Exercises of Chapter 1 (Part 3/6), Computational Geometry

Exercise 1.23. Is it possible to partition a cube into six congruent tetrahedra? Defend your answer.

Answer. Yes. The tetrahedralisation can be accomplished by performing the following procedures
1. Create a interior diagonal between two opposite vertices of the cube (any of the two most far apart)
2. Starting from each of the ends of the diagonal above created, make a diagonal on each of the three rectangular surfaces adjacent to the vertex.
3. All the above diagonals unite to characterise such a tetrahedralisation. □


Exercise 1.24. Find the six different tetrahedralisations of the cube up to rotation and reflection.


Answer. This is way harder.
It's worth classifying the tetrahedralisation first by the general way of dividing the cube, especially the first cut. We need to make sure that these classes don't overlap and also note that the shapes cut out will be triangulated individually but care needs to be taken when scoring out repeated count up to rotation and reflection.

One big class of tetrahedralisation of cube is made by dividing the cube by the surface diagonal into two congruent cheese-like prisms. The discussion of this class is ignored, it's not too hard to go through all the cases involved in it, and there should be 4 different ones up to rotation and reflection.
Naturally the other class is with tetrahedralisations that don't present separate double prisms. This class involves a special tetrahedralisation is the only one (up to rotation and reflection) that cuts the cube into 5 pieces, which consists a big tetrahedron  formed by four vertices of the cube any two of those sitting on two ends of a surface diagonal, along with the remaining pieces. The other triangulation in this class is the one that first takes away two tetrahedrons each of which is with equilateral triangle base and a top just above the base sending down edges perpendicular to each other to the base and having and has the base parallel to that of the other, and then adds an internal diagonal that connects the vertices, one from each of the two bases. □

Exercise 1.25. Classify the set of triangulations on the boundary of the cube that 'induce' the tetrahedralisations of the cube, where each such tetrahedralisation matches the triangulation on the cube surface.


Answer. As all possible triangulations were generally discussed in the above exercise, the detailed answer to this question is omitted. □

Exercise 1.25. Show that the n-dimensional cube can be triangulated into exactly n! simplices.


Proof. (far from complete) Simplice that consists of as many edges as possible of a cube with edges 1 unit in length is with an n-dimensional volume of 1 / n!. As the volume of the cube is 1, there can be as many as  n! simplices. □