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[GIF] Hamiltonian (Circuit of octahedra)

GROUPS:

Circuit of octahedra

Hamiltonian

Just as in Exchange, in which small cubes trace out a Hamiltonian cycle on the skeleton of a big cube, this shows 6 octahedra centered at the 6 vertices of a large octahedron which trace out a Hamiltonian cycle on the skeleton of the big octrahedron.

There are two different types of Hamiltonian cycle on the octahedron, and just by chance the first one FindHamiltonianCycle finds is of the type I like better (originally I wanted to do an icosahedron -- which has lots of distinct Hamiltonian cycles -- in this style, but the image was way too cluttered).

Anyway, here's the code:

smootherstep[t_] := 6 t^5 - 15 t^4 + 10 t^3;
smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4;

DynamicModule[{pol = "Octahedron", clist, v, e, graph, cycle, s,
  cols = Append[ColorData["BrightBands"] /@ Range[0, 1, 1/6], 
    RGBColor["#2A363B"]]},
 v = PolyhedronData[pol, "VertexCoordinates"];
 e = PolyhedronData[pol, "Edges"][[2]];
 graph = PolyhedronData[pol, "SkeletonGraph"];
 cycle = VertexList[PathGraph[First[FindHamiltonianCycle[graph]]]];
 Manipulate[
  s = smootheststep[Mod[t, 1]];
  Graphics3D[{Thickness[.007], 
    Table[{cols[[Mod[i - Floor[t], 6, 1]]],
      GraphicsComplex[
       2 ((1 - s) v[[cycle[[i]]]] + 
             s*v[[ RotateLeft[cycle][[i]]]]) + # & /@ 
        RotationTransform[(-1)^i π/2 s, {0, 0, 1}][2 v/3], e]}, 
       {i, 1, Length[v]}]},
   Boxed -> False, PlotRange -> 2, ImageSize -> 540, 
   ViewPoint -> 10 {.8, -2.8, .5}, ViewVertical -> {0, 0, 1}, 
   ViewAngle -> π/70, SphericalRegion -> True, 
   Background -> cols[[-1]]],
  {t, 0., 6}]
 ]
POSTED BY: Clayton Shonkwiler
Answer
1 year ago

enter image description here - another post of yours has been selected for the Staff Picks group, congratulations! We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

POSTED BY: Moderation Team
Answer
1 year ago

Group Abstract Group Abstract