Euler Characteristic
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1 Euler Characteristic
2 Face Classification set_view(gl_render); set_scene(gl_render); glgetdoublev(gl_modelview_matrix, modelview_matrix1); glgetdoublev(gl_projection_matrix, projection_matrix1); glgetintegerv(gl_viewport, viewport1); gluproject((gldouble) poly->tlist[i]->center.entry[0], (GLdouble) poly->tlist[i]- >center.entry[1], (GLdouble)poly->tlist[i]->center.entry[2], modelview_matrix1, projection_matrix1, viewport1, &face_norm_start.entry[0], &face_norm_start.entry[1], &face_norm_start.entry[2]);
3 Topics Today Platonic solids Corner structure
4 Topics Today Platonic solids Corner structure
5 Platonic Solids shiftingsands.com.au/platonicsolids.html
6 Platonic Solids shiftingsands.com.au/platonicsolids.html
7 Platonic Solids davidf.faricy.net/polyhedra/platonic_solids.html
8 Platonic Solids
9 Are We Missing Anything?
10 Are We Missing Anything? All regular polyhedron must be convex.
11 Are We Missing Anything? All regular polyhedron must be convex. When n=3?
12 Are We Missing Anything? All regular polyhedron must be convex. When n=3? m=3: tetrahedron
13 Are We Missing Anything? All regular polyhedron must be convex. When n=3? m=3: tetrahedron m=4: octahedron
14 Are We Missing Anything? All regular polyhedron must be convex. When n=3? m=3: tetrahedron m=4: octahedron m=5: icosahedron
15 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4?
16 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4? m=3, Hexahedron (cube)
17 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4? When n=5?
18 Are We Missing Anything? All regular polyhedron must be convex. When n=3? When n=4? When n=5? m=3: dodecahedron
19 Are We Missing Anything? For example, is it possible to have m=3 and n=3 but f<>4?
20 Euler Characteristics L=V-E+F=2 Why?
21 Elementary Collapse on Edges
22 Elementary Collapse for Edges
23 Elementary Collapse for Edges
24 Elementary Collapse for Edges
25 Elementary Collapse for Edges
26 Elementary Collapse for Edges
27 Elementary Collapse for Edges
28 Elementary Collapse for Edges
29 Elementary Collapse for Edges
30 Elementary Collapse for Edges
31 Elementary Collapse for Edges
32 Elementary Collapse for Edges
33 Elementary Collapse for Edges V=E for closed a simple planar curve.
34 Elementary Collapse for Edges V=E for closed a simple planar curve. What about 3D surfaces?
35 Elementary Collapse for Edges V=E for closed a simple planar curve. What about 3D surfaces? Need to consider merging faces.
36 Elementary Collapse for Faces
37 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C
38 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C
39 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C
40 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C
41 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C
42 Proof of Euler s Theorem on a Cube H G H G E E F D C A B A B C
43 Proof of Euler s Theorem on a Cube H G H G E E F C A B A B C
44 Proof of Euler s Theorem on a Cube G G E E F C A B A B C
45 Proof of Euler s Theorem on a Cube G G F C A B A B C
46 Proof of Euler s Theorem on a Cube F C A B A B
47 Proof of Euler s Theorem on a Cube F B B
48 Proof of Euler s Theorem on a Cube F
49 Proof of Euler s Theorem on a Cube F F
50 Another Look H G H G E E F D C A B A B C
51 Another Look H G H G E F E F D C D C A B A B
52 Dual of a Hexahedron H G E F A D B C
53 Dual of a Hexahedron H G E F A D C
54 Dual of a Hexahedron H G E F A D C
55 Dual of a Hexahedron H G E F A D C
56 Dual of a Hexahedron
57 Dual Shape What is the dual of Octahedron Icosahedron Dodecahedron Tetrahedron Does the dual operation change the Euler characteristic? What operations will change it?
58 Are We Missing Anything? For example, is it possible to have m=3 and n=3 but f<>4?
59 Are We Missing Anything? For example, is it possible to have m=3 and n=3 but f<>4? No, we are not.
60 Proof v-e+f=2 n=number of edges in the polyong m=number of faces (edges) meeting at a vertex
61 Proof v-e+f=2 n=number of edges in the polygon m=number of faces (edges) meeting at a vertex We have 2e=nf
62 Proof v-e+f=2 n=number of edges in the polygon m=number of faces (edges) meeting at a vertex We have 2e=nf mv=nf
63 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2
64 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2 3f/3-3f/2+f=2
65 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2 3f/3-3f/2+f=2 f/2=2
66 Proof v-e+f=2 2e=nf mv=nf When m=3, n=3 what is f? nf/m-nf/2+f=2 3f/3-3f/2+f=2 f/2=2 f=4
67 Any questions?
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