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# Resistance-network Problems

Fred Curtis - Feb 2000
[Last annotated November 2004]

## 1. Infinite Lattices

I became interested in resistance network problems from the recurring questions posted to the rec.puzzles archive.  In particular, the resistance between two adjacent nodes on an infinite 2D rectangular lattice of 1ohm resistors was given as ½ohm. The solution posted in the rec.puzzles FAQ (section "physics/resistors.p") was roughly

• a source current of 1amp injected at a vertex gives a current of ¼A flowing through each adjacent resistor by symmetry
• a sink of -1A at an adjacent vertex gives ¼A through each adjacent resistor
• superimposing these currents gives a total of ½A flowing through the resistor joining the vertices
• the potential difference between the vertices is ½V; effective resistance of network is ½ohm.

I was so skeptical of this reasoning that I wrote a C++ program, surl.cc, to examine the convergence in ever-larger finite lattices. Not surprisingly, the FAQ gave the correct answer.

I used the same reasoning (see References for more on the superposition principle) on some non-rectangular infinite lattices and found the following results. Simulations in ever-larger subsets of the lattices produced results which converged to the values predicted by the superposition principle:

• The general result for an infinite d-dimensional rectangular lattice is a resistance of (1/d)ohm.
• The result for an infinite lattice formed by tesselating equilateral triangles over the plane is 1/3ohm.
• The result for an infinite lattice formed by tesselating hexagons over the plane is 2/3ohm.

### 1.1 Why I am not comfortable with superposition

I'm still not comfortable with the superpositon principle as an explanation because:
• The superposition principle argues that a current injected at a vertex (with no sink specified) results in current flowing equally through all the edges joining the vertex. No current will flow without a sink, so the principle seems to assume a current sink "at infinity" -- think of the current sink as being an infinitely large circle or, to be more mathematically respectable, the point at infinity on the projective plane.

I don't see why the symmetry in currents flowing through the edges joining the source vertex would stay when the current sink changes from being an infinitely large circle to an adjacent vertex -- in fact it seems counter-intuitive since the sink at the adjacent vertex seems to introduce a clear asymetry.

• The superposition principle doesn't seem to give any information about current flows in any edges other than the single edge between the source and the sink currents. It only seems to apply to extremely regular graphs.

For example, consider an infinite half-plane (take an infinite lattice and throw away everything with a y coordinate less than 0). All the resistors along the bottom edge look equivalent, but we can't deduce anything about the resistance across one of these using the superposition principle. See Further Directions below for hints the result is 2/pi.

Even in the case of an infinite lattice where a current source is placed at a single vertex (and the sink is ignored or assumed to be a circle at infinity), there are no predictions about current flows in any edges except the 4 edges around the vertex.

Because the superposition principle only ever makes predictions for the resistance in one edge, and because the reasoning isn't strong, it seems plausible to me that the principle is just a coincidence. That is, the principle is a just story that agrees with observed results but doesn't explain the results. Contrast this with the works listed in the References below which mathematically prove the results.

There are some experiments I could perform (see Further Directions below) that might convince me to treat the superposition principle as more than a mathematical coincidence.

Note: [2004 November] I came across József Cserti's 1999 paper [see References], which deals with infinite lattice resistance problems. The techniques used in Cserti's paper [and, I'm guessing, many of the papers cited by Cserti] involve far more integral calculus than I can understand.

## 2. Finite Lattices

A general method in analysing finite lattices is to locate equipotential points and join them, yielding a simpler network.

### 2.1 Hypercube of Arbitrary Dimension

#### 2.1.1 Opposite vertices

When I read the rec.puzzles FAQ there was [November 2004: & still is] no general solution to the resistance between opposite corners of an arbitrary hypercube of 1-ohm resistors; solutions were only given for 3D and 4D hypercubes. This problem seemed quite tractable - just combinatorics. The same reasoning about symmetry for the cube case can be applied to the arbitrary d-dimensional case: joining the equipotential points results in d layers containing d(d-1 choose i) resistors in parallel in the i'th layer, giving an overall resistance of (sum from i=0 to d-1 of 1/(d-1 choose i)) / d. The values for the first few d are:

 d 2 3 4 5 6 7 8 9 10 11 resistance 1 5/6 2/3 8/15 13/30 151/420 32/105 83/315 73/315 1433/6930

As d gets larger, the resistance tends toward 2/d [more accurately, 2/d + 2/(d-1)^2] ; the dense layers in the middle become relatively perfect conductors compared to the outermost layers.

The resistance between adjacent vertices of a d-dimensional hypercube is (2^d - 1)/d(2^(d-1)) ohm, which has a nice geometric interpretation of (1-less-than-number-of-vertices)/(number of edges). This value, like the opposite-vertices resistance, converges to 2/d.

### 2.2 Simplex

A simplex is a d-dimensional analogue of the tetrahedron. It has d+1 vertices, each of which is joined to all the others by an edge. If a voltage v is applied across any two vertices of a d-dimensional simplex, the potential at each of the remaining vertices is v/2 by symmetry. Joining equipotential vertices gives a net resistance of (1/(d-1) series 1/(d-1)) parallel 1
= 2/(d-1) parallel 1
= 2/(d+1) ohms.

### 2.3 Cross Polytope

The cross polytope is the d-dimensional analogue of the octahedron. There are 2d vertices, two on each coordinate axis, each vertex joined to every other vertex except its opposite on the same axis.

#### 2.3.1 Opposite vertices

If a voltage v is applied between opposite vertices, the remaining vertices are at a potential of v/2 by symmetry, yielding a resistance of 1/(2d-2) series 1/(2d-2) = 1/(d-1) ohms.

If a voltage v is applied between adjacent vertices, the remaining vertices *not opposite* to the test points are at a potential of v/2 by symmetry yielding a simplified network:

``` A---------B
|\       /|
| \     / |
|  \   /  |
|   \ /   |
|   v/2   |
|  /   \  |
| /     \ |
|/       \|
0---------v
```

where the edge resistances are 1 ohm and the diagonal resistances are each 1/(2d-4). Further symmetry indicates the potential at B is (v-A). The resistance of the network, by tedious algebra, is (d-0.5)/(d(d-1)), which also yields the correct result when d=2.  The potential at vertex A is v(d-1)/(2d-1).

### 2.4 Square Lattices

Thanks to Arvind Giridhar posing the question about diagonal resistance on an NxN (2D) lattice. No exact solution to date.

A similar but more tractable model (test points at (0,N) and (N,0), all points on lines parallel to y=x connected to make them equipotential) has a diagonal resistance of (1 + 1/2 + 1/3 + ... 1/N). This underestimates the simulated diagonal resistance of an NxN lattice by about 10% for smaller N, rising to 13% for N=80.

NDiagonal resistance
on NxN lattice
(by simulation)
1 + 1/2 + ... 1/N
(From altered model with
equipotential diagonals)
111
21.51.5
31.8571.833
103.1332.929
304.4503.995
805.6734.965

## 3. Further Directions

Some things which would be interesting to investigate are:

• Do the current flows in all edges of a 2D retangular lattice agree with the superposition principle?
• How is the result for an infinite 2D rectangular lattice [quoted in the rec.puzzles entry physics/resistors.p] that "equivalent resistance between two nodes k diagonal units apart is (2/pi)(1+1/3+1/5+...+1/(2k-1))" derived?
• Generalised result for separated-by-n-steps vertices on a hypercube?
• Results for other regular polytopes in 4D?
• What would be the result for an edge resistor in an infinite half-plan network? [Thanks to 'Maxwell Smart' for his email with this question, in which he states an empirical guess of 2/pi. A simulation of a 31x15 grid I ran yielded 0.6377 ; a 51x25 simulation yielded 0.6370 ; 2/pi is roughly 0.6366]
• Are there any circumstances in which the superposition principle disagrees with (simulated) experiment? The principle only applies to highly regular infinite graphs: the examples above are all tesselations of euclidean spaces, but the principle could be tested on tesselations of hyperbolic spaces (visualise trying to connect regular polygons around each vertex such that the angle sum is greater than 360 degrees, e.g. 7 triangles, or 5 squares, or 4 pentagons about each vertex).
• When injecting a current source into a single vertex without a sink (i.e. the sink is an infinitely large circle), what are the current flows in edges that don't join the vertex? The convergence could be simulated with ever larger finite grids, and the results compared with the current flows in ever larger simulations where the sink is at an adjacent vertex.

## 4. Chronology

• 2000 January - Started this web page and wrote surl.cc
• 2002 November - Added question about resistance in an infinite half-plane network raised in email from 'Maxwell Smart'.
• 2004 November - Added references to the paper by József Cserti and the Google aptitude test question, prompted by an email from Michael Dagg.
• 2005 January - Listed the reasons for my discomfort with the superposition principle following an email from Chris Drost.
• 2005 August - Added section on square lattices following email from Arvind Giridhar posing the question about diagonal resistance on such a lattice.

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