A Rhombic Dodecahedron inscribed in a Truncated octahedron. This is a study of arrays.
Showing a single block, one of 12 that fill the space between the RD and TO.
An array of RD cells, connected at vertices, with spaces between them. Spaces can be filled in various ways, such as with equilateral triangular cross-section prisms.
Tetraxis Geometry is a video illustration and explanation of how these triangular spaces can be filled.
An extended array of RD cells. A question that can be asked is, what is the proportion of volume of cells to volume of triangular prisms as the array extends to infinity?
A study of RD in TO, to determine the ratio of volume of RD to volume of TO. As it presents on opening, to the left is a cube, showing it’s dissection into 6 pyramids that coincide with dissection of RD into 12 such pyramids. This says that the volume of an RD is the same as 2 cubes. To the right is shown a cube dissected into halves. It can be seen that the half cube that is panelled coincides with dissection of the TO into 8 such blocks. This says that the volume of the TO is the same as 4 cubes. Thus, the single block showing at the lower right, of which there are 12, in total have a volume the same as 2 cubes. This single block is shown within an equilateral triangular prism, with outlines of adjacent blocks. Given that this is a dissection of a TO, and that Truncated Octahedra fill space, This demonstrates that this array has equal volume of RD cells, and spaces between.
