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Polyhedral Volumes Visual Techniques Polyhedral Volumes Visual Techniques T. V. Raman & M. S. Krishnamoorthy Polyhedral Volumes – p.1/43 Locating coordinates of regular polyhedra. Using the cube to compute volumes. Volume of the dodecahedron. Volume of the icosahedron. Outline Identities of the golden ratio. Polyhedral Volumes – p.2/43 Using the cube to compute volumes. Volume of the dodecahedron. Volume of the icosahedron. Outline Identities of the golden ratio. Locating coordinates of regular polyhedra. Polyhedral Volumes – p.2/43 Volume of the dodecahedron. Volume of the icosahedron. Outline Identities of the golden ratio. Locating coordinates of regular polyhedra. Using the cube to compute volumes. Polyhedral Volumes – p.2/43 Volume of the icosahedron. Outline Identities of the golden ratio. Locating coordinates of regular polyhedra. Using the cube to compute volumes. Volume of the dodecahedron. Polyhedral Volumes – p.2/43 Outline Identities of the golden ratio. Locating coordinates of regular polyhedra. Using the cube to compute volumes. Volume of the dodecahedron. Volume of the icosahedron. Polyhedral Volumes – p.2/43 The Golden Ratio Polyhedral Volumes – p.3/43 The golden ratio and its scaling property. The scaling rule for areas and volumes. The Pythogorian theorem. Formula for pyramid volume. Basic Facts Dodecahedral/Icosahedral symmetry. Polyhedral Volumes – p.4/43 The scaling rule for areas and volumes. The Pythogorian theorem. Formula for pyramid volume. Basic Facts Dodecahedral/Icosahedral symmetry. The golden ratio and its scaling property. Polyhedral Volumes – p.4/43 The Pythogorian theorem. Formula for pyramid volume. Basic Facts Dodecahedral/Icosahedral symmetry. The golden ratio and its scaling property. The scaling rule for areas and volumes. Polyhedral Volumes – p.4/43 Formula for pyramid volume. Basic Facts Dodecahedral/Icosahedral symmetry. The golden ratio and its scaling property. The scaling rule for areas and volumes. The Pythogorian theorem. Polyhedral Volumes – p.4/43 Basic Facts Dodecahedral/Icosahedral symmetry. The golden ratio and its scaling property. The scaling rule for areas and volumes. The Pythogorian theorem. Formula for pyramid volume. Polyhedral Volumes – p.4/43 Basic Units Color Significance 1 2 Blue Unity 1 φ Red Radius of I1 sin 72 φ sin 72 Yellow Radius of C1 sin 60 φ sin 60 Green Face diagonal of C1 p2 φp2 Polyhedral Volumes – p.5/43 Form a Fibonacci sequence. Successive Powers Of The Golden Ratio 1 + φ = φ2 φ + φ2 = φ3 . = . n 2 n 1 n φ − + φ − = φ Polyhedral Volumes – p.6/43 Successive Powers Of The Golden Ratio 1 + φ = φ2 φ + φ2 = φ3 . = . n 2 n 1 n φ − + φ − = φ Form a Fibonacci sequence. Polyhedral Volumes – p.6/43 Golden Rhombus B1 A ¡ B2 φ B ¢ £ C 1 Polyhedral Volumes – p.7/43 Golden Rhombus B1 A ¡ B2 φ £ B ¢ C 1 Polyhedral Volumes – p.7/43 Golden Rhombus B1 A ¡ B2 φ £ B ¢ C 1 Polyhedral Volumes – p.7/43 2Y2 = φp3 Scaled Golden Rhombus B1 B ¡ A 2 B3 φ ¢ £ C 1 Polyhedral Volumes – p.8/43 Scaled Golden Rhombus B1 B ¡ A 2Y2 = φp3 2 B3 φ £ ¢ C 1 Polyhedral Volumes – p.8/43 Scaled Golden Rhombus B1 B ¡ A 2Y2 = φp3 2 B3 φ £ ¢ C 1 Polyhedral Volumes – p.8/43 Golden ratio and pentagon diagonal. Useful Identities 2 cos 36 = φ Polyhedral Volumes – p.9/43 Useful Identities 2 cos 36 = φ Golden ratio and pentagon diagonal. Polyhedral Volumes – p.9/43 Blue-red triangle. Useful Identities 2 2 1 + φ = 4R1 Polyhedral Volumes – p.10/43 Useful Identities 2 2 1 + φ = 4R1 Blue-red triangle. Polyhedral Volumes – p.10/43 Useful Identities cos 2 18 = 2 cos2 18 1 ∗ − = 2 sin2 72 1 − Combining these gives 1 + φ2 sin2 72 = 4 2 = R1 Polyhedral Volumes – p.11/43 Blue-yellow triangle. Useful Identities 1 + φ4 = 3φ2 Polyhedral Volumes – p.12/43 Useful Identities 1 + φ4 = 3φ2 Blue-yellow triangle. Polyhedral Volumes – p.12/43 Useful Identities p1 + φ2 sin 36 = 2φ Polyhedral Volumes – p.13/43 Locating Vertices Of Regular Polyhedra Polyhedral Volumes – p.14/43 Cube 1 1 1 ( ; ; ) : f 2 2 2 g Polyhedral Volumes – p.15/43 Tetrahedron (1; 1; 1) ( 1; 1; 1) 2 2 2 −2 −2 2 (1; 1; 1) ( 1; 1; 1) 2 −2 −2 −2 2 −2 Self dual. Polyhedral Volumes – p.16/43 Octahedron ( 1; 0; 0); (0; 1; 0); (0; 0; 1) : f g Dual To Cube Polyhedral Volumes – p.17/43 Rhombic Dodecahedron 1 1 1 ( ; ; ): 2 2 2 Vertices of cube and octahedron. ( 1; 0; 0); (0; 1; 0); (0; 0; 1) : f g Polyhedral Volumes – p.18/43 Cube-Octahedron (0; 1; 1); ( 1; 0; 1); ( 1; 1; 0) : f g Dual to rhombic dodecahedron. Faces of cube and octahedron. Polyhedral Volumes – p.19/43 Dodecahedron Cube vertices φ φ φ ( ; ; ) 2 2 2 Coordinate planes. 2 2 2 ( φ ; 1; 0) ( 1; 0; φ ) (0; φ ; 1) 2 2 2 2 2 2 Polyhedral Volumes – p.20/43 Icosahedron Dual to dodecahedron. ( φ; 1; 0) 2 2 (0; φ; 1) 2 2 (1; 0; φ) 2 2 Polyhedral Volumes – p.21/43 Using The Cube To Compute Volumes Polyhedral Volumes – p.22/43 Volume Of The Tetrahedron Constructing right-angle pyramids on tetrahedral faces forms a cube. 1 1 1 = : 2 3 6 3 4 VT = 1 − 6 1 = : 3 Polyhedral Volumes – p.23/43 Volume Of The Octahedron Place 4 tetrahedra on 4 octahedral faces to form a 2x tetrahedron. Octahedron is 4 times the tetrahedron. 8 1 VO = 4 3 − 3 4 = : 3 Polyhedral Volumes – p.24/43 Reflect these with respect to the cube faces to create a rhombic dodecahedron. Rhombic dodecahedron is twice the cube. Volume Of The Rhombic Dodecahedron Connect the center of the cube to its vertices. This forms 6 pyramids inside the cube. Polyhedral Volumes – p.25/43 Volume Of The Cube-octahedron Subtracting 8 right-angle pyramids from a cube gives a cube-octahedron. 1 V = 8 8 CO − 6 20 = : 3 Polyhedral Volumes – p.26/43 Volume Of The Dodecahedron Polyhedral Volumes – p.27/43 Diagonal of a pentagon is in the golden ratio to its side. Cube And The Dodecahedron Dodecahedron contains a golden cube. 8 of the 20 vertices determine a cube. Cube edges are dodecahedron face diagonals. Polyhedral Volumes – p.28/43 Constructing Dodecahedron From A Cube Consider again the golden cube. Construct roof structures on each cube face. Unit dodecahedron around a golden cube. Polyhedral Volumes – p.29/43 Decomposes into a pyramid and a triangular cross-section. Summing The Parts Volume of the golden cube is φ3. Consider the roof structure. Polyhedral Volumes – p.30/43 1 Height of pyramid is 2 Volume Of Pyramid Pyramid has rectangular base. Rectangle of side φ 1 . × φ 1 Volume is 6. Polyhedral Volumes – p.31/43 1 And height 2. Triangular Cross-Section Cross-section has length 1. Triangular face with base φ, φ Volume is 4 . Polyhedral Volumes – p.32/43 Dodecahedron Volume φ 1 φ3 + 6( + ) 4 6 Polyhedral Volumes – p.33/43 Volume Of The Icosahedron Polyhedral Volumes – p.34/43 Golden cube inside dodecahedron computes its volume. Volume Of The Icosahedron Icosahedron is dual to dodecahedron. Octahedron is dual to the cube. Octahedron outside icosahedron gives volume. Polyhedral Volumes – p.35/43 With the icosahedral edge as hypotenuse. 1 These have legs of length p2. Distance between opposite icosahedral edges is φ. Right-triangles with these lines as hypotenuse complete the square. Constructing The Octahedron Squares in XY , Y Z, and ZX planes. Consider a pair of opposite icosahedral edges, And construct right-triangles in their plane, Polyhedral Volumes – p.36/43 φ+1 φ2 Square of side p2 = p2 in the XY plane. Square In XY Plane ¡¢ ¢ ¡ £ § £¤ ¤ § ¥ ¦ ¥ ¦ Figure 1: Green square around a blue golden rectangle. Polyhedral Volumes – p.37/43 .37/43 p – olumes V al olyhedr P golden lue b a £ ¤ around £¨ ¥© ¢ ¤ ¥ ¡¢ ¦ ¡¨ ¦§© Plane square § Y X Green In . 1: e Figure rectangle Squar 4 Volume of octahedron of side p2 is 3. φ Scale this result by 2 . Complete The Octahedron Construct similar squares in the Y Z and ZX planes. φ2 Constructs an octahedron of side p2. φ6 Volume is 6 . Polyhedral Volumes – p.38/43 Residue consists of 12 pyramids, 2 per octahedral vertex. Computing The Residue Icosahedron embedded in this octahedron. Icosahedral volume found by subtracting φ6 residue from 6 . Polyhedral Volumes – p.39/43 φ Pyramid has height 2 . Pyramid Volume Observe pyramid with right-triangle base in XY plane. 1 Triangular base has area 4. φ Pyramid Volume is 24 Polyhedral Volumes – p.40/43 Icosahedral Volume φ6 φ 6 − 2 Polyhedral Volumes – p.41/43 Conclusion Polyhedral Volumes – p.42/43 Dedication To my Guiding-eyes, Bubbles. Polyhedral Volumes – p.43/43.
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