FIVE TETRAHEDRA COMPOUND

    Tetrahedra and the dodecahedron

We have seen how it is possible to inscribe a cube inside a dodecahedron, and also how it is possible to inscribe two tetrahedra inside a cube. If we choose just one of the inscribed tetrahedra from each cube in a consistent way, we get this very pretty compound of five tetrahedra. One of the attractions of this compound is that it has a definite orientation. If we look at one of the inner ‘dimples’ we see a clockwise(?) swirl of faces. The compound made from the other five tetrahedra has the swirl in the opposite direction. Each is the mirror image of the other, and the pair are said to be enantiomorphic. They are pictured below.

Five tetrahedra have 20 vertices; a dodecahedron has 20 vertices. Hence using the symmetry of the compound, each vertex of the dodecahedron occurs as the vertex of just one tetrahedron.   View this structure.

We can break the model down into its five component tetrahedra.

Red

Blue

Green

Yellow

Magenta

Another way of viewing the compound is to remove the cubes one by one:     

Five tetrahedra – RBGYM

Four tetrahedra – BGMY

Three tetrahedra – BGM

Two tetrahedra – BM

One tetrahedron – B

A pair of enantiomorphic solids

  Intersection and structure

Let us look at the red face, and the lines on this face formed by the intersecting faces. We get a figure which looks like this. The lines on this face determine an equilateral triangle. Since there are 20 triangular faces, there are 20 such equilateral triangles. These triangles form the faces of an icosahedron, which is the solid of intersection of the five tetrahedra.

We now show that each of the dimples of this compound occurs as the common intersection point of five diagonals of the defining dodecahedron. We shall need to use coordinates to do this, but the results are very pretty.

Using our knowledge of the defining dodecahedron, we can assign coordinates as follows:

A (–1, –1, 1), B (1, –1, 1), C (1, –1, –1), D (1, 1, –1), E ((1, 1, 1), F (–1, 1, 1), G(, –^–1, 0), H (, ^–1, 0), J (^–1, 0, ).

We seek the coordinates of the dimple X.

• (a) Find the equation of the light green plane. Show that it passes through red vertex G.
(In fact, it passes through two other vertices of the dodecahedron. Can you explain this?

(b) In the same way, the red plane passes through a vertex of the green tetrahedron. Deduce that dimple X lies on a diagonal of the dodecahedron, and in fact on five diagonals.

(c) Find the coordinates of the dimple X by using the fact that it is the intersection of diagonals AH and FG.
Show that the dimple coordinates are (^–1, 0, ^–2).

(d) In fact, the 12 dimple coordinate sets are (^–1, 0, ^–2), (^–2, ^–1, 0), (0, ^–2, ^–1).
Given that a representative set of vertex coordinates for an icosahedron is (, 0, 1,), (1, , 0), ( 0, 1, ), deduce that the set of dimples in our model do in fact determine an icosahedron.

Check your answers ...

Restore the diagram ...

  Making the model

This is a very pretty model to make, and well worth the effort involved. For instructions on preparing and assembling the pieces, check out this link.

  Playing with the applet

It is profitable to spend some time playing with the applet, and enjoying this beautiful compound polyhedron. Use the applet to check out the following statements:

• There is a clockwise swirl of faces within each pentagonal face.
• The three closest vertices to any given vertex form an equilateral triangle.
• The dimples are vertices of a net of equilateral triangles (which form an icosahedron).
• Placing the dodecahedron so that its boundary is a (non-regular) hexagon gives two triangular faces lying with parallel edges (at least in projection).
• If the dodecahedron is placed with parallel faces top and bottom, then the middle zigzag path touches ten vertices, two of each different colour.

  Vertex coordinates

We have also seen that the five inscribed cubes can be taken to have vertices (1, 1, 1),   

The vertices of the five tetrahedra will then be chosen from these, but in which order?

• (a) Two tetrahedra can be inscribed in each cube. List the possible vertex sets for these.

(b) Choose one vertex set from the first cube. Now which vertex sets will you take from the other cubes to form a nice symmetric five tetrahedra compound?

Now check your answers ...

(1, 1, 1), (1, –1, –1), (–1, 1, –1), (–1, –1, 1)  and  (–1, –1, –1), (–1, 1, 1), (1, –1, 1), (1, 1, –1);      

(1, 1, 1), (0, ^–1, –), (–, 0, ^–1 ), ( ^–1, –, 0)  and  (–1, –1, –1), (0, – ^–1, ), (, 0, – ^–1 ), (– ^–1, , 0);

(–1, 1, 1), (0, ^–1, –), (, 0, ^–1 ), (– ^–1, –, 0),  and  (1, –1, –1), (0, – ^–1, ), (–, 0, – ^–1 ), ( ^–1, , 0);     

(1, –1, 1), (0, – ^–1, –), (–, 0, ^–1 ), ( ^–1, , 0) and   (–1, 1, –1), (0, ^–1, ), (, 0, – ^–1 ), (– ^–1, –, 0);  

(1, 1, –1), (0, ^–1, ), (–, 0, – ^–1 ), ( ^–1, –, 0)  and  (–1, –1, 1), (0, – ^–1, –), (, 0, ^–1 ), (– ^–1, , 0).    

(1, 1, 1), (1, –1, –1), (–1, 1, –1), (–1, –1, 1);  
(–1, –1, –1), (0, – ^–1, ), (, 0, – ^–1 ), (– ^–1, , 0);
(–1, 1, 1), (0, ^–1, –), (, 0, ^–1 ), (– ^–1, –, 0);
          (1, –1, 1), (0, – ^–1, –), (–, 0, ^–1 ), ( ^–1, , 0);    and
(1, 1, –1), (0, ^–1, ), (–, 0, – ^–1 ), ( ^–1, –, 0). 

  References