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 √2 φ√2

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 = φ√3

Scaled Golden Rhombus

B1 B ¡ A

2 B3 φ

¢ £ C 1

Polyhedral Volumes – p.8/43 Scaled Golden Rhombus

B1 B ¡ A

2Y2 = φ√3

2 B3 φ £ ¢ C 1

Polyhedral Volumes – p.8/43 Scaled Golden Rhombus

B1 B ¡ A

2Y2 = φ√3

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 ( , , ) . { 2 2 2 }

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) . {    } 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) . {    }

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 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 √2. 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 √2 = √2 in the XY plane. 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 4 Volume of octahedron of side √2 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 √2.

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