Octahedra+tetrahedra filling tessellations in 3D

1

2

This is called the quasiregular space-filling tessellations in 3D:

formed by

and

Are there any smart ways to do a Mathematica figure drawing this?

enter image description here

wonderich

Posted 2020-02-18T03:47:30.320

Reputation: 603

1You can draw elements of your composition using Octahedron[{x,y,z}] and Tetrahedron[{x1, y1, z1}] where coordinates are centers of figures. So, you need calculate them as well as the rotation angles of figures with Rotate – Rom38 – 2020-02-18T05:15:28.357

Answers

3

It was a bit of a hack to get the rotations almost right (If I was you I would check the rotations properly), but something like this could work:

nx = 1;
ny = 1;
nz = 1;
xvec = Sqrt[2]/2*{1, 1, 0};
yvec = Sqrt[2]/2*{1, 0, 1};
zvec = Sqrt[2]/2*{0, 1, 1};
coords = Flatten[
Table[xvec*i + j*yvec + k*zvec, {i, 0, nx}, {j, 0, ny}, {k, 0, nz}], 2];
octas = Graphics3D[Map[{Red, EdgeForm[Thick], Octahedron[#]} &, coords]];
tetra = Translate[{Yellow, EdgeForm[Thick], Rotate[Rotate[Tetrahedron[], 
  1*\[Pi]/4, {0, 0, 1}], -4 \[Pi]/13, {-1, 1, 0}]}, Sqrt[2]/4*{1, 1, 1}];
tetras = Graphics3D[Map[Translate[tetra, #] &, coords]];
Show[{tetras, octas}]

Tiling

Dunlop

Posted 2020-02-18T03:47:30.320

Reputation: 3 292