Polyhedral VolumesVisual Techniques

T. V. Raman & M. S. Krishnamoorthy

Polyhedral Volumes – p.1/43

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

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

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

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

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.

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 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 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 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 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

Successive Powers Of The Golden Ratio

1 + φ = φ2

φ + φ2 = φ3

... =...

φn−2 + φn−1 = φn

Form a Fibonacci sequence.

Polyhedral Volumes – p.6/43

Successive Powers Of The Golden Ratio

1 + φ = φ2

φ + φ2 = φ3

... =...

φn−2 + φn−1 = φn

Form a Fibonacci sequence.

Polyhedral Volumes – p.6/43

Golden Rhombus

�

�

AB1

B2

�B1

�

C

φ

Polyhedral Volumes – p.7/43

Golden Rhombus

�

�

AB1

B2

�B1

�C

φ

Polyhedral Volumes – p.7/43

Golden Rhombus

�

�

AB1

B2

�B1

�C

φ

Polyhedral Volumes – p.7/43

Scaled Golden Rhombus

�

� AB1

B3

�

B

1� C

φ2

2Y2 = φ√

3

Polyhedral Volumes – p.8/43

Scaled Golden Rhombus

�

� AB1

B3

�

B

1

� C

φ22Y2 = φ

√3

Polyhedral Volumes – p.8/43

Scaled Golden Rhombus

�

� AB1

B3

�

B

1

� C

φ22Y2 = φ

√3

Polyhedral Volumes – p.8/43

Useful Identities

2 cos 36 = φ

Golden ratio and pentagon diagonal.

Polyhedral Volumes – p.9/43

Useful Identities

2 cos 36 = φ

Golden ratio and pentagon diagonal.

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Useful Identities

1 + φ2 = 4R12

Blue-red triangle.

Polyhedral Volumes – p.10/43

Useful Identities

1 + φ2 = 4R12

Blue-red triangle.

Polyhedral Volumes – p.10/43

Useful Identities

cos 2 ∗ 18 = 2 cos2 18 − 1

= 2 sin2 72 − 1

Combining these gives

sin2 72 =1 + φ2

4= R2

1

Polyhedral Volumes – p.11/43

Useful Identities

1 + φ4 = 3φ2

Blue-yellow triangle.

Polyhedral Volumes – p.12/43

Useful Identities

1 + φ4 = 3φ2

Blue-yellow triangle.

Polyhedral Volumes – p.12/43

Useful Identities

sin 36 =

√

1 + φ2

2φ

Polyhedral Volumes – p.13/43

Locating Vertices Of RegularPolyhedra

Polyhedral Volumes – p.14/43

Cube

{(±1

2,±1

2,±1

2)}.

Polyhedral Volumes – p.15/43

Tetrahedron

(1

2, 1

2, 1

2) (−1

2,−1

2, 1

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.

(±φ2

2,±1

2, 0) (±1

2, 0,±φ2

2) (0,±φ2

2,±1

2)

Polyhedral Volumes – p.20/43

Icosahedron

Dual to dodecahedron.

(±φ

2,±1

2, 0)

(0,±φ

2,±1

2)

(1

2, 0,±φ

2)

Polyhedral Volumes – p.21/43

Using The Cube To ComputeVolumes

Polyhedral Volumes – p.22/43

Volume Of The Tetrahedron

Constructing right-angle pyramids on tetrahedralfaces forms a cube.

1

2

1

3=

1

6.

VT = 13 − 4

6

=1

3.

Polyhedral Volumes – p.23/43

Volume Of The Octahedron

Place 4 tetrahedra on 4 octahedral faces to forma 2x tetrahedron.Octahedron is 4 times the tetrahedron.

VO =8

3− 4

1

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.

Reflect these with respect to the cube facesto create a rhombic dodecahedron.

Rhombic dodecahedron is twice the cube.

Polyhedral Volumes – p.25/43

Volume Of The Cube-octahedron

Subtracting 8 right-angle pyramids from a cubegives a cube-octahedron.

VCO = 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 facediagonals.

Diagonal of a pentagon is in the golden ratioto its side.

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 φ3.

Consider the roof structure.

Decomposes into a pyramid and a triangularcross-section.

Polyhedral Volumes – p.30/43

Volume Of Pyramid

Pyramid has rectangular base.

Rectangle of side φ × 1

φ.

Height of pyramid is 1

2

Volume is 1

6.

Polyhedral Volumes – p.31/43

Triangular Cross-Section

Cross-section has length 1.

Triangular face with base φ,

And height 1

2.

Volume is φ

4.

Polyhedral Volumes – p.32/43

Dodecahedron Volume

φ3 + 6(φ

4+

1

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.

Golden cube inside dodecahedron computesits volume.

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 icosahedraledges,

And construct right-triangles in their plane,

With the icosahedral edge as hypotenuse.

These have legs of length 1√2.

Distance between opposite icosahedraledges is φ.

Right-triangles with these lines ashypotenuse complete the square.

Square of side φ+1√2

= φ2

√2

in the XY plane.

Polyhedral Volumes – p.36/43

Square In XY Plane

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��

��

��

��

��

��

��

�Figure 1: Green square around a blue golden

rectangle.

Polyhedral Volumes – p.37/43

Square In XY Plane

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��

� �

��

��

��

��

��

�

� �

��

� �

�

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 φ2

√2.

Volume of octahedron of side√

2 is 4

3.

Scale this result by φ

2.

Volume is φ6

6.

Polyhedral Volumes – p.38/43

Computing The Residue

Icosahedron embedded in this octahedron.

Icosahedral volume found by subtractingresidue from φ6

6.

Residue consists of 12 pyramids, 2 peroctahedral vertex.

Polyhedral Volumes – p.39/43

Pyramid Volume

Observe pyramid with right-triangle base inXY plane.

Triangular base has area 1

4.

Pyramid has height φ

2.

Pyramid Volume is φ

24

Polyhedral Volumes – p.40/43

Icosahedral Volume

φ6

6− φ

2

Polyhedral Volumes – p.41/43

Conclusion

Polyhedral Volumes – p.42/43

Dedication

To my Guiding-eyes, Bubbles.

Polyhedral Volumes – p.43/43