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Visualization of Conway Polyhedron Notation World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:3, No:2, 2009 Visualization of Conway Polyhedron Notation Hidetoshi Nonaka Abstract—This paper presents an interactive modeling system of TABLE II THE LIST OF PLATONIC SOLIDS, WHERE pqr,, ARE THE NUMBER OF polyhedra using the isomorphic graphs. Especially, Conway VERTICES, EDGES, AND FACES, RESPECTIVELY. polyhedron notation is implemented. The notation can be observed as p q r interactive animation. Symbol Name of polyhedron P33 Tetrahedron 4 6 4 Keywords—Conway polyhedron notation, Polyhedral graph, P 3 Cube 8 12 6 Visualization. 4 P34 Octahedron 6 12 8 I. INTRODUCTION P53 Dodecahedron 20 30 12 ONWAY polyhedron notation is used for describing P35 Icosahedron 12 30 20 Cvarious polyhedra based on Platonic solids as seed polyhedra [1-2]. The author has recently developed an TABLE III THE LIST OF ARCHIMEDEAN SOLIDS, WHERE pqr,, ARE THE interactive modeling system of polyhedra by means of graph NUMBER OF VERTICES, EDGES, AND FACES, RESPECTIVELY. drawing and simulated elasticity, mainly for educational Symbol Name of polyhedron p q r purpose [10-13]. In this paper, we present an implementation of A 2 Cuboctahedron 12 24 14 the full set of the notation on the system. Using this (3¸ 4) implementation, we can observe all of the operations of the A4610¸¸ Great Rhombicosidodecahedron 120 180 62 notation as animation interactively. A468¸¸ Great Rhombicuboctahedron 48 72 26 A 2 II.CONWAY POLYHEDRON NOTATION (3¸ 5) Icosidodecahedron 30 60 32 John Conway has proposed a notation for describing various A3454¸¸¸ Small Rhombicosidodecahedron 24 48 26 polyhedra based on five Platonic solids as seed polyhedra. A343 Small Rhombicuboctahedron 24 60 38 Table 1 shows the complete set of operations of the notation. ¸ A 4 34¸ Snub Cube 24 36 14 TABLE 1THE LIST OF OPERATORS OF CONWAY POLYHEDRON NOTATION. A 4 Snub Dodecahedron 60 150 92 Operator Name Description 35¸ Each vertex becomes a new face and A 2 Truncated Cube 24 36 14 d dual 38¸ each face becomes a new vertex A 2 Truncated Dodecahedron 60 90 32 t truncate Vertices are truncated 310¸ A 2 Truncated Icosahedron 60 90 32 Vertices are truncated to the edge mid 56¸ a ambo point A 2 46¸ Truncated Octahedron 24 36 14 New faces are added in place of edges b bevel A 2 Truncated Tetrahedron 12 18 8 and vertices 36¸ Each vertex makes a new face and e expand Five Platonic solids and thirteen Archimedean solids are each edge makes a new quadrangle listed in Tables 2-3. For example, A 2 indicates that two Each vertex makes a new face and (3¸ 4) s snub each edge makes a pair of triangles regular triangles and two squares are gathered alternately on each vertex. Archimedean solid can be expressed using k kiss Dual of truncate International Science Index, Mathematical and Computational Sciences Vol:3, No:2, 2009 waset.org/Publication/14426 Conway notation as follows: j join Dual of ambo AaPaP43, AbPbP5 3 , m meta Dual of bevel (3¸ 4) 344610¸¸ 3 5 AbPbP43, AaPaP235 , o ortho Dual of expand 468¸¸ 34(3¸ 5)3 5 AePeP5 3 , AePeP343, g gyro Dual of snub 3454¸¸¸ 3 5 34¸ 3 4 AsPsP443, AsPsP435 , AtP23, 34¸ 3 4 35¸ 3 5 38¸ 4 AtP23 , AtP2 5 , AtP24 , AtP23 , H. Nonaka is with Hokkaido University, Sapporo 060 0814, Japan 310¸ 5 56¸ 3 46¸ 3 36¸ 3 (corresponding author to provide; e-mail: [email protected]). and additionally, PaP3343 , PsP35 33 . (1) International Scholarly and Scientific Research & Innovation 3(2) 2009 126 scholar.waset.org/1307-6892/14426 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:3, No:2, 2009 III. GRAPH OPERATION FOR POLYHEDRA An edge contraction is a graph contraction of an edge The author previously presented an operation-based notation (Figure 1). A vertex splitting is defined conventionally as the for Archimedean graph [12]. Three operations are defined for reverse of edge contraction, but in this paper, we define a polyhedral graph: edge contraction, vertex splitting, and vertex splitting of vV as the composition of the operations diagonal addition (Figure 1 – 3). of subdivision of incident edges on v, connecting the new vertices in a proper order, and deleting the vertex v (Figure 2). A diagonal addition to a face in F4 is an edge addition between a pair of non-adjacent vertices in a quadrangular face (Figure 3). The operation of diagonal addition is applied only when the graph G is obtained by applying “expand” to other graph H. Fig. 1. An example of edge contraction operations. Following statement holds for an arbitrary polyhedral graph G: GHG.( eH ) GG01 G uvw. [ ()uVº uV01 uV ()uV01 uV ((,)()())uvº E u V01 v V u V 10 v V (deg()3)wV0 w Fig. 2. An example of vertex splitting operations by the present (deg()4)uVº1 u ], (3) definition in this paper. where GVEF {, , }is the dual of GVEF {, , }. The first three clauses inside of the bracket mean that G is a bipartite graph and V is subdivided to V0 and V1 . The fourth clause denotes that G0 includes at least one vertex with degree 3, and G0 contains at least one triangle. The last clause denotes that G is a 4-regular graph, and all faces in G are quadrangles, Fig. 3. An example of diagonal addition operations. 1 1 which will be applied diagonal addition. Quadrangles in A polyhedral graph GVEF {, , } is defined by the set of G0 should not be applied. Consequently, the operation of diagonal addition is performed by following algorithm: vertices Vv {,01" , vp }, edges Ee {,01" , eq }, and faces 1. Find a triangle f in G, and if not found, return false. Ff {,01" , fr }. G is planar and 3-connected graph. The set 2. Search the dual graph G from f in breadth-first way. F is subdivided as follows: 3. If the depth is odd number, in other words, if the distance FFFFjkG , (2) (path length) from f is odd number, add a diagonal to f. *i3,4,5," ijk 4. If not all the nodes in G are traversed, go to step 2. where Fn denotes the set of faces with n sides. 5. Return true. vs ec ec vs ec P34 A4 .6 2 vs A(3 .4) 2 A3 .8 2 P43 ec da da A3 .6 2 A4 ..6 8 A3 .4 3 A4 .3 4 vs ec ec vs vs International Science Index, Mathematical and Computational Sciences Vol:3, No:2, 2009 waset.org/Publication/14426 P35 A5 .6 2 vs A(3 .5) 2 A3 .10 3 P5 3 ec da P33 A4 ..6 10 A3 ...4 5 4 A3 4 .5 Fig. 4. Relations of 5 Platonic graphs and 13 Archimedean graphs induced by edge contraction (ec), vertex splitting (vs), and diagonal addition (da). International Scholarly and Scientific Research & Innovation 3(2) 2009 127 scholar.waset.org/1307-6892/14426 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:3, No:2, 2009 The operators of Conway notation can be expressed using graph operations for the isomorphic graph as follows. Truncate “t” is equivalent to vertex splitting. Ambo “a” is equivalent to the composition of vertex splitting and edge contraction for the original edges before applying vertex splitting. Bevel “b” is equivalent to “ta”. Expand “e” is equal to two consecutive ambos “aa”. Snub “s” is applying “e” followed by diagonal addition. The operators of the remainder are duals of above operators. Figure 4 shows the relations of 5 Platonic graphs and 13 Archimedean graphs induced by the three graph operations, and they correspond to the expressions in (1). IV. INTERACTIVE MODELING SYSTEM OF POLYHEDRA The interactive modeling system of polyhedron consists of three subsystems: graph input subsystem, wire-frame subsystem, and polygon subsystem [11]. Figure 5 shows a screen shot of graph input subsystem. The first step of the modeling of polyhedron is drawing a polyhedral graph isomorphic to the polyhedron. In the subsystem, vertex addition, vertex deletion, edge addition, and edge deletion are Fig. 6. Screen shot of wire-frame subsystem. implemented as fundamental operations. Figure 6 shows a screen shot of wire-frame subsystem. After constructing a polyhedral graph, the next step is arranging vertices in 3D space with virtual springs and Hooke’s law. Wire-frame polyhedron can be formed by controlling the natural length of virtual spring corresponding to three types of binary relations between pairs of vertices. Figure 7 shows a screen shot of polygon subsystem. After arranging vertices in 3D space, the last step is detecting faces, selecting appropriate faces, and rendering the solid. Detecting n-regular polygon is equivalent to finding simple closed path with length n. Fig. 7. Screen shot of polygon subsystem. Figure 8 is a screen shot of the tool box for Conway International Science Index, Mathematical and Computational Sciences Vol:3, No:2, 2009 waset.org/Publication/14426 polyhedron notation. The captions of buttons indicate the corresponding operations of Conway notation. These operations can be applied in both the wire-frame subsystem and the polygon subsystem. Fig. 5. Screen shot of graph input subsystem. Fig. 8. Tool box for Conway polyhedron notation International Scholarly and Scientific Research & Innovation 3(2) 2009 128 scholar.waset.org/1307-6892/14426 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:3, No:2, 2009 By pressing a button on the tool box, corresponding REFERENCES operation is applied to the polyhedron with animation.
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