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Answer 1: Look what happens on a face of the big tetrahedron where some edges of small ones come together: you have to make angle 180° out of some dihedral angles of the tetrahedra (which is about 70°) --- that is impossible.

Answer 2: There is the so-called Dehn invariant. If a polyhedron $X$ is split into a number of polyhedra $Y_1,Y_2,\dots,Y_n$ then the Dehn invariant of $X$ is equal to the sum of the Dehn invariants of $Y_i$.

For the regular tetrahedron, the Dehn inveriant is nonzero and proportional to the length of a side. Suppose you could split a regular tetrahedron with side $a$ into a number of tetrahedra with sides $a_1, a_2,\dots, a_n$. Then from the volume you have $$a_1^2+a_2^2+\dots+a_n^2=a^2$$ and from the Dehn invariant you have $$a_1+a_2+\dots+a_n=a.$$ It follows that there is no nontrivial splitting.

Answer 1: Look what happens on a face of the big tetrahedron where some edges of small ones come together: you have to make angle 180° out of some dihedral angles of the tetrahedra (which is about 70°) --- that is impossible.

Answer 2: There is the so-called Dehn invariant. If a polyhedron $X$ is split into a number of polyhedra $Y_1,Y_2,\dots,Y_n$ then the Dehn invariant of $X$ is equal to the sum of the Dehn invariants of $Y_i$.

For the regular tetrahedron, the Dehn inveriant is nonzero and proportional to the length of a side. Suppose you could split a regular tetrahedron with side $a$ into a number of tetrahedra with sides $a_1, a_2,\dots, a_n$. Then from the volume you have $$a_1^3+a_2^3+\dots+a_n^3=a^3$$ and from the Dehn invariant you have $$a_1+a_2+\dots+a_n=a.$$ It follows that there is no nontrivial splitting.

Answer 1: Look what happening on the face of big tetrahedron where few edges of small ones come together you have to make angle 180° out of few Dihedral angles of tetrahedra (which is about 70°) --- that is impossible.

Answer 2: There is so called Dehn invariant. If a polyhedron $X$ is splited in a number of polyhedra $Y_1,Y_2,\dots,Y_n$ then Dehn invariant of $X$ is equal to the sum of the Dehn invariants of $Y_i$.

The regular tetrahedron, it is nonzero it is proportional to the length of side. If could split regular tetrahedron with size $a$ in a number of tetrahedra with side $a_1, a_2,\dots, a_n$. Then from volume you have $$a_1^2+a_2^2+\dots+a_n^2=a^2$$ and from the Dehn invariant you have $$a_1+a_2+\dots+a_n=a.$$ It follows that there is no nontrivial splitting.

Answer 1: Look what happens on a face of the big tetrahedron where some edges of small ones come together: you have to make angle 180° out of some dihedral angles of the tetrahedra (which is about 70°) --- that is impossible.

Answer 2: There is the so-called Dehn invariant. If a polyhedron $X$ is split into a number of polyhedra $Y_1,Y_2,\dots,Y_n$ then the Dehn invariant of $X$ is equal to the sum of the Dehn invariants of $Y_i$.

For the regular tetrahedron, the Dehn inveriant is nonzero and proportional to the length of a side. Suppose you could split a regular tetrahedron with side $a$ into a number of tetrahedra with sides $a_1, a_2,\dots, a_n$. Then from the volume you have $$a_1^2+a_2^2+\dots+a_n^2=a^2$$ and from the Dehn invariant you have $$a_1+a_2+\dots+a_n=a.$$ It follows that there is no nontrivial splitting.

Answer 1: There is so called Dehn invariant. If a polyhedron $X$ is splited in a number of polyhedra $Y_1,Y_2,\dots,Y_n$ then Dehn invariant of $X$ is equal to the sum of the Dehn invariants of $Y_i$.

The regular tetrahedron, it is nonzero it is proportional to the length of side. If could split regular tetrahedron with size $a$ in a number of tetrahedra with side $a_1, a_2,\dots, a_n$. Then from volume you have $$a_1^2+a_2^2+\dots+a_n^2=a^2$$ and from the Dehn invariant you have $$a_1+a_2+\dots+a_n=a.$$ It follows that there is no nontrivial splitting.

Answer 2: Look what happening on the face of big tetrahedron where few edges of small ones come together you have to make angle 180° out of few Dihedral angles of tetrahedra (which is about 70°) --- that is impossible.

Answer 1: Look what happening on the face of big tetrahedron where few edges of small ones come together you have to make angle 180° out of few Dihedral angles of tetrahedra (which is about 70°) --- that is impossible.

Answer 2: There is so called Dehn invariant. If a polyhedron $X$ is splited in a number of polyhedra $Y_1,Y_2,\dots,Y_n$ then Dehn invariant of $X$ is equal to the sum of the Dehn invariants of $Y_i$.

The regular tetrahedron, it is nonzero it is proportional to the length of side. If could split regular tetrahedron with size $a$ in a number of tetrahedra with side $a_1, a_2,\dots, a_n$. Then from volume you have $$a_1^2+a_2^2+\dots+a_n^2=a^2$$ and from the Dehn invariant you have $$a_1+a_2+\dots+a_n=a.$$ It follows that there is no nontrivial splitting.