ 6¼ ð Þ identical regular polygons. At each vertex of a Platonic 50 2121 where solid the same number of faces meet. There are exactly five 51 Platonic solids, the tetrahedron, octahedron, hexahedron 52 u x; y; z x2 y2 z2; ð Þ¼ þ þ (or cube), icosahedron, and dodecahedron. See figure 2. 53 v x; y; z z 2x z x4 x2z2 z4 2 x3z xz3 Two Platonic solids are dual to each other if one is the 54 ð Þ¼ À ð þ Þð À þ þ ð À Þ 55 5 y4 y2z2 10 xy2z x2y2 : 2 convex hull of the centers of the faces of the other. The þ ð À Þþ ð À ÞÞ ð Þ octahedron and the cube are dual to each other, as are the 56 2323 For any value c [ 0, the zero set of this polynomial, such icosahedron and the dodecahedron. The tetrahedron is 57 24 as the one displayed in figure 1, is an example of a surface dual to itself. Dual Platonic solids have the same symmetry 58 25 that we want to call a ‘‘Platonic star’’. This particular group. For a more rigorous and more general definition of 59 26 example we call a ‘‘dodecahedral star’’ because it has its duality of convex polytopes see [7, p. 77]. 60 27 cusps at the vertices of a regular dodecahedron and has the The Platonic solids are vertex-transitive polyhedra: their 61 28 same symmetries. We refer to the familiar Platonic solid symmetry group acts transitively on the set of vertices. This 62 29 with 12 regular pentagons as faces, 30 edges, and 20 ver- means that for each pair of vertices there exists an element 63 30 tices. See figure 2e. of the symmetry group that transports the first vertex to the 64 31 The following article deals with the construction of sur- second. One says that all vertices belong to one orbit of the 65 32 faces such as the one above. We will always use polynomials action of the symmetry group. 66

Citation of Phaedrus from [8]. Supported by Project 21461 of the Austrian Science Fund FWF. Figures 12 and 13UNCORRECTED are generated with Wolfram Mathematica 6 for Students. All the other figures are produced with the free ray-tracing software Povray, http://www.povray.org.

1FL01 1Of course a lot of people have been working on construction of surfaces with many singularities, also via symmetries. We want to mention, for example, Oliver Labs and 1FL02 Gert-Martin Greuel.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK singular point,orsingularity, of an algebraic surface is a 85 point where the surface is locally not a manifold. This signi- 86 fies that the first partial derivatives of the defining polynomial 87 disappear at the point. Isolated means that in a neighborhood 88 of the singularity there are no other singular points. 89 An isolated surface singularity is said to be of type A2 if it 90 has (up to local analytic coordinate transformations) the 91 equation x3 + y2 + z2 = 0. The corresponding zero set is a 92 two-dimensional cusp Y as displayed in figure 3a. Note that 93 the cusp, in these coordinates, is a surface of rotation. Its 94 axis of rotation is the x-axis. We call that axis the tangent- 95 line of the cusp Y at the origin. (Clearly it is not the tangent- 96 line in the usual, differential-geometric sense. The origin is a 97 singularity of the cusp, that is, the surface is not a manifold 98 there, so that differential-geometric methods fail there.) One 99 can also view this ‘‘tangent-line’’ as the limit of secants of Y 100 joining one point of intersection at the singular point 0 to 101 another point of intersection moving toward 0. Now if X is 102 Figure 1. Dodecahedral star with parameter value c = 81. any variety with a singularity of type A2 at a point p, then we 103 define the tangent-line at this point analogously. Note that 104 67 A convex polyhedron that has regular polygons as faces we are no longer dealing with a surface of rotation. 105 68 106 Author Proof and that is vertex-transitive is either a Platonic solid, a prism, We will choose the location of the singular points so that 69 an antiprism, or one of 13 solids called Archimedean solids.2 they all form one orbitPROOF of the action of the selected group. If 107 70 One can extend the notion of duality as we defined it to we use the symmetry group of a Platonic solid, we can 108 71 Archimedean solids. Their duals are not Archimedean any choose, for example, the vertices of the corresponding 109 72 longer; they are called Catalan solids3 or just Archimedean Platonic or Archimedean solid. 110 73 duals. Each Archimedean solid has the same symmetries as Now we are ready to define our ‘‘object of desire’’, the 111 74 one of the Platonic solids, but with this proviso: in two ‘‘Platonic star’’. We want to emphasize that the following is 112 75 cases we do not get the full symmetry group but just the not a rigorous mathematical definition. 113 76 rotational symmetries. Let S be a Platonic (Archimedean) solid and m the 114 77 Here we will deal with just three groups: the symmetry number of its vertices. Denote its symmetry group in O3 R 115 ð Þ 116 78 group of the tetrahedron Td, that of the octahedron and by G. An algebraic surface X that is invariant under the 117 79 cube Oh, and that of the icosahedron and dodecahedron Ih. action of G and has exactly m isolated singularities of type 80 The Catalan solids are not vertex-transitive but are obvi- A2 at the vertices of the solid, is called a Platonic (Archi- 118 81 ously face-transitive. medean) star. We require that the cusps point outward, 119 82 By ‘‘prescribing singularities’’ of a surface we mean that otherwise we speak of an anti-star. In both cases for all 120 83 we insist the zero set should have a certain number of isolated singular points p the tangent-line of X at p should be the 121 84 singularities of fixed type, at a priori chosen locations. A line joining the origin to p. 122 ......

ALEXANDRA FRITZ is a Master’s Degree HERWIG HAUSER studied in Innsbruck and candidate at the University of Innsbruck. Paris; he is now a Professor at the University She spent last year at the University of of Vienna. He has done research in algebraic AUTHORS Vienna, where she worked on algebraic and analytic geometry, especially in resolution stars, as reported here, under the supervi- of singularities. Among his efforts in presenting sion of Herwig Hauser. mathematics visually is a movie, ‘‘ZEROSET – I spy with my little eye’’. Fakulta¨t fu¨r Mathematik Universita¨t Wien Fakulta¨t fu¨r Mathematik A-1000 Vienna Universita¨t Wien Austria A-1000 Vienna e-mail: [email protected] Austria e-mail: [email protected]

2FL01 2Often the Archimedean solids are defined as polyhedra that have more than one type of regular polygons as faces but do have identical vertices in the sense that the 2FL02 polygons are situatedUNCORRECTED around each vertex in the same way. This definition admits (besides the Platonic solids, prisms, and antiprisms) an additional 14th polyhedron 2FL03 called the pseudo-rhombicuboctahedron. This is a fact that has often been overlooked. The sources we use, namely [3, p. 47–59] and [4, p. 156 and p. 367], are not very 2FL04 clear about it. See [4]. 3FL01 3Named after Euge` ne Charles Catalan, who characterized certain semi-regular polyhedra.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK

(a) Tetrahedron. (b) Octahedron. (c) Hexahedron. (d) Icosahedron. (e) Dodecahedron.

Figure 2. The five Platonic solids.

Author Proof PROOF

(a) The A2-singularity, (b) Dodecahedral star and x3 + y2 + z2 =0. dodecahedron.

Figure 3. The two-dimensional cusp and the dodecahedral star.

(a) c = −300. (b) c = −30. (c) c = −15. (d) c = −3. (e) c = −1.5. 7(f) c = −2 /32.

(g) c = −0.6. (h) c = −0.3. (i) c = −0.03. (j) c =0. (k) c =0.003. 0(l) c = .03.

UNCORRECTED (m) c =0.3. (n) c =3. (o) c = 30. (p) c = 81. (q) c = 300. (r) c = 3000.

Figure 4. Dodecahedral star with varying parameter value c, for c B-27/32 the surfaces are clipped by a sphere of radius 4.5.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 123 Later we will see that the algebraic surfaces defined by understand these relations. As a first result, it can be shown 174 124 equation (1) from the introductory example satisfy by that C x G always contains some n algebraically independent 175 ½ Š 125 construction the conditions of the definition above. For elements, say u1, ..., un. These need not generate the whole 176 126 now we ask the reader to consider the illustrations, espe- ring. But it turns out that u1, ..., un canbechosensothat 177 127 cially figure 3b, that suggest that this claim is true. If we C x G is an integral ring extension of its subring C u ; ...; u . 178 ½ Š ½ 1 nŠ 128 choose c [ 0 we get stars, for c \ 0 anti-stars. The choice This is Noether’s Normalization Lemma. 179 129 c = 0 yields an ordinary sphere. See figure 4 for the effect In particular, C x G will be a finite C u ; ...; u -module. 180 ½ Š ½ 1 nŠ 130 of varying the parameter c. Note that the singularities stay A theorem that probably first appeared in an article by 181 131 fixed on a sphere of radius one for all parameter values, so Hochster and Eagon [5] asserts that for finite groups G, the 182 132 for c \ 0 we have to zoom out to be able to show the invariant ring is even a free C u1; ...; un -module (one says 183 G ½ Š 133 whole picture. For c =-27/32 the anti-star has a point at that C x is a Cohen-Macaulay module). That is to say, 184 ½ Š G G 134 infinity in the direction of the z-axis, which is among the there exist elements s1; ...; sl C x such that C x 185 135 l C 2 ½ Š ½ Š ¼ 186 normals of the faces of the dodecahedron. By symmetry it aj 1 sj u1; ...; un . This decomposition is called the ¼ Á ½ Š 136 will also have points at infinity in the direction of the Hironaka decomposition; the ui are called primary 187 4 5 137 normals of the remaining faces. The pictures suggest that invariants and the sj secondary invariants. Therefore 188 138 for c [ -27/32 the dodecahedral anti-stars and stars are each invariant polynomial f has a unique decomposition 189 139 bounded while they remain unbounded for c \ -27/32. It 140 l might be interesting to refine the definition of stars and f s P u ; ...; u ; 141 j j 1 n anti-stars by demanding that the surfaces be bounded. In ¼ j 1 ð Þ 142 this article we shall not consider this question. X¼

for some polynomials Pj C x1; ...; xn . 191191

Author Proof 2 ½ Š 143 Some Basics from Invariant Theory Things are even better if G is a reflection group. An 192 n 193 144 In order to explain our construction of the equations for the element M GL C PROOFis called a reflection if it has exactly 2 ð Þ 145 stars we need a few results from invariant theory. Those one eigenvalue not equal to one. A finite subgroup of 194 n 146 who are familiar with the topic can proceed to the next GL C is called a reflection group if it is generated by 195 ð Þ G 147 section; those who want to know more details than we give reflections. In a reflection group, C x is even generated 196 ½ Š 197 148 can refer to [10]. by n algebraically independent polynomials u1, ..., un and 149 For ease of exposition, we work over the complex vice versa (Theorem of Sheppard-Todd-Chevalley) – so that 198 150 numbers C. Let there be given a finite subgroup G of the decomposition reduces to 199 151 GLn C . Typically, this will be the symmetry group of a f P u1; ...; un 152 Platonicð Þ solid, allowing also reflections. ¼ ð Þ 153 The group G acts naturally on Cn by left-multiplication. for a uniquely determined polynomial P. 201201 154 This induces an action of G on the polynomial ring Here is how we shall go about constructing the equations 202 155 C x ; ...; x , via p f(x) = f(p x). A polynomial f is called 1 n for our Platonic stars: Find a polynomial in the invariant 203 156 invariant½ withŠ respectÁ to G ifÁ p f = f for all p [ G. For generators such that f has the required geometric properties. 204 157 instance, if G is the permutation groupÁ S on n elements, n (Remember that when we speak of symmetry groups we do 205 158 the invariant polynomials are just the symmetric ones. not restrict to proper rotations. The symmetry groups of the 206 159 The collection of all invariant polynomials is clearly Platonic solids as we defined them are reflection groups. By 207 160 closed under addition and multiplication, and thus forms the Sheppard-Todd-Chevalley Theorem, this can be checked 208 161 the invariant ring by calculating the primary and secondary invariants.) Even 209 C x G : f C x ; f p f ; for all p G : though, for each f, the polynomial P is unique, there could be 210 ½ Š ¼f 2 ½ Š ¼ Á 2 g several f sharing the properties. This phenomenon will 211 163163 In the nineteenth century it was a primary goal of actually occur; it is realized by a certain flexibility in choosing 212 164 invariant theory to understand the structure of these rings. the parameters of our equations. The families of stars which 213 165 Hilbert’s Finiteness Theorem asserts that for finite groups, are thus obtained make certain parameter values look more 214 166 C x G is a finitely generated C-algebra: There exist invariant 215 ½ Š natural than others. This is the case for the plane symmetric 167 polynomials g x ; ...; g x such that any other invariant 216 1ð Þ kð Þ star with four vertices, where only one choice of parameters 168 polynomial h is a polynomial in g , …, g , say h x 217 1 k ð Þ¼ yields a hypocycloid, the famous Astroid (see example 9). 169 P g x ; ...; g x . Said differently, 218 ð 1ð Þ kð ÞÞ For surfaces, the appropriate choice of parameters is still an G open problem. This raises also the question of whether (in 219 C x C g1; ...; gk : ½ Š ¼ ½ Š analogy to the rolling small circle inside a larger one for the 220 171171 In general, the generators may be algebraically dependent, Astroid) there is a recipe for contructing the Platonic stars 221 172 that is, may satisfy an algebraic relation R(g1, ..., gk) = 0 with distinguished parameter values. We don’t know the 222 173 for some polynomial R y ; ...; y 0. It is important to answer. 223 ð 1 kÞ 6

4 4FL01 In the following chapterUNCORRECTED on the construction and in the examples, we write u, v, w instead of u1, u2, u3. Note that sometimes we do not need all three of them, as in the 4FL02 introductory example of the dodecahedron; but a general invariant polynomial may depend on all three. 5FL01 5There exist algorithms to calculate these invariants. One is implemented in the free Computer Algebra System SINGULAR. See http://www.singular.uni-kl.de/index.html 5FL02 for information about SINGULAR and http://www.singular.uni-kl.de/Manual/latest/sing_1189.htm#SEC1266 for instruction.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 224 Construction of Stars coefficients of f from (3), that is, in our notation the 278 279 225 In this section the group G O3 R we consider once parameters aijk. Additionally we obtain inequalities that 226 again one of the three real symmetry ð groupsÞ of the Platonic give us information about whether we will obtain a star or 280 227 solids. If the scalars of the input of the algorithms for the an anti-star. In general this system of equations will be 281 228 calculation of primary and secondary invariants are con- underdetermined. We will be left with free parameters, as 282 229 tained in some subfield of C, then the scalars of the output we already saw in the introductory example of the 283 230 are also contained in this subfield, see [10, p.1]. In our dodecahedral star. 284 231 examples the inputs are real matrices (the generators of G) Evidently, in this construction we have to choose the 285 232 and the outputs are the primary and secondary invariants degree d of the indetermined polynomial f. If we choose it 286 287 233 that generate the invariant ring as a subring of C x1; ...; xn . too small, the system of equation may not have a solution; 234 They even generate the real invariant ring, R x½ ; ...; x GŠ. but we want d to be as small as possible subject to this. The 288 ½ 1 nŠ 235 See the last section ‘‘Technical Details’’ for a proof. degree d has to be greater or equal to three, clearly. It 289 290 236 The symmetry groups of the Platonic solids are reflec- depends on the degrees of the primary invariants ui,aswe 237 tion groups. This implies that we have primary invariants will see in the examples. 291 238 u; v; w R x; y; z such that R x; y; z G R u; v; w .In The same construction should work for any dimension 292 239 fthe followingg ½ we alwaysŠ assume½ thatŠ we¼ have½ alreadyŠ n. The case of plane curves, n = 2, is easier to handle. 293 240 constructed a set of homogeneous primary invariants Even there the results are quite nice, as we will see in the 294 241 u; v; w R x; y; z . section on ‘‘plane dihedral stars’’. An interesting general- 295 242 f Ourg aim is½ to constructŠ a polynomial f in the invariant ization for n = 4 would be to calculate ‘‘Schla¨fli stars’’, 296 243 ring of G with prescribed singularities. By the results from corresponding to the six convex regular polytopes in four 297 244 the previous section we may write the polynomial uniquely dimensions, which were classified by Ludwig Schla¨fli, [2,p. 298 299 Author Proof 245 in the form 142]. We now conclude this section by demonstrating the 300 f u; v; w a uivj wk; 3 PROOF 301 ð Þ¼ ijk ð Þ above procedure in detail in the example of the octahedron id1 jd2 kd3 d 302 þ Xþ  and the cube. More examples will follow in the next sec- 303 247247 where d1 deg u ; d2 deg v ; d3 deg w ,andaijk R. tion, namely, the remaining Platonic stars and some 248 Such a polynomial¼ ð Þ has¼ the desiredð Þ symmetries,¼ ð Þ so we2 may Archimedean stars. We will also present some selected 304 249 move on and prescribe the singularities. They should lie at surfaces with dihedral symmetries in real three-space. 305 250 the vertices of a Platonic or an Archimedean solid. Let S be 251 a fixed Platonic (or Archimedean) solid. In the introduction EXAMPLE 2 (Octahedral and Hexahedral Star). The 306 252 we mentioned that these solids are vertex-transitive. This octahedron (the Platonic solid with 6 vertices, 12 edges, 307 253 implies that the algebraic surface corresponding to the and 8 faces) and its dual the cube (or hexahedron - with 8 308 254 polynomial (3), which is an element of the invariant ring of vertices, 12 edges, and 6 faces) have the same symmetry 309 255 the symmetry group of S, has to have the same local group Oh, of order 48. We choose coordinates x, y, and z of 310 256 geometry at each vertex of S. Therefore it is sufficient to R3 such that in these coordinates the vertices of the octa- 311 257 choose one vertex and impose conditions on f(u, v, w) hedron are (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1). Then O is 312 258 h guaranteeing an A2-singularity there. generated by two rotations r , r around the x- and 313 259 1 2 We can always suppose that S has one vertex at the y-axes by p/2, together with the reflection s in the 314 260 p := (1, 0, 0), otherwise we perform a coordinate change xy-plane: 315 261 to make this true. Having a singularity is a local property of 262 10 0 00 1 the surface, so we have to look closer at f at the point p.We À 263 do that by considering the Taylor expansion at p, that is, r1 0 00 1 1; r2 0 01 01; 264 + ¼ À ¼ substitute x 1 for x in f(u(x, y, z), v(x, y, z), w(x, y, z)). 01 0 10 0 265 We have the following necessary condition for a singularity B C B C @ 10 0A @ A 266 of type A2, with c1 and c2 being real constants not equal to 267 s 01 0 : zero, see [1, p.209]. ¼ 0 1 00 1 F x; y; z B À C ð Þ @ A : f u x 1; y; z ; u x 1; y; z ; u x 1; y; z These matrices are the input for the algorithm imple- 317317 ¼ ð 1ð þ Þ 2ð þ Þ 3ð þ ÞÞ 2 2 3 mented in SINGULAR that computes the primary and 318 c1 y z c2x higher order terms: 4 ¼ ð þ Þþ þ ð Þ secondary invariants. In this example the primary invariants 319 269269 ‘‘Higher order terms’’ here refers to all terms that have that generate the invariant ring are the following (although 320 270 weighted order, with weights (1/3, 1/2, 1/2), greater than it is easy to see that these three polynomials are invariant, it 321 271 1—that is, all monomials xi yj zk with i/3 + j/2 + k/2 [ 1. is not evident that they are primary invariants, that is, 322 272 If c1 and c2 have the same sign, the cusps will ‘‘point generate the invariant ring as an algebra): 323 273 outward’’, that is, we obtain a star. If they have different 274 signs the cusps will ‘‘point inward’’. u x; y; z x2 y2 z2; 275 UNCORRECTEDð Þ¼ þ þ Now expanding F(x, y, z) and comparing the coeffi- v x; y; z x2y2 y2z2 x2z2; 5 276 cients of x, y, and z with the right-hand side of equation ð Þ¼ þ þ ð Þ 277 w x; y; z x2y2z2: (4), we obtain a system of linear equations in the unknown ð Þ¼

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 325325 Now how low can the degree be of our indeterminate 326 polynomial? (Here and in the rest of this article degree 327 means the usual total degree in x, y, z.) Clearly it must be 328 even. A degree four polynomial yields no solvable system 329 of equations. Let us try a polynomial of degree six, f u; v; w 1 a u a u2 a u3 a uv a v a w: ð Þ¼ þ 1 þ 2 þ 3 þ 4 þ 5 þ 6 331331 We substitute x + 1 for x and expand the resulting 332 polynomial F(x, y, z) = f(u(x + 1, y, z), v(x + 1, y, z), 333 w(x + 1, y, z)). Next we collect the constant terms and 334 the linear, quadratic, and cubic terms, and compare them (a) Octahedral star, a4 = −100, (b) Hexahedral star, a1 = −100, 335 with the right-hand side of (4). This yields the following a5 =0,a6 =0. a2 =0,a3 =0. 336 system of linear equations: Figure 5. Octahedral and hexahedral star. Constant term of F : 1 a a a 0; þ 1 þ 2 þ 3 ¼ Coefficient of x : 2a1 4a2 6a3 0; 2 þ þ ¼ 374 Coefficient of x : a1 6a2 15a3 0; so we perform a rotation to achieve this, and write the 2 2 þ þ ¼ invariants in the new coordinates. With these invariants we 375 Coefficient of y and z : a5 a1 a4 2a2 3a3 c1; þ þ þ þ ¼ can proceed as in the example of the octahedron. Again we 376 Coefficient of x3 : 4a 20a c : 2 þ 3 ¼ 2 get no solution with degree four and must use a polynomial 377 6 ð Þ of degree six. After solving the system of equations we 378 338338 perform the inverse coordinate change and obtain the 379 Author Proof Since the monomials y, z, xy, xz, yz do not appear, we do 380 339 not obtain further equations from them. following polynomials (8) as candidates for hexahedral PROOF 381 340 Solving the first three equations from the above system stars or anti-stars: 341 yields the polynomial (7) with three free parameters. In 2 3 f u; v; w 1 3u a1u a2u a3uv 9 3a1 v 342 addition we get an inequality from the condition that the ð Þ¼ À þ þ þ þð À Þ 3 9 3 a 3a w; 8 343 coefficient of x must have the same sign as the coefficient of À ð þ 3 þ 2Þ ð Þ 344 y2 and z2 if we want to obtain a star. Substituting the solution with 3a1 + 9a2 + 2a3 = 0. For a1 = 3, a2 =-1, and 383383 345 of the first three equations yields c1 = a4 + a5 and c2 =-8. a3 = 0 we obtain the sphere. Other choices such that 384 346 We impose a4 + a5 = 0 to obtain a star or an anti-star: 3a1 + 9a2 + 2a3 = 0 may give singularities but cannot give 385 3 A2-singularities. Again there is a choice of parameters, 386 f u; v; w 1 u a4uv a5v a6w; ð Þ¼ð À Þ þ þ þ namely a = 3, a =-1 and a = c = 0, for which the 387 with a a 0: 7 1 2 3 4 þ 5 6¼ ð Þ surface has too many singularities. We obtain the same 388 389 348348 If we allowed all three parameters to be zero we would object as in the example of the octahedral star, with equa- 390 349 obtain the sphere of radius one. We have already made clear tion (9) below. In the other cases we obtain a hexahedral \ 391 350 that for a + a = 0 the zero set of (7) cannot have singu- star for 3a1 + 9a2 + 2a3 0 (figure 5b), or anti-star for 4 5 [ 392 351 larities of type A , so it must either be smooth or have 3a1 + 9a2 + 2a3 0, even though, as in the example of the 2 393 352 singularities of a different type. If we choose a = c, a = 0 octahedral stars, additional components may appear.// 4 5 394 353 and a =-9c, c = 0, the zero set is again not an octahedral Before proceeding, let us say more about the surface (9) 6 395 354 star, for it has too many singularities; we will describe this that emerged as a special case both of the octahedral and 396 355 phenomenon in more detail after the example of the hexa- the hexahedral stars. It has 14 singularities, exactly at the 397 356 hedral star. For the other choices of parameters the vertices of the octahedron and the cube, see figure 6, 357 corresponding zero sets are octahedral stars for a4 + a5 \ 0, 3 358 f u; v; w 1 u cuv 9cw; with c 0: 9 or anti-stars for a4 + a5 [ 0. See figure 5a. Sometimes ð Þ¼ð À Þ þ À 6¼ ð Þ 359 additional components appear and the stars or anti-stars We will call this object a 14-star or 14-anti-star for c \ 0 399399 360 become unbounded. In all examples presented in this or c [ 0, respectively. The parameter value c = 0 yields 400 361 article, especially when there is more than one free param- obviously a sphere. See figure 7 for an illustration of the 401 362 eter, special behaviors (such as additional components, dependence on the parameter. 402 363 unboundedness, or maybe more singularities than expected) This star does not correspond to a Platonic or Archi- 403 364 may occur for special choices of the free parameters. Most medean solid, but to the polyhedron S that is the convex 404 365 pictures presented are merely based on (good) choices of hull of the vertices of a hexahedron and an octahedron that 405 366 parameters. As we already mentioned, it would be interest- have all the same Euclidean diameter. This polyhedron has 406 367 ing to find conditions that prevent this behavior so that we 14 vertices, 36 edges, and 24 faces, which are isosceles 407 368 could prescribe boundedness as well as irreducibility in the triangles. See figure 6b. It is remarkable that it appears 408 369 definition of a star. here, for the symmetry group Oh does not act transitively 409 370 Now we turn to the Platonic solid dual to the octahe- on its vertices! The vertices of the hexahedron form one 410 371 dron, namelyUNCORRECTED the cube. If we use the same coordinates as orbit, the vertices of the octahedron another. If we fol- 411 372 1 1 1 lowed the program of this paper and sought such a star, we 412 before, it has vertices at p3 ; p3 ; p3 . But as we 373 already mentioned, we preferðÆ to haveÆ aÆ vertexÞ at (1, 0, 0), would need to fix two points, one in each orbit, and 413 ffiffi ffiffi ffiffi

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK Here a degree three polynomial yields no solution but 434434 degree four already suffices: 435 f u; v; w 1 2u cu2 8v 3c 1 w; ð Þ¼ À þ þ Àð þ Þ with c 1: 11 6¼ ð Þ For c \ 1 we obtain a star, for c [ 1 an anti-star. Its 437437 singular points (for c = 1) are (1, 1, 1), (-1, -1, 1), (1, 438 -1, -1), and (-1, 1, -1). If we choose c = 1in(11) the 439 polynomial f has four linear factors, see figure 8j: 440 f x 1 z y x 1 z y x 1 z y (a) 14-star, c = −50. (b) Polyhedron S corresponding ¼ð À þ À Þð À À þ Þð þ À À Þ x 1 z y : to the 14-star. Áð þ þ þ Þ 442442 Figure 6. 14-star and the corresponding convex polyhedron. For very small c-values there seem to appear four additional cusps at the vertices of a tetrahedron dual to the 443 first one; but these points stay smooth for all c R. For 444 414 prescribe singularities at both. This would lead to a larger 0 \ c \ 1 the zero set of our polynomial has additional2 445 415 system of linear equations. components besides the desired ‘‘star shape’’. For c [ 1we 446 416 Note, by the way, that if the vertices of the octahedron get anti-stars, see figure 8. Note that for c [ 0 the surfaces 447 417 and the cube have different Euclidean norms of a certain are unbounded. So unlike the previous examples, there are 448 418 ratio, namely 2=p3, the convex hull is a Catalan solid, no bounded anti-stars. 449 419 called the rhombic dodecahedron (14 vertices, only 12 Author Proof 420 faces because theffiffiffi triangles collapse in pairs into rhombi, 421 EXAMPLE 4 (Icosahedral star). The icosahedron is the 450 and 24 edges). This is the dual of the Archimedean solid PROOF 451 422 called the cuboctahedron that will be discussed later. Platonic solid with 12 vertices, 30 edges, and 20 faces. The symmetry group Ih of the icosahedron and its dual, the 452 dodecahedron, has 120 elements. Its invariant ring is gen- 453 423 Further Platonic and Archimedean Stars erated by the polynomials (2) from the first example, 454 424 EXAMPLE 3 (Tetrahedral star). The tetrahedron is the together with a third one (12), 455 425 Platonic solid with 4 vertices, 6 edges, and 4 faces. Its w x;y;z 4x2 z2 6xz 426 symmetry group Td has 24 elements. If we choose coordi- ð Þ¼ þ À 427 z4 2z3x x2z2 2zx3 x4 25y2z2 nates x, y, z such that one vertex is (1, 1, 1), the invariant ÀÁÁ À À þ þ À 428 ring is generated by the primary invariants displayed in 30xy2z 10x2y2 5y4 À À À þ 429 (10). One could also choose (1, 0, 0) as a vertex to avoid a 4 3 2 2 3 4 2 2 z 8z x 14x z 8zxÁ x 10y z 430 coordinate change, but then the invariants would be more Á þ þ À þ À 431 10x2y2 5y4 : 12 complicated. Note how different the primary invariants are À À þ Þ ð Þ 432 from those of the octahedron and the hexahedron (5). We point out that both invariants v and w factorize (over 457457 R 458 u x; y; z x2 y2 z2; ) into six, respectively ten, linear polynomials. The zero sets ð Þ¼ þ þ of these linear polynomials are related to the geometry. To 459 v x; y; z xyz; 10 ð Þ¼ ð Þ explain this, we introduce a new terminology: Given a Pla- 460 w x; y; z x2y2 y2z2 z2x2: tonic solid P, we call a plane through the origin a centerplane 461 ð Þ¼ þ þ

(a) c = −10000. (b) c = −1000. (c) c = −100. (d) c = −10. (e) c = −1. (f) c = −0.1.

UNCORRECTED (g) c =0. (h) c =0.1. (i) c =1. (j) c =3.5. (k) c =4. (l) c =5.

Figure 7. 14-star and anti-star, for c C 4 the surfaces are clipped by a sphere with radius 5.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK

(a) (b) c = −1000. (c) c = −100. (d) c = −3. (e) c = −1. (f) c =0. c = −100000.

(g) c =0.3. (h) c =0.6. (i) c =0.98. (j) c =1. (k) c =1.02. (l) c =3.

Figure 8. Tetrahedral star (and anti-star) with varying parameter value c; for c [ 0 the images are clipped by a sphere with radius 5.

Author Proof 462 of P if it is parallel to a face of the solid. The dodecahedron radius one. For all c = 0 the 12 singularities lie on this 475 463 has twelve faces and six pairs of parallel faces, so it has six sphere. For c = 27/32PROOF the surface has points at infinity in the 476 464 centerplanes. They correspond to the six linear factors of the direction of normals to the faces of the corresponding ico- 477 465 second invariant v. Analogously the icosahedron has ten sahedron. Note that this is just the negative value of c for 478 466 centerplanes, which give the linear factors of w. We have which the dodecahedral stars are unbounded. The illustra- 479 467 written out the factorization in the last section, see (38). tions suggest that for c [ 27/32 the surfaces become 480 468 For the dodecahedral and the icosahedral star the unbounded while they are bounded for c \ 27/32. 481 469 ‘‘smallest possible degree’’ is six. The third invariant has 470 degree ten, so we do not use it in either case. An equation EXAMPLE 5 (Cuboctahedral star). The cuboctahedron is 482 471 for the icosahedral star is the following: the Archimedean solid with 14 faces (6 squares and 8 483 equilateral triangles), 24 edges, and 12 vertices. See fig- 484 f u; v; w 1 u 3 cu3 cv; with c 0: 13 ð Þ¼ð À Þ þ þ 6¼ ð Þ ure 11b. Its symmetry group is that of the octahedron and 485 473473 Figure (9) shows icosahedral stars (c \ 0) and anti-stars cube. We use the invariants (5). Our construction yields a 486 474 (c [ 0) for various c-values. For c = 0 we get a sphere of polynomial of degree six, with three free parameters: 487

(a) c = −1000. (b) c = −100. (c) c = −10. (d) c = −0.1. (e) c =0.

UNCORRECTED(f) c =0.1. (g) c =0.5. (h) c =0.8. (i) c =27/ 32. 0(j) c = .9. Figure 9. Icosahedral star and anti-star, with varying parameter c; for c [ 27/32 the surfaces are clipped by a sphere with radius 11.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK f u; v; w ð Þ 128565 115200p5 49231296000p5 93078919125 1 þ c À ¼ þ 1295029 3 þ 15386239549  ffiffiffi ffiffiffi c c 3c 3 u 4 À 1 À 2 À þ  230400p5 257130 238926989250 126373248000p5 À À c À þ 1295029 3 þ 15386239549  ffiffiffi ffiffiffi c 3c 8c 3 u2 4 þ 1 þ 2 þ þ  p p 3 2 2 115200 5 128565 91097280000 5 172232645625 Figure 10. x + y - z = þ c3 À The zero set of 0. þ 1295029 þ 15386239549  ffiffiffi ffiffiffi c 3c 6c 1 u3 4 À 1 À 2 À þ 2 3  f u; v; w 1 3u au 12 4a v bu 121075 51200p5 102400p5 242150 ð Þ¼ À þ þð À Þ þ 14 c À c v À 2c 4 4b uv cw; ð Þ þ 3 þ 11881 4 þ 11881 À 3 Àð þ Þ þ ffiffiffi ffiffiffi uv c u4 c u5 c u2v c w; 489489 Á þ 1 þ 2 þ 3 þ 4 with c 0 and b c ; c ; c ; c 490 with a + b = 2 and 8(a + b) - c = 16. For a = 3, 4 6¼ ð 1 2 3 4Þ : 991604250 419328000p5 c 20316510c 491 b =-1, and c = 0 we obtain a sphere. In this example we ¼ð À Þ 4 þ 3þ 492 135776068 121661440ffiffiffip5 c2 have a new kind of behavior. We always got inequalities þð À Þ 493 3 2 2 from the condition that the coefficients of x and y + z in 33944017 30415360p5 cffiffiffi1 30415360p5 þð À Þ þ Author Proof 494 the Taylor expansion of f in (1, 0, 0) should have the same 33944017 0: ffiffiffi ffiffiffi 15 495 sign. In this case the coefficient of x3 is -8, but y2 and À 6¼ ð Þ 2 PROOF 517517 496 z have different coefficients, namely a + b - 2 and We obtain stars if we choose c1, c2, c3, and c4 such that 518 497 16 - 8(a + b) + c, respectively. So if both are negative we c4 and b(c1, c2, c3, c4) have the same sign. Otherwise we 519 498 obtain stars, see figure 11a; if both are positive, anti-stars; obtain anti-stars. See figure 11c. 499 but if they have different signs, we will have a ‘‘new’’ 500 object, whose singularities look, up to local analytic coor- Plane Dihedral Stars 520 501 dinate transformations6, such as x3 + y2 - z2 = 0, see Analogous to the Platonic and Archimedean stars in three 521 502 figure 10. The singularities always lie on a sphere of radius dimensions, we will define plane stars. Let P be a regular 522 503 one. polygon with m vertices. Its symmetry group in O2 R is 523 ð Þ 524 504 EXAMPLE 6 (Soccer star). The truncated icosahedron is the dihedral group denoted by Dm. It is of order 2m.A 525 505 the Archimedean solid which is obtained by ‘‘cutting off the plane m-star is a plane algebraic curve that is invariant D 526 506 vertices’’ of an icosahedron. It is known as the pattern of a under the action of the dihedral group m and has exactly 3 527 507 m singularities of type A2 (that is, with equation x + soccer ball. It has 32 faces (12 regular pentagons and 2 528 508 y = 0) at the vertices of P, ‘‘pointing away form the origin’’; 20 regular hexagons), 60 vertices, and 90 edges. See 529 509 otherwise, that is, if the cusps ‘‘point towards the origin’’, we figure 11d. Its symmetry group is the icosahedral group 530 510 speak of an m-anti-star. In the examples presented here the Ih. For this example we do need the third invariant, because 531 511 singularities will be at the mth roots of unity. the first polynomial that yields a solvable system of If we consider the dihedral groups as subgroups of 532 512 equations is of degree ten. We obtain the following equa- O R , they are reflection groups. This is not true if we 533 513515 2 tion with four free parameters, in the invariants (2) and viewð Þ them as subgroups of O R , as we do in the next 534 514 3 (12): batch of examples. ð Þ 535

(a) Cuboctahedral star, (b) Cuboctahedron. (c) Soccer star, (d) Truncated

a =0,b =0,c = −100. c1 = −100, c2 = −100, icosahedron. UNCORRECTEDc3 = −100, c4 = −100. Figure 11. Two Archimedean solids and stars.

6 6FL01 But if we allow complex local analytic coordinate transformations, the singularities are still of type A2.

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 1.0 1.0 1.0

0.5 0.5 0.5 y y 0.0 0.0 y 0.0

0.5 0.5 0.5

1.0 1.0 1.0 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 x x x (a) A 2-star (19), c1 = −4. (b) The Deltoid. (c) The Astroid.

Figure 12. Some plane dihedral stars.

536 There is another way to construct plane stars, namely as figure 12a. For c1 [ 0 it is an unbounded anti-star. In both 568 537 hypocycloids. A hypocycloid is the trace of a point on a cases it has two singularities at (±1 , 0). 569 538 = 570 Author Proof circle of radius r that is rolling within a bigger circle of The hypocycloid for k 2 is parametrized by (2r 539 radius R. If the ratio of the radii is an integer, R : r = k, then cosu, 0) where u is in [0, 2p]. So it is not an algebraic curve 571 PROOF 572 540 the curve is closed and has exactly k cusps but no self- but an interval on the x-axis. 541 intersections. Hypocycloids have a quite simple trigono- 573 542 metric parametrization (16): EXAMPLE 8 (3-star). The primary invariants of D3 are u k 1 r cos u r cos k 1 u; u x; y x2 y2; 7! ð À Þ þ ð À Þ ð Þ¼ þ 20 k 1 r sin u r sin k 1 u ; u 0; 2p : 16 v x; y x3 3xy2: ð Þ ð À Þ À ð À Þ Þ 2½ Š ð Þ ð Þ¼ À 544544 There are algorithms for the implicitization of trigono- In this case a degree four polynomial suffices to generate a 575575 545 metric parametrization, see [6]. It turns out that hypocycloids star, see figure 12b. The polynomial (21) is completely 576 546 are stars in our sense: they have the correct symmetries and determined, we have no free parameters. It coincides with 577 578 547 singularities of type A2. In the construction of stars via pri- the hypocycloid for k = 3, which is also called Deltoid: 548 mary invariants we always try to find a polynomial of f u; v 1 6u 3u2 8v: 21 549 minimal degree that satisfies these properties. We will see ð Þ¼ À À þ ð Þ 550 that sometimes the hypocycloids coincide with the stars we EXAMPLE 9 (4-star). The dihedral group of order eight, 580580 551 obtain that way. In one of the examples presented here, D4, has primary invariants 581 552 namely the 5-star, the degree of the implicitization of the 553 u x; y x2 y2; hypocycloid is higher than the degree of the polynomial our ð Þ¼ þ 22 554 construction yields. This suggests that we define a ‘‘star’’ as v x; y x2y2: ð Þ 555 the zero set of the polynomial of minimal degree satisfying all ð Þ¼ 583583 556 other conditions. Our construction yields the following polynomial of degree six with two free parameters: 584 557 EXAMPLE 7 (2-star). The group D2 has primary invariants f u; v 1 u 3 c v c uv; with c c 0; 23 ð Þ¼ð À Þ þ 1 þ 2 1 þ 2 6¼ ð Þ 586586 2 u x; y x ; + \ + [ 587 ð Þ¼ 17 For c1 c2 0 we obtain stars, for c1 c2 0 anti- v x; y y2: ð Þ stars. In both cases additional components might appear. 588 ð Þ¼ The curves become unbounded for c [ 4. 589 559559 2 Our constructions yields the degree six polynomial with The hypocycloid with four cusps is also called Astroid. 590 560 six free parameters: Its implicit equation is (1 - u)3 - 27v = 0. So if we choose 591 3 2 2 2 c =-27 and c = 0in(23) we obtain the same curve. See 592 f u; v 1 u c1v c2uv c3v c4uv c5u v 1 2 ð Þ¼ð À Þ þ þ þ þ þ figure 12c. 593 c v3; with c c c 0: 18 þ 6 1 þ 2 þ 5 6¼ ð Þ EXAMPLE 10 (5-star). The primary invariants of D5 are 594 562562 The choice c1 = 0 and the remaining parameters equal to 563 zero yield the simple equation u x; y x2 y2; ð Þ¼ þ 24 3 5 3 2 4 f uUNCORRECTED; v 1 u c1v; with c1 0: 19 v x; y x 10x y 5xy : ð Þ ð Þ¼ð À Þ þ 6¼ ð Þ ð Þ¼ À þ 566 \ 565565 For c1 0 we obtain a 2-star. The corresponding curve If we try a degree five polynomial, we obtain (25) with 596596 567 runs through the points 0; 1 and is bounded. See p c1 no free parameters. It only permits anti-stars. 597 ð Æ À Þ ffiffiffiffiffiffi THE MATHEMATICAL INTELLIGENCER Journal : Large 283 Dispatch : 8-3-2010 Pages : 14

Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 621 10 2 8 singularities’’ are of type A1, that is, they have, up to ana- f u; v 1 u 5u v: 25 2 2 ð Þ¼ À 3 þ À 3 ð Þ lytic coordinate transformations, equation x + y = 0. 622 One could call this curve an algebraic pentagram. For 623 599599 So let us use degree six. This yields the following [ 624 600 c 80 the curve has two components, see figure 13e. polynomial for plane 5-stars or anti-stars: The implicit equation of the hypocycloid with five cusps 625 c 10 8 is already of degree eight, while the polynomial we found 626 f u; v 1 þ u 2c 5 u2 1 c v cu3; ð Þ¼ À 3 þð þ Þ À 3 ð þ Þ þ with our construction has degree six. The two equations 627 628 with c 1; 5: 26 cannot coincide for any choice of the free parameter c. 6¼À ð Þ 602602 Here, as the parameter value c varies we observe a quite 3 603 curious behavior. For c \ -1 one obtains a star, the Dihedral ‘‘Pillow Stars’’ in R 629 630 604 smaller c gets, the smaller is its ‘‘inner radius’’, see fig- If we consider the dihedral groups Dm as subgroups of 605 ure 13a. The choice c =-1 yields a circle with radius O3 R , they cease to be reflection groups, so we have to 631 ð Þ 606 one—the circle containing the five singularities of the (26) consider the secondary invariants as well. The number of 632 607 for other c. For -1 \ c \ 5 the cusps of (26) point inward, secondary invariants depends on the order of the group 633 608 that is, we have anti-stars. For - 1 \ c \ 0 the curve has and the degrees of the primary invariants, see [10, p.41]. In 634 609 one bounded component; for c = 0, it is unbounded with the examples we give here there are always two secondary 635 636 610 five components, figure 13b; for 0 \ c \ 5 the curve is invariants. The first one, s1, is always 1, so we do not 637 611 again bounded, but has five components, like drops falling mention it every time but just give the second one, s2. 612 away from the center, figure 13c. For c = 5 only finitely In this section our aim is to construct surfaces that are 638 639 613 many points satisfy the equation, the five points that are invariant under the action of Dm with singularities at the 614 [ mth roots of unity in the xy-plane, and which in addition 640

Author Proof singular in the other cases. If we choose c 5 we obtain 615 stars again, that is, the cusps point outward, even though pass through the points (0, 0, ±c) with 0 c R. Intui- 641 616 \ \ PROOF6¼ 2 642 for 5 c 80 the curve also has five components, like tively the resulting surface should look like a pillow. To 617 drops falling towards the origin, figure 13d. The curve we lead to such a shape, more conditions, such as bounded- 643 618 obtain for c = 80 is special in that it has self-intersections, ness and connectedness, would have to be imposed. We 644 619 that is, five additional singularities. They lie on a circle with do not have a systematic theory, but our experimental 645 620 radius one quarter, on a regular pentagon. These ‘‘extra results seem promising. In these examples we have a large 646

1.0 c 0.4 c 0

4 c 0 4 c 0.5 c 0.6 0.5 2 2 1 0.9 c c 1 c c 1.1 c 2 c 2

y 0 y 0 y 0.0 c 5 c 1 c 10 c 100 2 2 0.5 4 4

1.0 1.0 0.5 0.0 0.5 1.0 4 2 0 2 4 4 2 0 2 4 x x x (a) 5-stars. (b) 5-stars. (c) 5-stars.

1.0 1.0

0.5 0.5 c 10

c 30 c 90 c 70 c 120 y 0.0 c 80 y 0.0 c 80 c 200

0.5 0.5

1.0 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 UNCORRECTEDx x (d) 5-anti-stars. (e) 5-anti-stars.

Figure 13. 5-stars and anti-stars.

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(a) D3-star. (b) D4-star. (c) D5-star.

Figure 14. Pillow stars.

1 c c4 c c6 f u; v 1 þ 1 þ 4 u 3v c u2 c uv 3v2 ð Þ¼ À c2 À þ 1 þ 2 þ c w c u3 v3 c uw c vw c uv2 þ 3 þ 4 À þ 5 þþ 6 þ 7 c u2v; 32 þ 8 ð Þ Figure 15. Zitrus for c =-4. 676676 4 6 2 with c3 + c6 \ 0 and -(1 + c1c + c4c ) + c (c2 + 677 647 number of free parameters, unfortunately. We have tried to c7) \ 0. See figure 14b for the resulting surface, where we 678 648 choose values giving attractive results. chose c = 1/3, c3 =-27 and set all the other parameters 679 to zero. 680 649 EXAMPLE 11 (D ). The primary invariants of D O R 3 3  3ð Þ Author Proof 650 are EXAMPLE 13 (D5). The primary invariants of D5 are 681 u x; y; z z2; PROOF ð Þ¼ u x; y; z 0z2; v x; y; z x2 y2; 27 ð Þ¼ ð Þ¼ þ ð Þ v x; y; z x2 y2; 33 w x; y; z x3 3xy2; ð Þ¼ þ ð Þ ð Þ¼ À w x; y; z x5 10x3y2 5xy4: 652652 its secondary invariant is ð Þ¼ À þ Its secondary invariant is 683683 2 3 s2 x; y; z 3x yz y z: 28 ð Þ¼ À ð Þ 4 2 3 5 s2 x; y; z 5x yz 10x y z y z: 34 654654 A polynomial of degree three yields no solution. The ð Þ¼ À þ ð Þ 655 general equation of a degree four polynomial in the A degree five polynomial already produces a solvable 685685 686 656 invariant ring of D3 is f1(u, v, w) + bs2, where f1(u, v, w) system of equations, but the resulting polynomial with 657 is an indeterminate polynomial of degree four in R u; v; w three free parameters only permits anti-stars. So we choose 687 ½ Š 688 658 as in the previous examples, and b is a constant. A degree a polynomial of degree six; again s2 does not appear: 659 four polynomial suffices to obtain a solvable system of 1 c c4 c c6 10 c 660 equations. It yields the following polynomial with three f u; v; w 1 þ 1 þ 3 u þ 4 v c u2 c2 1 661 free parameters: ð Þ¼ À À 3 þ 2 8 4 c2uv 5 2c4 v 1 c4 w 1 c1c 2 2 þ þð þ Þ À 3 ð þ Þ þ f u; v; w 1 þ 2 u c1u c2uv 6v 3v 8w; 3 3 2 2 ð Þ¼ À c þ þ À À þ c3u c4v c5uv c6u v: 35 29 þ þ þ þ ð Þ ð Þ The zero sets of these polynomials are stars for 690690 663663 with -(1 + c c4) + c2c \ 0. Note that the secondary 1 2 4 6 2 664 c4 1\0 and 1 c1c c3c c c2 c5 \0; or invariant s2 does not appear in the above polynomial, its þ Àð þ þ Þþ ð þ Þ 665 4 6 2 coefficient b is zero. We obtain a nice result for c1 = c4 5 [ 0 and 1 c1c c3c c c2 c5 [ 0: 666 c = c = À Àð þ þ Þþ ð þ Þ 2 0, 1/3, see figure 14a. 36 ð Þ 667 EXAMPLE 12 (D4). The group D4 has the following = = 692692 668 primary and secondary invariants: A nice choice for the free parameters is c 1/3, c4 -3, setting all other parameters equal to zero. See figure 693 u x; y; z z2; 14c. 694 ð Þ¼ v x; y; z x2 y2; 30 ð Þ¼ þ ð Þ w x; y; z x2y2; 695 ð Þ¼ EXAMPLE 14 (Zitrus). The last example we want to 670670 present is the surface Zitrus. It is the plane 2-star rotated 696 s x; y; z x3yz xy3z: 31 2ð Þ¼ À ð Þ around the x-axis (figure 15). Its equation is 697 672672 Our constructionUNCORRECTED yields a degree six polynomial; as in the 2 2 2 3 2 2 673 f x; y; z 1 x y z c y z ; previous example, the secondary invariant s2 happens to ð Þ¼ð Àð þ þ ÞÞ þ ð þ Þ 674 with c\0: 37 drop out: ð Þ

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 699699 Outlook 721721 700 In all the examples presented above we observed ‘‘unwan- 5yp5p3 5p3y 2 75 30p5z À þ þ þ 701 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ted’’ behavior for special choices of the free parameters: the ffiffiffi ffiffiffi ffiffiffi ffiffiffi 702 surfaces became unbounded at some point, or additional x 75 30p5 x 75 30p5p5 Á À þ þ þ 703 components appeared. Sometimes we even had more sin-  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi ffiffiffi 704 gularities, or singularities of a different type than we 5yp5p3 5p3y 2 75 30p5z þ À þ þ 705 expected. Further investigations would be necessary to find qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffiffiffi ffiffiffi ffiffiffi ffiffiffi 706 conditions preventing such behavior. After this is done, one x 75 30p5 x 75 30p5p5 5yp5p3 Á À þ À þ 707 could refine the definition of ‘‘stars’’ and ‘‘anti-stars’’ by  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi 708 demanding that the surfaces be bounded and irreducible. 5p3y 2 75 30p5z þ À À 709 Dual (Platonic) solids have the same symmetry group, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffiffiffi ffiffiffi 710 hence the same primary invariants were used to construct the x 75 30p5 x 75 30p5p5 5yp5p3 711 Á À þ À À corresponding stars. But there seems to be no obvious duality  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi 712 between the stars such as occurs for dual (Platonic) solids. 5p3y 2 75 30p5z : À À À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffiffiffi ffiffiffi 38 ð Þ 713 Technical Details 724 G The invariant ring R x1; ...; xn 723723725 Rn ½ Š 726 714 Factorization of the primary invariants of Ih Let G GL be a finite subgroup. Then there exist n  ð Þ 727 715 In example (4) of the icosahedral stars, we claimed that two homogeneous, algebraically independent polynomials u ; ...; u C x ; ...; x (called the primary invariants of 728 Author Proof 716 1 n 1 n of the primary invariants of Ih factor into linear polynomials 2 ½ Š 729 717 corresponding to the centerplanes of the dodecahedron G) and l (depending on the cardinality of G and the degrees PROOFC 730 718 of the ui) polynomials s1; ...; sl x1; ...; xn (the sec- and icosahedron, respectively, and we promised to give the 2 ½ Š 731 719 factorizations explicitly. Here they are: ondary invariants of G) such that the invariant ring C G l C 732 decomposes as x1; ...; xn aj 1sj u1; ...; un . There v x; y; z are algorithms to½ calculateŠ these¼ primary¼ ½ and secondaryŠ 733 ð Þ 1 invariants, see [10, p.25]. Also in [10, p.1] it is claimed that if 734 z 2x z p5 1 x 10 2p5y 2z 735 ¼À16 ð þ Þð þ Þ À À À the scalars of the input for these algorithms are contained in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 736 ffiffiffi ffiffiffi a subfield K of C, then all the scalars in the output will also p5 1 x 10 2p5y 2z be contained in K. So in our case with G GL Rn , the 737 Áð þ Þ þ À À  ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi primary and secondary invariants will be real polynomials: 738 ffiffiffi ffiffiffi u ; ...; u ; s ; ...; s R x ; ...; x . 739 p5 1 x 10 2p5y 2z 1 n 1 l 2 ½ 1 nŠ Áð À Þ À þ þ Now the claim is, in the notation above: R x1; ...; 740 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ffiffiffi ffiffiffi G R ½ 741 xn aj 1sj u1; ...; un . p5 1 x 10 2p5y 2z ; Š ¼ ¼ ½ R Š ... G l R ... 742 Áð À Þ þ þ þ The first inclusion x1; ; xn aj 1sj u1; ; un qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is trivial. We prove the opposite½ inclusion:Š  Let¼ f ½ R x ; ...;Š 743 ffiffiffi 1 ffiffiffi 2 ½ 1 w x; y; z 3x xp5 z 3x xp5 z x G C x ; ...; x G be an invariant polynomial. As C x ; 744 ð Þ¼À20250000 À þ þ þ À nŠ  ½ 1 nŠ ½ 1 G l C ÀÁffiffiffi ÀÁffiffiffi ...; xn equals aj 1sj u1; ...; un , we can write f 745 2x 75 30p5 x 75 30p5p5 5p3y Š ¼ ½ Š Á À þ þ þ þ uniquely in the following form: 746  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi l 75 30p5z a À þ f x1; ...; xn sj cjau ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð Þ¼ j 1 a A ffiffiffi ¼ 2 2x 75 30p5 x 75 30p5 p5 X X Á À þ þ þ Á where c = d + ie are complex constants, and A is 748748  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ja ja ja ffiffiffi ffiffiffi ffiffiffi some finite subset of Nn. Then 749 5p3y 75 30p5z À À þ l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a a ffiffiffi ffiffiffi f x1; ...; xn sj djau i ejau 2x 75 30p5 x 75 30p5p5 ð Þ¼ j 1 a A þ a A ! Á À þ À X¼ X2 X2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l l 39 ffiffiffi ffiffiffi ffiffiffi a a 5p3y 75 30p5z sj djau i sj ejau ð Þ À þ À ¼ j 1 a A þ j 1 a A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  X¼ X2 X¼ X2 ffiffiffi ffiffiffi ...... 2x 75 30p5 x 75 30p5p5 f1 x1; ; xn if2 x1; ; xn : Á À þ À ¼ ð Þþ ð Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffi ffiffiffi Here f1 and f2 are real polynomials. Since f is also contained 751751 5p3y 75 30p5z 752 þ þ À in the real polynomial ring, f2 must be equal to zero. But UNCORRECTEDqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  l a l ffiffiffi ffiffiffi from f2 x1; ...; xn j 1 sj a A ejau a A j 1 sj eja 753 ð Þ¼ ¼ 2 ¼ 2 ð ¼ Þ x 75 30p5 x 75 30p5p5 a [ l Á À þ þ þ u 0 it would followP that forP all a A the sumP j P1 sj eja must 754  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ ffiffiffi ffiffiffi ffiffiffi P

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Article No. : 9147 h LE h TYPESET MS Code : TMIN-200 h44CP h DISK 771 755 be equal to zero, because the ui are algebraically inde- [2] H. S. M. Coxeter. Regular polytopes. Methuen, London, 1948. l a [3] P. R. Cromwell. Polyhedra. Cambridge University Press, Cam- 772 756 pendent. Hence f f1 x1; ...; xn j 1 sj a A djau ¼ ð Þ¼ ¼ 2 2 bridge, 1997. 773 l R ... 757 aj 1 sj u1; ; un . P P [4] B. Gru¨ nbaum. An enduring error. Elemente der Mathematik, 774 ¼ ½ Š 64(3):89–101, 2009. 775 758 ACKNOWLEDGMENTS [5] M. Hochster and J. A. Eagon. Cohen-Macaulay rings, invariant 776 759 We thank Frank Sottile for valuable suggestions and C. theory, and the generic perfection of determinantal loci. American 777 760 Bruschek, E. Faber, J. Schicho, D. Wagner, and D. Westra J. of Mathematics, 93(4):1020–1058, 1971. 778 761 for productive discussions. We also thank Manfred Kuhn- [6] H. Hong and J. Schicho. Algorithms for trigonometric curves 779 762 kies and all of FORWISS, University Passau, for their (simplification, implicitization, parametrization). J. Symbolic Com- 780 763 motivating enthusiasm. (We recommend looking at some putation, 26:279–300, 1998. 781 764 of the beautiful 3D-prints of algebraic surfaces produced by [7] J. Matousˇ ek. Lectures on discrete geometry. Springer, 2002. 782 765 FORWISS and Voxeljet http://www.forwiss.uni-passau.de/ [8] Plato. Phaedrus. Project Gutenberg, http://www.gutenberg.org/ 783 766 de/projectsingle/64/main.html/). etext/1636, October 2008. Translated by B. Jowett. 784 [9] T. Roman. Regula¨ re und halbregula¨ re Polyeder. Kleine Er- 785 ga¨ nzungsreihe zu den Hochschulbu¨ chern fu¨ r Mathematik; 21. 786 767 REFERENCES Deutscher Verlag der Wissenschaften, 1968. 787 768 [1] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singu- [10] B. Sturmfels. Algorithms in Invariant Theory. Springer, second 788 769 larities of differentiable maps. Vol. I, volume 82 of Monographs in edition, 2008. 789 770 Mathematics. Birkha¨ user, Boston, 1985. 790 Author Proof PROOF

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