Platonic Solids are 3D shapes which meet the following conditions. Each face needs to be the same regular polygon and the same number of polygons need to meet at each vertex. Platonic Solids include the tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron.

Tetrahedron

The tetrahedron is made up of four triangular faces. It has four vertices (three triangles meet at each vertex) and six edges. It has seven axes of symmetry. Instructions on how to create an origami tetrahedron can be found here.

Cube or Hexahedron

The cube is made up of six quadratic faces. It has eight vertices (three squares meet at each vertex) and twelve edges. It has thirteen axes of symmetry. Instructions on how to create an origami cube can be found here.

Octahedron

The octahedron is made up of eight triangular faces. It has six vertices (four triangles meet at each vertex) and twelve edges.

Dodecahedron 

The dodecahedron is made up of twelve pentagonal faces. It has twenty vertices (three pentagons meet at each vertex) and thirty edges. It has thirty-one axes of symmetry. Instructions on how to create an origami tetrahedron can be found here.

Icosahedron 

The icosahedron is made up of twenty triangular faces. It has twelve vertices (five triangles meet at each vertex) and thirty edges.

 

Using origami, one can combine platonic solids to create intricate stars or compounds. An example of this is the interlocking tetrahedra which is created from five tetrahedra. Instructions on how to create this can be found here.

Interlocking Tetrahedra 

Icosahedron and Dodecahedron 

References

“Mathematical Origami.” Mathigon. Web. 15 Oct. 2015. http://mathigon.org/mathigon_org/origami/

“Platonic Solids.” Platonic Solids. Web. 15 Oct. 2015. https://www.mathsisfun.com/platonic_solids.html