Cuboctahedron as Vector Equilibrium

Wednesday, July 16, 2008

I found an interesting article on the cuboctahedron as vector equilibrium:

Amy C. Edmondson
A Fuller Explanation
Chapter 7, Vector Equilibrium
pages 90 through 93

Cuboctahedron as Vector Equilibrium

We can understand the symmetry of the plane by observing that although any polygon can be made to have equal edge lengths, only the regular hexagon can have edges equal in length to the distance between the polygon's center and its vertices. In the same way, although there are many regular and seniiregular polyhedra with equal edge lengths, there is only one spatial configuration in which the length of each polyhedral edge is equal to that of the radial distance from its center of gravity to any vertex: the cuboctahedron (Fig. 7-5). (3) This shape therefore is the only one that allows the requisite arrangement of vectors to demonstrate equilibrium.

Vector equilibrium

Fig. 7-5. Vector equilibrium.
Click on thumbnail for larger image.


Cuboctahedron, twist cuboctahedron
Fig. 7-6. (a) Cuboctahedron; (b) twist cuboctahedron.
Click on thumbnail for larger image.


We first saw the cuboctahedron as the degenerate truncation of both the cube and the octahedron, but at that point in our investigation we were only looking at surface topology. Now diving into the interior shape, we discover this unique property of equivalence. Table IV compares the radial lengths of various familiar polyhedra given unit edge lengths. Only in the cuboctahedron—hereafter referred to by Fuller's term, vector equilibrium or VE—can the radius be of unit length.



Table IV
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Unit-edge polyhedronRadiusCentral AngleAxial Angle
Tetrahedron
0.6124
109.47°
35.26°
Octahedron
0.7071
90.00°
45.00°
Icosahedron
0.9511
63.43°
58.28°
Cube
0.8660
70.53°
54.76°
Pentadodecahedron
1.4012
41.81°
69.04°
VE
1.0000
60.00°
60.00°
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Posted by Jonah Dempcy at 12:30 AM  

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