In his book 'More Mathematical Puzzles and Diversions' Martin Gardner posed the following puzzle "The electrical puzzle involves the network depicted in Figure 4. If each edge of the cube has a resistance of one ohm, what is the resistance of the entire structure when current flows from A to B? " (a copy of the relevant chapter 'The Five Platonic Solids' is available here; http://assets.cambridge.org/97805217/56105/excerpt/9780521756105_excerpt.pdf). In the same book, in a different chapter titled 'Squaring the Square', written by W.T. Tutte is an explanation of how the square was squared using electrical network theory. Tutte mentions "it was a pleasing recreation to work out perfect rectangles corresponding to networks with a high degree of symmetry", this activity and the ensuing analysis of network symmetry led ultimately to the solution of the problem of squaring the square. However we never heard anything further about the squared rectangles of networks with a high degree of symmetry, except for some tantalizing hints from time to time , eg http://www.math.indiana.edu/gallery/color.phtml So I decided to create the squared rectangles of some symmetrical networks. One way of creating a squared rectangle is to select a polyhedral graph, with unit resistances in each edge, choose one of the edges of the graph, replace that edge with an electromotive force and calculate the node voltages and edge currents to give the positions and sizes of squares. The obvious place to start is with regular polyhedra such as the Platonic Solids, as they are highly symmetrical, and there is really only one kind of edge under symmetry, so they (and their duals) will produce only one squared rectangle each, being the same squared rectangle for graph and dual. Following wikipedia; "In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) has two types of edges: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge." Perhaps we can call regular polyhedra which are not isotoxal, 2-isotoxal or 3-isotoxal depending on the number of edge transitive classes that the edges belong to. I have been able to work out the squared rectangles belonging to the Platonic solids, for these, there are only 3 rectangles in total for all 5 polyhedra. They are not perfect (meaning the squares are not all different sizes). In the terminology of squared rectangles, they are often imperfect and compound. Due to symmetry some of the square sizes come out as zero, reducing the number of squares in the rectangles. One edge is the battery edge and does not appear in any tiling, unless one adds it back in to create a trivial squared cylinder. Next I tried the Archimedean solids. I took the graphs from here ; http://mathworld.wolfram.com/ArchimedeanGraph.html The squared rectangles I created are here; http://www.squaring.net/sq/sr/platonic--archimedean-sq-rects.pdf The only one I havnt worked out yet is the Great Rhombicosidodecahedron, my C++ software is failing due to numerical precision errors. I have a Mathematica program which should calculate the square sizes correctly, this version calculates Duijvestijn's squared square, I will need to insert the graph of the Great Rhombicosidodecahedron in place of Duijvestijn's graph; 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. "] a = { {-1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, { 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, -1, 0, 0, 0, 0, -1, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, -1, 0, -1, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1}}; A = Drop[Aa, {1, 1}] ; MatrixForm[A] Text["Multiply A by its transpose, Transpose[A] to get the Laplacian \ (or Kirchhoff) matrix K"] K = A.Transpose[A]; MatrixForm[K] 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]; MatrixForm[V] Text["Dividing rows or columns by GCD , we get reduced voltages - \ this step is not necessary"] 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; MatrixForm[F] 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 non-diagonal rows are squared \ rectangle element 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 easily draw or construct the squared square."] B = MatrixForm[F/R]