# How to generate a 3-d simple cubic lattice of length 4 in each dimension?

LatticeData["SimpleCubic", "Image"]


The above line gives me a single cube (2 lattice points in each direction).

But I want a lattice with 4 lattice points in each direction. How do I generate that?

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related Q/A: How to make a 3D grid – kglr Feb 6 '13 at 16:39

It is easy to generate coordinates for these using Tuples:

coords = Tuples[Range[4], 3]

Graphics3D@Point[coords]


If you need the grid lines too, you can generate a 2D lattice for each face in a similar way and extrude them into lines.

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Is there a way in which i can connect those dots and replace the dots by spheres. – gforce89 Feb 6 '13 at 16:37
@gforce89 Yes. Just use Sphere[#, radius]& in place of Point. For the lines you need to generate a 2D grid for each face. – Szabolcs Feb 6 '13 at 16:38
what i finally used was this : tpls = Tuples[Range[4], 3]; lines3 = Table[Partition[RotateRight[#, k] & /@ tpls, 4], {k, 0, 2}]; Graphics3D[{{RGBColor[#], Sphere[#, 1/10]} & /@ tpls, Tube /@ lines3}, Boxed -> False] – gforce89 Feb 6 '13 at 16:42

The comments I got for this post are about speed basically. I improved it greatly because I removed opacity objects (which are nice visually but cause slow down). So let me just show the final result of fast and much more complicated (Tetrahedral) lattice than yours (which means your simple cubic will work too but with a bonus). So get your data

cell = LatticeData["TetrahedralPacking", "Image"];


Build extended lattice:

Graphics3D[Translate[DeleteCases[cell, {_, _, Polygon[_]}, Infinity][[1]],
2 Tuples[Range[3], 3]], Boxed -> False]


Now, let me go back in time and explain from the very beginning how it works. A brute force approach. Here is a single lattice cell:

cell = LatticeData["SimpleCubic", "Image"];


Now use Translate:

Graphics3D[Translate[cell[[1]], 2 Tuples[Range[4], 3]], Boxed -> False]


Even so the method is a bit wasteful (you repeating overlapping spheres), it is very simple and works for much more complicated crystal structures where creating custom grid is tedious - plus the style of lattice data is nice:

cell = LatticeData["TetrahedralPacking", "Image"]


Graphics3D[Translate[cell[[1]], 2 Tuples[Range[2], 3]], Boxed -> False]


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This is good but it takes to much time to compute and animation is very slow. The tuples method as suggested by @Szabolcs is very fast. – gforce89 Feb 6 '13 at 16:51
Very nice, +1. It would be even nicer to remove the faces of the cube, if easily possible. – Szabolcs Feb 7 '13 at 0:21
@Szabolcs thanks, your comment convinced me to do some work on this ;) – Vitaliy Kaurov Feb 7 '13 at 1:57
@gforce89 improved. – Vitaliy Kaurov Feb 7 '13 at 1:59
It got interesting when you started putting others kinds of cells in the lattice :) – Szabolcs Feb 7 '13 at 3:07