The regular dodecahedron has $20$ vertices, and $12$ faces which are (congruent) regular pentagons; three pentagons meet at a vertex.
Fixing a vertex $v$ of dodecahedron, we count the other vertices, whose minimum distance from $v$ is $3$. One can see the Schlegel diagram of the dodecahedron here.
There are exactly $6$ vertices at distance $3$ from $v$: call them $x,x',y,y',z,z'$.
With figure below (Schlegel diagram), we can see that we can partition these six vertices into two sets, such that vertices in each set are at distance $3$ from each other.
Thus, from the vertex $v$, we produced $2$ tetrahedrons, with one vertex common.
But, I was unable to see from this construction, why do we get only $10$ regular tetrahedrons inside regular dodecahedron? Can anybody explain it?
(Note: I know some other way - going via cubes, and then tetrahedrons; but, here I am trying to understand it from the Schlegel diagram, for better understanding of the diagram and explanation.)
why this question came? There are $20$ vertices in Dodecahedron, and tetrahedron has $4$ vertices, so if we want to get $10$ tetrahedrons, I was unable to see how the repetition of vertices is coming in tetrahedrons while distributing vertices of dodecahedron.