The Platonic solids or Platonic polyhedra are the convex polyhedra where all faces are copies of the same regular polygon, and the same number of edges meet at every vertex. There are five of these shapes: the tetrahedron (like a pyramid but with a triangular base): cube, octahedron, dodecahedron and icosahedron.
Recent versions of Blender include an addon called “Regular Solids”, which lets you instantly generate these and a whole bunch of other similar shapes. However, the following steps do not require any addons.
The Tetrahedron
Bring up the Add Mesh menu (Shift+A), and select a Cone. Set the number of Vertices to 3, leave Radius 1 at its default value of 1.000 and Radius 2 at 0.000. Now, set the Depth to . To make sure that you have a regular tetrahedron, you can check the lengths of the edges (in Edit Mode, press N to open the Properties panel and locate the checkbox Length in the section Edge Info).
The Cube
This happens to be a built-in shape in Blender. Just bring up the Add Mesh menu, and select Cube. Done!
The Octahedron
This shape is the dual of the cube—it has vertices where the cube has faces, and faces where the cube has vertices. To make it, first create a cube. Press Tab to switch to Edit mode. All the vertices should already be selected. Press W to bring up the Specials menu, and select the Bevel function (or select it directly with CTRL + B ). As you move the mouse, you will see each vertex of the cube turn into a triangular face; don’t bother getting the shape exactly right, simply press LMB to finish the drag. Then, look in the panel that should have appeared at the bottom of the Toolshelf on the left of the 3D view (press T to toggle its visibility); you should see an editable numeric field labelled “Offset”. Type the value 1.0 into this field, and that should exactly form the octahedron shape.
Finally, bring up the Specials menu again, and this time select Remove Doubles.
The Icosahedron
Bring up the Add Mesh menu, and select an Icosphere. Set the Subdivision to 1. Simple!
The Dodecahedron
This shape is the dual of the icosahedron. To create it, make an icosahedron as above. Then do what you did to make an octahedron out of a cube: press Tab to switch to Edit mode. All the vertices should already be selected. Press W to bring up the Specials menu, and select the Bevel function. As with the octahedron, press LMB to finish the drag. Then set the Offset value to 0.607, the closest I could find.
Then, bring up the Specials menu again, and this time select Remove Doubles. However, the default Merge Distance of 0.0001 will not be enough to remove all the doubled vertices; change this to 0.001, and you should see the message “Removed 40 vertices” briefly flash up.
Exercise
What’s the dual of the tetrahedron? Try applying the Bevel operation to one of those; what do you end up with?