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Home , Archimedean solid, Cube, Cuboctahedron, Dodecahedron, Icosidodecahedron, Platonic graph

Operation-Based Notation for Archimedean Graph

Hidetoshi Nonaka Research Group of Mathematical Information Science Division of Computer Science, Hokkaido University N14W9, Sapporo, 060 0814, Japan

ABSTRACT Table 1. The list of Archimedean solids, where p, q, r are the number of vertices, edges, and faces, respectively. We introduce three graph operations corresponding to Symbol Name of p q r polyhedral operations. By applying these operations, thirteen Archimedean graphs can be generated from Platonic graphs that A(3⋅4)2 12 24 14 are used as seed graphs. A4610⋅⋅ Great 120 180 62 A468⋅⋅ Great 48 72 26 Keyword: Archimedean graph, , Polyhedron A 2 30 60 32 notation, Graph operation. (3⋅ 5) A3454⋅⋅⋅ Small Rhombicosidodecahedron 60 120 62 1. INTRODUCTION A3⋅43 Small Rhombicuboctahedron 24 48 26 A34 ⋅4 24 60 38

Archimedean graph is a simple isomorphic to the A354 ⋅ Snub 60 150 92 skeleton or wire-frame of the . There are A38⋅ 2 24 36 14 thirteen Archimedean solids, which are semi-regular polyhedra A3⋅102 60 90 32 and the subset of uniform polyhedra. Seven of them can be A56⋅ 2 Truncated 60 90 32 formed by of Platonic solids, and all of them can be A 2 Truncated 24 36 14 formed by polyhedral operations defined as Conway polyhedron 4⋅6 A 2 Truncated 12 18 8 notation [1-2]. 36⋅

The author has recently developed an interactive modeling system of uniform polyhedra including Platonic solids, Archimedean solids and Kepler-Poinsot solids, based on graph drawing and simulated elasticity, mainly for educational purpose [3-5]. Obviously, it is possible to draw a polyhedral graph only with two graph operations: addition and addition. However there is an individual difference, it might be A(3⋅ 4)2 A4610⋅⋅ A468⋅⋅ time-consuming to input a graph with steady steps, especially in the case of Archimedean graph.

In this paper, three graph operations corresponding to polyhedral operations are introduced, and it is shown that every Archimedean graph can be generated using these operations and a as the seed graph.

2. ARCHIMEDEAN SOLIDS A(3⋅ 5)2 A3454⋅⋅⋅ A34⋅ 3

Thirteen Archimedean solids are listed in Table 1, and illustrated in Figures 1-2. The symbols in the table stand for the vertex configurations. For example, A(3⋅ 4)2 indicates that two regular and two are gathered alternately on each vertex. Archimedean solids are surrounded by several sorts of congruent regular , and their vertex figures are not regular but congruent polygons. The term was introduced by H. A344 ⋅ A354 ⋅ Coxeter as the segment joining the mid- of the two sides through a vertex [6]. Figure 1. Appearance of thirteen Archimedean solids (1/2). 3. GRAPH OPERATIONS

Thirteen Archimedean graphs isomorphic to the skeletons of Archimedean solids are depicted in Figure 3. A graph GVE= (, ) is defined by the set of vertices

Vvv= {,01" ,p− } and edges Eee= {,01" ,q− }. All graphs considered in this paper are 3-, 4-, or 5-regular polyhedral graphs, which are simple, planar and 3-connected graphs [7]. Detection A 2 A 2 A 2 38⋅ 310⋅ 56⋅ of faces with n sides is equivalent to finding n-cycles. After

selecting faces Fff= {,01" ,r− }, a polyhedral graph is redefined as GVEF = {, , }[3]. The set of faces is subdivided as follows,

FFFFFFF=∪∪∪∪∪3456810,: ∀≠ ijFFij ∩=φ ,

where Fn denotes the set of faces with n sides.

We define three graph operations for GVEF = {, , }: A46⋅ 2 A36⋅ 2 addition, edge contraction, and vertex splitting. Examples are Figure 2. Appearance of thirteen Archimedean solids (2/2). shown in Figures 4-6. A diagonal addition to a in F4 is an edge addition between a pair of non-adjacent vertices in a quadrangular face. An edge contraction is a graph contraction of an edge. A vertex splitting is defined conventionally as the reverse of edge contraction, but in this paper, we define a vertex splitting of v as the composition of the operations of subdivision of incident edges on v , connecting the new vertices in a proper order, and deleting the vertex v . Conventional vertex splitting is equivalent to the present vertex splitting followed by

A(3⋅ 4)2 A4610⋅⋅ A468⋅⋅ several edge contractions.

A(3⋅ 5)2 A3454⋅⋅⋅ A34⋅ 3 Figure 4. An example of diagonal addition operations.

Figure 5. An example of edge contraction operations. A344 ⋅ A354 ⋅ A38⋅ 2

A310⋅ 2 A56⋅ 2 A46⋅ 2 A36⋅ 2 Figure 6. An example of vertex splitting operations by the present Figure 3. Thirteen Archimedean graphs drawn as plane graphs. definition in this paper. 4. OPERATION-BASED NOTATION OF APP(3⋅ 5)23==a 6, 6bσσ35 a 10, 10 b 5 ARCHIMEDEAN GRAPH AA4610⋅⋅ ==σσσσσ(3⋅ 5)23a 6, 6b P35 =a 10,10 b P 5 AA3454⋅⋅⋅==a 6,10b 4610 ⋅⋅ a 10,6bσσa 6,6 b P35 The symbol δ stands for applying diagonal addition to each = a 6,10bσσa 10,10 b P53 quadrangular face of F4 which is adjacent to two triangular AA354 ⋅ ==δδσσδσσ3454⋅⋅⋅ a10, 6b a 6, 6b P35 =a 6, 10b a 10, 10 b P53 faces of F3 . The operator ababmn,(= nm , )stands for applying ab3, 6 AP36⋅ 23= 3 , ab6, 6 AP36⋅ 24= 3 edge contraction to each edge incident on two faces of Fm and ab3, 8 AP38⋅ 23= 4 , ab3,10 AP310⋅ 23= 5 , Fn . The operator σ stands for applying vertex splitting to every a 3, 4 bAP33= , a 4, 4bAAP344==a 3, 4 b vertex. 34⋅ 4 34⋅⋅ 3 4 3 a 5, 6 bAP56⋅ 2 = 35 , a 4, 6 bAP38⋅ 23= 5 , The operator “truncate” in the Conway polyhedron notation [1-2] a 3, 4 bAP3454⋅⋅⋅ = 53 , a 4, 5bAAP3454⋅⋅⋅ ==a 3, 5 b 354 ⋅ 35 corresponds to the graph operation σ . The operator “ambo”, a 3, 4 bAP3444⋅ = 3, ab3, 5 AP354 ⋅ = 35 “expand” and “snub” are expressed by amn, bσ , Figure 7 shows the relations of 5 Platonic graphs and 13 akl,,bσσa mnb , and δσakl,,b a mnb σ, respectively. In the Archimedean graphs. There are five more possibilities of edge case of Archimedean graph, equalities lmn== hold. We contraction operations to Archimedean graphs: ab3, 4 A 2 , use following notations for Platonic graphs and Platonic solids: (3⋅ 4) a 3, 5 bA(3⋅ 5)2 , ab4, 4 A34⋅ 3 , ab3, 3 A43⋅ 4 , and ab3, 3 A53⋅ 4 , cube P 3 , dodecahedron P 3 , icosahedron P 5 , octahedron P 4 , 4 5 3 3 which lead to trivial graph with one isolated vertex. and tetrahedron P33 . As a consequence, every Archimedean graph can be expressed 5. ADDITION OF THE GRAPH OPERATIONS TO AN using one Platonic graph and graph operations as follows. INTERACTIVE MODELING SYSTEM

AP36⋅ 23= σ 3 AP46⋅ 24==σσσ 3a 4, 8b a 8, 8 b P 4 3 An interactive modeling system has been developed by the AP38⋅ 23==σσσ 4a 4, 6b a 6, 6 b P 3 4 author [3-5]. The coordinate of vertices are computed without the APP(3⋅ 4)24==a 6, 6bσσ 3a 8, 8 b 4 3 knowledge of metric information, but only with the structure of AA468⋅⋅ ==σσσσσ(3⋅ 4)24a 6, 6b P 3 =a 8, 8 b P 4 3 the isomorphic polyhedral graph. The system consists of three AA34⋅ 343==a 6, 8b 468⋅⋅ a 8, 6bσσa 6, 6b P 3 =a 6, 8b σσa 8, 8 b P 4 subsystems: graph input subsystem, wire-frame subsystem, and AA3443⋅⋅==δδσσδσσ 34a 8, 6b a 6, 6b P 3 4 =a 6, 8b a 8, 8 b P 4 3polygon subsystem. The overviews of the subsystems are AP56⋅ 23==σσσ35 a 4,10b a 10,10 b P 5 described in the continuing subsections. AP38⋅ 23==σσσ 5 a 4, 6b a 6, 6 b P35

Figure 7. Relations of 5 Platonic graphs and 13 Archimedean graphs induced by diagonal addition, edge contraction and vertex splitting. 5.1 Graph Input Subsystem 5.4 Addition of the graph operations Figure 8(a) shows a screen shot of GUI of graph input subsystem. The graph operations of vertex splitting and edge contraction Drawing a planar graph isomorphic to polyhedron is the first step have been already implemented in the previous version of the of polyhedron modeling. In the subsystem, vertex addition, graph input subsystem, but they are applied to individual vertex vertex deletion, edge addition, and edge deletion are and edge respectively. In the present version, simultaneous vertex implemented as fundamental operations. Vertices can be moved splitting of all the vertices is realized, and simultaneous edge attended with the incident edges. In addition, edge contraction contraction of all the edges between specified faces are also and vertex splitting are introduced. implemented. Such operations require 3-dimensional coordinate of each vertex and of each face, therefore, they are 5.2 Wire-Frame Subsystem available in the wire-frame subsystem and subsystem. In Figure 8(b) shows a screen shot of GUI of wire-frame subsystem. these subsystems, vertex splitting and edge contraction are After constructing a polyhedral graph, the next step is arranging displayed as animation. vertices in 3-dimesional space. We define three binary relations between two vertices: adjacent, neighbor, and . The 6. REFERENCES relation adjacent corresponds to the length of an edge in a 3-dimensional space. The relation neighbor means that the length [1] G. W. Hart, “Conway Notation for Polyhedra”, of path between two vertices is 2, and two vertices are URL: http:/ /www.georgehart.co/virtual-polyhedra/ neighborhood of another vertex. It corresponds to the of conway_notation.html. vertex figure in a 3-dimentional space. The relation diameter [2] E.W.Weisstein,“Conway Polyhedron Notation”, MathWorld. means that the length of path between two vertices is the [3] H. Nonaka, “Detection of Faces in Wire-Frame Polyhedra”, diameter of the graph. It corresponds to the circum- of Proceedings of Int. Conf. on Computer Graphics Theory and polyhedron. Virtual elastic forces are assumed between vertices Applications (GRAPP 2008), 134-137, Madeira, Jan. 2008. according to these three relations and Hooke’s law. Wire-frame [4] H. Nonaka, “Modeling of Kepler-Poinsot Solid Using polyhedron can be formed semi-automatically by controlling the Isomorphic Polyhedral Graph”, International Journal of natural length of virtual springs corresponding to the three types Computer Science, 3, 1, 64-67, 2008. of binary relations. URL: http:/www.waset.org/ijcs/former.html [5] H. Nonaka, “Interactive Modeling of Uniform Polyhedra by 5.3 Polygon Subsystem Means of Graph Drawing”, Int. Conf. on Information Figure 8(c) shows a screen shot of GUI of polygon subsystem. Technology and Applications (ICITA 2008), Cairns, June After arranging vertices in 3-dimensional space, the last step is 2008 (to appear). detecting faces, selecting faces, and rendering the solid [3]. In the [6] H. S. M. Coxeter, “Regular ”, Dover Pub., 1973. case of Platonic solids, Archimedean solids, prisms, and [7] Branko Grünbaum, “Graphs of polyhedra; polyhedra as anti-prisms, common routine is used. The faces of Kepler-Poinsot graphs”, , 307, 445-463, 2007. solids are detected by separate routine.

(a) Graph input subsistem (b) Wire-frame subsystem (c) Polygon subsystem

Figure 8. Screen shot of GUI of the interactive modeling system.