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Calculate Squared Rectangles of GreatRhombicosidodecahedralGraph?

Posted 7 years ago

Hi, I have been calculating the voltage and current through nodes and edges of the edge skeletons of polyhedra. In particular, the Platonic solids and the Archimedean solids. One edge is treated as a direct current 'battery' or EMF, all edges have unit resistance, and using Kirchhoff's Laws of Voltage and Current, all the node voltages and edge currents can be calculated. The graph, labeled with edge currents and flow directions is called a Smith diagram. Once this is done, a 'Squared Rectangle' ( a dissection of a rectangle into squares) can be constructed with the edge currents as the square sizes. Squared rectangle and its electrical network Depending on the edge which is chosen for the battery, different Squared Rectangles may result. The Platonic solids have edges which are invariant under symmetry, so only produce one unique squared rectangle. Also the cube and octahedron, when considered as graphs are dual to each other, dual graphs produce the same squared rectangle, same for the icosahedron and dodecahedron. The tetrahedron is of course dual to itself. So from the Platonic solids we can derive only 3 unique squared rectangles. Due to the symmetry of the Platonic solids, some edges have zero currents and so have vanishing zero size squares. With the Archimedean solids, there are 1,2 or 3 different types of edge under symmetry depending on the solid. So quite a few more squared rectangles to discover, many with quite intricate patterns. I had written a C++ program some time ago which used node analysis to generate the squared rectangle 'Bouwkampcode' from the Archimedean solids. Here's a PDF which is the collection of squared rectangles I obtained. I got all except 3 belonging to the GreatRhombicosidodecahedron and it's graph. The double precision overflowed when I tried to calculate it. Recently it occurred to me it might be easier to use the precision, inbuilt data and functions of Mathematica to perform these calculations. I started testing out functions I might use. I tried;

 GraphData["GreatRhombicosidodecahedralGraph", "LaplacianMatrix"], but got an error;

Not sure what the error means, the matrix is large but not massive. I am still finding my way around Mathematica so my question is how do I get around this error? and how do I use mapping of functions to do the Johnson solids as well?

POSTED BY: Stuart Anderson

To give a bit more detail on the calculations using Mathematica, this is an example of code which I used to calculate Duijvestijn's 112x112 Perfect Squared Square with 21 squares (the Perfect Squared Square with the fewest squares);
I have modified the code slightly to insert the graph of the GreatRhombicosidodecahedron in place of Duijvestijn's graph;

It appears the original error has disappeared and this code now works;

Text[" Incidence matrix Aa[i,j] -> graph G(i,j); i nodes, j edges
Aa[i,j]={ 1, if edge j is directed towards node i,
Aa[i,j]={-1, if edge j is directed from node i,
Aa[i,j]={ 0, if edge j is not incident to node i. "]
Text[" The Mathematica Archimedean Graphs are undirected and need to turned into directed graphs (any direction will work)]
Ga = DirectedGraph[GraphData["GreatRhombicosidodecahedralGraph"], "Random"]
Aa = IncidenceMatrix[Ga]
Text[" A row and column of the incidence matrix need to be removed"]
A = Drop[Aa, {1, 1}] ;
Text["Multiply A by its transpose, Transpose[A] to get the Laplacian (or Kirchhoff) matrix K"]
K = A.Transpose[A];
Text["Perform LU decomposition on K, to factor it into an upper (u) and a lower (l) triangular matrix and diagonal matrix"]
{lu, p, c} = LUDecomposition[K];
det = Tr[lu, Times]
Text["The V matrix is equal to determinant* Inverse matrix of K and gives the (full)voltages for nodes satisfying Kirchhoff's 1st law"]
V = det*Inverse[K];
Text["Dividing rows or columns by GCD , we get reduced voltages - this step is useful for calculating the Bouwkampcode "]
W = Apply[GCD, V, {1}]
Y = MatrixForm[V/W]
Text["The triple matrix product A.V.A gives the edge full current solutions matrix F"]
F = Transpose[A].V.A;
Text["Dividing rows or columns of F by the GCD of each row we get a reduction vector, needed to reduce the full currents so we get tilings in smallest solutions"]
R = Apply[GCD, F, {1}]
Text["Dividing the full currents matrix F by reduction vector R gives the reduced currents matrix B, which provides all the solutions in terms of the sizes of the edge currents. Edge currents become square sizes in the squared square. The last row of the B matrix is Duijvestijn's 112 squared square, with all the square sizes in reduced (GCD=1) size. The other rows are other squared rectangles, and the non-diagonal entries are element (square) sizes, and the width of the rectangle is the diagonal entry in each row, once we know the width we can work out the height from the determinant. In the case of a squared square w = h = diagonal entry."]
Text["Now we substitute the values of the edge currents from matrix B, one row at a time, into the edges of the graph we started with, this gives us the 'Smith Diagram' of the squared square, and we can now draw or construct the squared square. A separate program to produce Bouwkampcode from the reduced currents and voltages is how this is usually done."]
B = MatrixForm[F/R]

POSTED BY: Stuart Anderson
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