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Platonic Solids

I'm spending a little time doing some maths with my eldest daughter (nearly five). One of the tasks we're looking at currently is constructing Platonic solids from their nets. The net of the octahedron is shown below. It was a great little exercise constructing the solid, but trying to effectively glue the flaps was tricky. PVA glue was probably a better plan than a glue stick. An interesting thing we discovered is that its possible, using the same net, to produce the concave rather than the convex octahedron

Construction of an octahedron

The octahedron is composed from two square based pyramids. The square based pyramids in turn are made from cutting away sections from a solid cube using planes. The trick aspect is calculating the correct equations for the planes!

The ground and sky are added to help distinguish the various faces of the octahedron. Below is the code for the construction of the pyramid

Resistor matrices (square matrix)
The three matrices of resistors have inspired me to consider the $n\times n$ case. So far, I have reduced the problem to quite a neat set of recursive relations. $$V = 2(I_1 + I_2 + \dots + I_n),\; I = 2I_1$$ We then have $I_k = \beta_kI_{k-1}$, with $\beta_n =\frac{1}{2}$ and $\beta_{k-1} = \frac{2}{4 - \beta_k}$ These together give $$R = 1 + \beta_2 + (\beta_2\beta_3) + \dots +(\beta_2\beta_3\beta_4\dots\beta_n)$$

So for example, in the $2\times 2$ case, we get $\beta_2 = \frac{1}{2}$, and $\beta_i =0\; \forall\;i\gt 2$.

Cuboctahedron construction using Povray

The cuboctahedron is an Archimedean solid with 6 square faces and 8 triangular ones. The associated question (in the puzzles section) was inspired by a construction using magna-tiles.

The construction of the cuboctahedron was done using Povray. The key object construction from the scene file is given below; essentially we have taken a cube and sliced the corners off using planes!

Tubes!

The Mathematica code below inflates a curve to produce a tube!

(* Refactoring of Curve-to-tube recipe *)
CoreObj[t_]  = {Sin[4t], Cos[3t], Sin[5t]};
(* This defines the curve to be tubed *)
FirstD[t_] = D[CoreObj[t],t];
(* This defines the tangent vector at t *)
FirstNV[t_] = {FirstD[t][[2]] - FirstD[t][[3]], FirstD[t][[3]] - FirstD[t][[1]],FirstD[t][[1]] - FirstD[t][[2]]};
(* this defines a normal vector to the curve*)
SecondNV[t_] = FirstD[t] \[Cross] FirstNV[t];
(* this defines a mutually orthogonal vector to both the tangent and normal vector *)
NNV1[t_] = Normalize[FirstNV[t]];
NNV2[t_] = Normalize[SecondNV[t]];
(* creates unit vectors *)
(* defines radius of tube *)
TubeObj[u_,v_] = CoreObj[u] + radius *(Sin[v]NNV1[u] + Cos[v]NNV2[u]);

ParametricPlot3D[TubeObj[u,v],{u,0,2 \[Pi]},{v,0,2 \[Pi]},Mesh->100,MeshFunctions->Automatic]



A few zoomed in variants of the spectacular mandelbrot set - generated using ultra Fractal 6. (links to the 5k versions at the bottom of the page!)

Cantor set - from a Julia set

Cantor sets are sets of points that have some very strange properties. Each point is a limit point - that is, every point is discrete, no point has a connected patch around it.

Every Cantor set is the set of limit points of a set of affine contractions on some initial set - though quite what these sets would be is pretty difficult to determine!

Ultra Fractal

Largely because of discussions with friends at Puritanical Creative, I rediscovered the joy of playing with ultra fractal.

Ultra fractal is as its name suggests an excellent program for quickly generating fractals. Basic things like Julia sets can be generated in seconds and much more complicated structures easily explored too.