Post on 27-Sep-2020
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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
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AB1
B2
�B1
�
C
φ
Polyhedral Volumes – p.7/43
Golden Rhombus
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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.
Polyhedral Volumes – p.9/43
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