Polyhedra

Here is a simple property of some regular objects, including polyhedra. Let us define the "connectance" between two points in a network. It combines like electrical conductance between nodes in networks of identical resistors. Let us define "pseudo-distance" as the reciprocal of connectance. Pseudo-distance combines like electrical resistance between nodes in network of identical resistors.

The diagrams show some simple examples of connectances and pseudo-distances. Basically connectances in parallel add, while those in series are found from the reciprocal of the sum of the reciprocals. Pseudo-distances add simply in series and add reciprocally in parallel.

For any network on a surface, in which all the lines are in equivalent positions, which means that there can be no boundaraies, the connectance and pseudo-distance between neighbouring vertices are given by equations 1 and 2 below. C is the connectance, and P is the pseudo-distance. N is the number of lines, V is the number of vertices, and F is the number of facets. D is the difference V - F. H is the number of holes in the surface, 1 for toroids, 0 for the other examples.

C = N / (V - 1) equation 1.1

P = (V - 1) / N equation 1.2

P = 0.5 + (D - 2H) / 2N equation 1.2A

For a network M and its dual M_D,

P(M) + P(M_D) = 1 equation 1.3

For a self-dual network,

C = 2 and P = 0.5 equation 1.4

These formulae work for networks in which all lines are in equivalent positions -

N-polygon and N-digon (The N-digon needs curved edges to draw it - it is N resistors in parallel.)

Infinite square network - self-dual

Infinite triangular network and infinite hexagonal network

Infinite network of rhombuses and infinite network of triangles and hexagons

Tetrahedron - self-dual Cube and octahedron Dodecahedron and icosahedron

Cuboctahedron and rhombic dodecahedron Icosidodecahedron and rhombic triacontahedron

Toroids with rectangular network with same number of vertices around both axes - self-dual

Toroids with rhombic network with both diagonals parallel to main axes.

Example - if you were to make a dodecahedron from high precision 1 kilohm resistors, the resistance measured across any one of them would be, from equation 1.2, 19/30 X 1 kilohm, and for an icosahedron it would be 11/30 X 1 k ohm. The sum of these dual values is 1 kilohm

For any network in which not all the lines are in equivalent positions, the formulae give the result for the average over all possible neighbouring pairs of vertices.

David Eppstein http://www.ics.uci.edu/~eppstein/ told me about http://www.faqs.org/faqs/puzzles/archive/physics/, which discusses some similar topics, including N-dimensional networks.

Infinite square resistance network

You can easily fit a part of an infinite square resistance network into a spread-sheet. Only one quadrant need be put in. In fact only one octant is strictly necessary. You can take one vertex as a current source or sink, and let the current come in or out at infinity. Or you can put a sink and a source at neighbouring nodes.

If we just take the single source, we can find out the currents in all the links.

Going along a diagonal path from the source point we find the current values in the nearby links corresponding to the resistances given the reference cited above http://www.faqs.org/faqs/puzzles/archive/physics/ -

0, 1/pi, 1/3pi. 1/5pi. etc.

Going along a straight path we find currents of -

0, 1/4 - 0/3pi, 3/4 - 6/3pi. 13/4 - 30/3pi, 63/4 - 148/3pi, etc.

The numbers get bigger by about a factor of five at each step, but the subtractions leave smaller and smaller values. In the limit the resulting potentials tend to inverse proportionality with the distance from the sink. Adding up the currents gives the potentials at the nodes. By combining a sink and a source at different nodes we can work out the equivalent resistance between any two points.

Correspondence to polyhedra@branta.co.uk please

Polyhedron Links

http://www.ics.uci.edu/~eppstein/junkyard/polytope.html

http://www.faqs.org/faqs/puzzles/archive/physics/

http://www.slip.net/~danielgr/geometry/geometry.htm

There are also some formulae about the resistance per square of regular meshes, if the mesh is much smaller than the square.

Let us define -

R is the resistance of each mesh line, and r is the resistance per square. N is the number of resistors per unit area, or per unit cell, and D is V - F, the difference between numbers of vertices and facets, again per unit are or per unit cell. Obviously N and D have to be defined in the same way. L is the length of a resistor, A is the area of one facet, and a is the normalised area of one facet, equal to a / L². M is a mesh, and M_D is its reciprocal or dual mesh.

Then we find -

r / R = a (1 - D / N) independent of orientation equation 2.1

r(M) . r(MD) = R² equation 2.2

This idea can be generalised to the resistivity in ohms-metre for three-dimensional objects generated by space-filling regular networks. Can you find the formula?

Note that for the resistance across a resistor the duals generated a sum, and for resistance per square the duals generated a product. So what about a dual pair of space-filling networks?

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