Platonic solid exists in a 3-dimensional space, constructed by congruent(identical in shape and size), regular(all angles equal and all sides equal), polygonal faces with the same number of faces meeting at each vertex. There only five solids that meets these criteria.
- Icosahedron / 24 faces / each face is a triangle
- Dodecahedron / 12 faces / each face is a pentagon
- Cube / 6 faces / each face is a square
- Octahedron / 8 faces / each face is a triangle
- Tetrahedron / 4 faces / each face is a triangle
There are relationship between each solids. So you can draw all the five platonic solid with just one. But there is a way you don’t need to know any numeric information to draw any of these solids. This is how:
First, draw a Icosahedron.
This is how it will look like from the top.
So, everything starts with a simple circle. Then divide the circle in five equal length. mark the points and connect the ones next to each other. These lines will be the bottom part of each triangle that forms a Icosahedron. If you’ve done this, based on the definition of platonic solid, every lines of this Icosahedron has the same length of the lines you just drew. So circle is perfect to maintain this length. First, draw a circle on one of the points you’ve just made with a radius of this length. Then rotate it perpendicular to the line of the center of the fist circle and the circle you just drew. Then rotate it based on the center o the very first circle. There will be two overlapped point. Pick one of the points, and connect it with the five points from the original first circle. Then you will have five triangles connected to each other. That is a part of the Icosahedron. Now, to make the rest, you should know this component: 3-point-orient. Platonic Solids are constructed with same angle, same length, same face. So with this 3-point-orient tool, you’ll be able to finish the Icosahedron.
If you’ve done this part, the rest is now simply connecting and dividing.
Icosahedron -> Dodecahedron :
Find the center of every triangle face of the Icosahedron. Connect the points next to each other to a pentagon. Those pentagon faces will turn into a Dodecahedron.
Dodecahedron -> Cube :
Reforming Dodecahedron to a Cube is more simple. Cube has 6 faces and Dodecahedron has 12 faces. Pick two faces next to each other. Then pick the next points to the ones that is overlapped. There are 4 points. Then connect these points and make a square. Then, chose another pentagon and the one on the next in the perpendicular side of the previous two pentagon. Repeat drawing squares. After that you will get a perfect Cube.
Cube -> Octahedron :
If you have a cube, find every center of the solid. then connect the center of the three faces sharing two edges. This will give you a triangle. And if you connect all the points like this, you will get a Octahedron.
Octahedron -> Cube -> Tetrahedron :
If you repeat the process with four faces then three with Octahedron, you will get a Cube again. Then with this Cube, connect the two diagonal points form one square of a Cube. Then another diagonal from the connected face with one point you have already chose. Then automatically you will get another two points that are in a diagonal position. Connect that too, and you will get a triangle. Make the same triangle around the cube sharing edges, you will get a nice little Tetrahedron.
I’ve drew all these with Rhino, maybe that is why I could make Icosahedron with the 3-point-orient component. I concentrated how to use the exact angle that I have made without using numbers, because by writing down it in numbers, the accuracy immediately become inaccurate.