Cubic Tetra

   
MMT meshing always starts from this type of tetra. The tetra is known from the
BCC mesh and from crystal mesh theory. We name it cubic.

You can tile the entire 3-space with cubic tetras.
But more:
You can divide the cubic tetra to 8 smaller ones of the same type.



For all tetras: from divideability follows tileability   but   from tileability does not follow divideability

Which tetras can be devided to 8 smaller ones of the same type?

A simple answer could be to look at the list of tileable tetras.
But today a complete list is not known.

Goldberg (1972) extended Sommervilles' (1923) list of tileable tetras and added 3 new parameter dependant classes.
quality
Sommerville 1 (cubic) sqrt 0.5
Sommerville 2 0.35355
Sommerville 3 0.25
Sommerville 4 0.17678
Sommerville 5 0.35355
Goldberg 1 ...
Goldberg 2 ...
Goldberg 3 ...

Tetra quality is defined as Volume / max edge length **3 and is normalized such that the quality of the regular tetra is 1.0 .


Two views of Sommerville 1-5 Tetras


Sommerville 3 is a half of a pyramid. It can be divided to 8 smaller ones of the same type by "3 fold: split the longest edge". If you divide this tetra as you divide the cubic one, you get 8 tetras of 2 kinds of an average quality of 0.364 which is better than 0.25 .