Polyhedral Volumes Visual Techniques T. V. Raman & M. S. - PowerPoint PPT Presentation

Polyhedral Volumes Visual Techniques T. V. Raman & M. S. Krishnamoorthy Polyhedral Volumes – p.1/43

Outline Identities of the golden ratio. Polyhedral Volumes – p.2/43

Outline Identities of the golden ratio. Locating coordinates of regular polyhedra. Polyhedral Volumes – p.2/43

Outline Identities of the golden ratio. Locating coordinates of regular polyhedra. Using the cube to compute volumes. 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. 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

Basic Facts Dodecahedral/Icosahedral symmetry. Polyhedral Volumes – p.4/43

Basic Facts Dodecahedral/Icosahedral symmetry. The golden ratio and its scaling property. Polyhedral Volumes – p.4/43

Basic Facts Dodecahedral/Icosahedral symmetry. The golden ratio and its scaling property. The scaling rule for areas and volumes. 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. 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 sin 72 φ sin 72 Red Radius of I 1 Yellow Radius of C 1 sin 60 φ sin 60 √ √ Green Face diagonal of C 1 2 2 φ Polyhedral Volumes – p.5/43

Powers Of The Golden Ratio 1 + φ = φ 2 φ + φ 2 = φ 3 . . = . . . . φ n − 2 + φ n − 1 = φ n Polyhedral Volumes – p.6/43

Powers Of The Golden Ratio 1 + φ = φ 2 φ + φ 2 = φ 3 . . = . . . . φ n − 2 + φ n − 1 = φ n Form a Fibonacci sequence. Polyhedral Volumes – p.6/43

� ✁ ✂ ✄ Golden Rhombus B 1 A φ B 2 B C 1 Polyhedral Volumes – p.7/43

� ✁ ✂ ✄ Golden Rhombus B 1 A φ B 2 B C 1 Polyhedral Volumes – p.7/43

� ✁ ✂ ✄ Golden Rhombus B 1 A φ B 2 B C 1 Polyhedral Volumes – p.7/43

� ✁ ✂ ✄ Scaled Golden Rhombus B 1 B A φ 2 B 3 C 1 Polyhedral Volumes – p.8/43

� ✁ ✂ ✄ Scaled Golden Rhombus B 1 B A √ 2 Y 2 = φ 3 φ 2 B 3 C 1 Polyhedral Volumes – p.8/43

� ✁ ✂ ✄ Scaled Golden Rhombus B 1 B A √ 2 Y 2 = φ 3 φ 2 B 3 C 1 Polyhedral Volumes – p.8/43

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

Useful Identities 1 + φ 2 = 4 R 1 2 Polyhedral Volumes – p.10/43

Useful Identities 1 + φ 2 = 4 R 1 2 Blue-red triangle. Polyhedral Volumes – p.10/43

Useful Identities cos 2 ∗ 18 = 2 cos 2 18 − 1 = 2 sin 2 72 − 1 Combining these gives sin 2 72 = 1 + φ 2 4 = R 2 1 Polyhedral Volumes – p.11/43

Useful Identities 1 + φ 4 = 3 φ 2 Polyhedral Volumes – p.12/43

Useful Identities 1 + φ 4 = 3 φ 2 Blue-yellow triangle. Polyhedral Volumes – p.12/43

Useful Identities � 1 + φ 2 sin 36 = 2 φ Polyhedral Volumes – p.13/43

Locating Vertices Of Regular Polyhedra Polyhedral Volumes – p.14/43

Cube { ( ± 1 2 , ± 1 2 , ± 1 2) } . Polyhedral Volumes – p.15/43

Tetrahedron ( 1 2 , 1 2 , 1 ( − 1 2 , − 1 2 , 1 2 ) 2 ) ( 1 2 , − 1 2 , − 1 2 ) ( − 1 2 , 1 2 , − 1 2 ) Self dual. Polyhedral Volumes – p.16/43

Octahedron { ( ± 1 , 0 , 0) , (0 , ± 1 , 0) , (0 , 0 , ± 1) } . Dual To Cube Polyhedral Volumes – p.17/43

Rhombic Dodecahedron ( ± 1 2 , ± 1 2 , ± 1 2) . Vertices of cube and octahedron. { ( ± 1 , 0 , 0) , (0 , ± 1 , 0) , (0 , 0 , ± 1) } . Polyhedral Volumes – p.18/43

Cube-Octahedron { (0 , ± 1 , ± 1) , ( ± 1 , 0 , ± 1) , ( ± 1 , ± 1 , 0) } . Dual to rhombic dodecahedron. Faces of cube and octahedron. Polyhedral Volumes – p.19/43

Dodecahedron Cube vertices ( ± φ 2 , ± φ 2 , ± φ 2) Coordinate planes. ( ± φ 2 2 , 0 , ± φ 2 2 ) (0 , ± φ 2 2 , ± 1 2 , 0) ( ± 1 2 , ± 1 2 ) Polyhedral Volumes – p.20/43

Icosahedron Dual to dodecahedron. ( ± φ 2 , ± 1 2 , 0) (0 , ± φ 2 , ± 1 2 ) 2 , 0 , ± φ ( 1 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 3 = 1 6 . 2 V T = 1 3 − 4 6 = 1 3 . Polyhedral Volumes – p.23/43

Volume Of The Octahedron Place 4 tetrahedra on 4 octahedral faces to form a 2 x tetrahedron. Octahedron is 4 times the tetrahedron. V O = 8 3 − 41 3 = 4 3 . Polyhedral Volumes – p.24/43

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. V CO = 8 − 81 6 = 20 3 . Polyhedral Volumes – p.26/43

Volume Of The Dodecahedron Polyhedral Volumes – p.27/43

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

Summing The Parts Volume of the golden cube is φ 3 . Consider the roof structure. Polyhedral Volumes – p.30/43

Volume Of Pyramid Pyramid has rectangular base. Rectangle of side φ × 1 φ . Volume is 1 6 . Polyhedral Volumes – p.31/43

Triangular Cross-Section Cross-section has length 1 . Triangular face with base φ , Volume is φ 4 . Polyhedral Volumes – p.32/43

Dodecahedron Volume 4 + 1 φ 3 + 6( φ 6) Polyhedral Volumes – p.33/43

Volume Of The Icosahedron Polyhedral Volumes – p.34/43

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

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

� �✁ ✁✂ ✂✄ ✄☎ ☎✆ ✆✝ ✝✞ ✞ Square In XY Plane Figure 1: Green square around a blue golden rectangle. Polyhedral Volumes – p.37/43

✠ ✠ ✌ ☞ ☞ ☛ ☛ ✡ ✡ � ✟ ✍ ✟ ✞ ✝✞ ✆✝ ☎✆ ✄☎ ✂✄ ✁✂ �✁ ✌ Square In XY Plane Figure 1: Green square around a blue golden rectangle. Polyhedral Volumes – p.37/43

Complete The Octahedron Construct similar squares in the Y Z and ZX planes. Constructs an octahedron of side φ 2 2 . √ Volume is φ 6 6 . Polyhedral Volumes – p.38/43

Computing The Residue Icosahedron embedded in this octahedron. Icosahedral volume found by subtracting residue from φ 6 6 . Polyhedral Volumes – p.39/43

Pyramid Volume Observe pyramid with right-triangle base in XY plane. Triangular base has area 1 4 . Pyramid Volume is φ 24 Polyhedral Volumes – p.40/43

Icosahedral Volume φ 6 6 − φ 2 Polyhedral Volumes – p.41/43

Conclusion Polyhedral Volumes – p.42/43