5 Platonic Solids

There are 5 recognized Platonic Solids
tetrahedron, octahedron, icosahedron, cube, and a dodecahedron
A platonic solid satisfy these three characteristics
All faces are regular polygons
All faces are congruent
All corners are congruent

A cube has 3 squares at each corner and 6 square faces

A tetrahedron has 3 triangles at each corner and 4 triangle faces

An octahedron has 4 equilateral triangles at each corner and 8 triangle faces

Icosahedron . has 5 triangles at each vertex and 20 triangle faces

Dodecahedron has 3 5 sided polygons at each vertex and 12 faces

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[viratwisetsingh]

3 triangles at vertex = tetrahedron
4 triangles at vertex = octahedron
5 triangles at vertex = icosahedron
6 triangles at vertex lies flat
7 triangles at vertex is impossible
3 squares at vertex = cube (hexahedron)
4 squares at vertex lies flat
5 squares at vertex is impossible
3 pentagons at vertex = dodecahedron
4 or more than 4 pentagons is impossible
3 hexagons at vertex lies flat
3 heptagons or octagons at vertex is impossible

[DRStidom]

I am not a mathematician, but can someone explain to me how an octahedron qualifies as a platonic solid (P.S.)? The other four can only be formed by using only the initial shape to form the P.S., but the cross section of the octahedron is a square. It would seem to me that the octahedron would be formed by putting two tetrahedrons together. However, using the property of the prime number, wouldn't the very mathematical definition disqualify it from being a P.S.? I am rather curious. Thank you.