JADUAL: 13 PEPEJAL ARCIMEDES Pepejal Nama Pepejal Simbol Muka, sisi, bucu Pepejal kuboktahedron (3,4,3,4) (F=14), (V=12), (E=24) Ikosidodekahedron (3,5,3,5) (F=32), (V=30), (E=60) Truncated tetrahedron (3,6,6) (F=8), (V=12), (E=18) Truncated octahedron (4,6,6) (F=14), (V=28), (E=44) Nama Pepejal Simbol Muka, sisi, bucu Pepejal Truncated kiub (3,8,8) (F=14), (V=24), (E=36) Truncated ikosahedron (5,6,6) (F=32), (V=60), (E=90) Truncated dedokahedron (3,10,10) (F=32), (V=60), (E=90) Rhombikub oktahedron (3,4,4,4) (F=26), (V=24), (E=48) Nama Pepejal Simbol Muka, sisi, bucu Pepejal Truncated kubotahedron (4,6,8) (F=26), (V=48), (E=72) Truncated ikosidodekahedron (4,6,10) (F=122), (V=120), (E=240) Snub kiub, snub kuboktahedron (3,3,3,3,4) (F=38), (V=24), (E=60) Rhombikosi dodekahedron (3,4,5,4) (F=62), (V=60), (E=120) Nama Pepejal Simbol Muka, sisi, bucu Snub dodekahedron (3,3,3,3,5) (F), (V), (E) JADUAL : 5 PEPEJAL PLATONIK Pepejal Nama pepejal Bilangan Permukaan pada satu bucu Simbol Schlafli (p,q) Bilangan permukaan (F) Bilangan Bucu (V) Bilangan Sisi (E) Dual Tetrahedron Kiub (Hexahedron) Octahedron Dodecahedron Icosahedron 3 3 4 3 5 (3,3) (4,3) (3,4) (5,3) (3,5) 4 6 8 12 20 4 8 6 20 12 4 Tetrahedron 12 Octahedron 12 Kiub (Hexahedron) 30 Icosahedron 20 Dodecahedron Archimedean Solid The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides the same length (Cromwell 1997, pp. 91-92). The Archimedean solids are distinguished by having very high symmetry, thus excluding solids belonging to a dihedral group of symmetries (e.g., the two infinite families of regular prisms and antiprisms), as well as the elongated square gyrobicupola (because that surface's symmetry-breaking twist allows vertices "near the equator" and those "in the polar regions" to be distinguished; Cromwell 1997, p. 92). The Archimedean solids are sometimes also referred to as the semiregular polyhedra. The Archimedean solids are illustrated above. Nets of the Archimedean solids are illustrated above. The following table lists the uniform, Schläfli, Wythoff, and Cundy and Rollett symbols for the Archimedean solids (Wenninger 1989, p. 9). solid 1 2 3 4 5 6 7 cuboctahedron great rhombicosidodecahedron great rhombicuboctahedron t icosidodecahedron small rhombicosidodecahedron small rhombicuboctahedron r snub cube s 3.4.5.4 r 4.6.10 t 4.6.8 uniform polyhedron Schläfli symbol Wythoff symbol Cundy and Rollett symbol 8 9 10 11 12 13 snub dodecahedron s truncated cube truncated dodecahedron truncated icosahedron truncated octahedron truncated tetrahedron t t t t t The following table gives the number of vertices , edges , and faces , together with the number of gonal faces for the Archimedean solids. The sorted numbers of edges are 18, 24, 36, 36, 48, 60, 60, 72, 90, 90, 120, 150, 180 (Sloane's A092536), numbers of faces are 8, 14, 14, 14, 26, 26, 32, 32, 32, 38, 62, 62, 92 (Sloane's A092537), and numbers of vertices are 12, 12, 24, 24, 24, 24, 30, 48, 60, 60, 60, 60, 120 (Sloane's A092538). Solid 1 cuboctahedron 12 24 14 8 6 30 12 12 20 8 6 12 2 great rhombicosidodecahedron 120 180 62 3 4 great rhombicuboctahedron icosidodecahedron 48 30 72 26 60 32 20 5 small rhombicosidodecahedron 60 120 62 20 30 12 6 7 8 9 10 11 12 13 small rhombicuboctahedron snub cube snub dodecahedron truncated cube truncated dodecahedron truncated icosahedron truncated octahedron truncated tetrahedron 24 24 48 26 8 18 60 38 32 6 12 6 12 12 20 6 4 8 4 60 150 92 80 24 60 60 24 12 36 14 8 90 32 20 90 32 36 14 18 8 http://mathworld.wolfram.com/ArchimedeanSolid.html Euler's Formula For any polyhedron that doesn't intersect itself, the Number of Faces plus the Number of Vertices (corner points) minus the Number of Edges always equals 2 This can be written: F + V - E = 2 Try it on the cube: A cube has 6 Faces, 8 Vertices, and 12 Edges, so: 6 + 8 - 12 = 2 To see why this works, imagine taking the cube and adding an edge (say from corner to corner of one face). You will have an extra edge, plus an extra face: 7 + 8 - 13 = 2 Likewise if you included another vertex (say halfway along a line) you would get an extra edge, too. 6 + 9 - 13 = 2. "No matter what you do, you always end up with 2" (But only for this type of Polyhedron ... read on!) Example With Platonic Solids Let's try with the 5 Platonic Solids (Note: Euler's Formula can be used to prove that there are only 5 Platonic Solids): Name Faces Vertices Edges F+V-E Tetrahedron 4 4 6 2 Cube 6 8 12 2 Octahedron 8 6 12 2 Dodecahedron 12 20 30 2 Icosahedron 20 12 30 2 The Sphere All Platonic Solids (and many other solids) are like a Sphere ... you can reshape them so that they become a Sphere (move their corner points, then curve their faces a bit). For this reason we know that F+V-E = 2 for a sphere (BE careful, you can not simply say a sphere has 1 face, and 0 vertices and edges, for F+VE=1) So, the result is 2 again ... ... But Not Always ... Now that you see how this works, I am going to show you how it doesn't work ...! What if I joined up two opposite corners of the icosahedron? It is still an icosahedron (but no longer convex). In fact it looks a bit like a drum where someone has stitched the top and bottom together. Now, there would be the same number of edges and faces ... but one less vertex! So: F+V-E=1 Oh No! It doesn't always add to 2! The reason it didn't work was that this new shape is basically different ... that joined bit in the middle means that two vertices get reduced to 1. Euler Characteristic So, F+V-E can equal 2, or 1, and maybe other values, so the more general formula is F+V-E= Where Here are a few examples: χ χ is called the "Euler Characteristic". Shape χ Sphere 2 Torus 0 Mobius Strip 0 And the Euler Characteristic can also be less than zero. This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so: F + V - E = -2 In fact the Euler Characteristic is a basic idea in Topology (the study of the Nature of Space). http://www.mathsisfun.com/geometry/eulers-formula.html