24/03/2020 - Platonic Solids

On this class, we built some platonic solids:

Tetrahedron - 4 faces (green)
Hexahedron (Cube) - 6 faces (cyan)
Octahedron - 8 faces (yellow)
Dodecahedron - 12 faces (pink)
Icosahedron - 20 faces (orange)

The first solid we built was the tetrahedron. For that, the first step is to draw the triangular base of this solid. This base is composed by a equilateral triangle with 10 units as the sides length. Then, we create a 2D plan of all the faces disassembled.

After this, we can now find out the center of the solid and the way that the faces are going to be folded.

For that, we draw a vertical line passing the center of the base triangle and a circunference with radius from the middle of the common side between the base and one face to the vertex of that same face.

Using the command 3DRotate, we rotate that circunference for it to be perpendicular to the face that we want to fold.

We can now fold the first face, using the command Align.

Since the faces of the tetrahedron are all equal, we can use the command 3DArray, that copies this face using a 360º angle to repeat it the times we want to. In this case, we want to repeat it 3 times in a 360º angle.

The next solid with built was the hexadron, usually called cube.

This one is a bit more simple than the last one, sinces the faces are all folded with a 90º angle.

First, we draw the 2D plan of the solid.

Then, we start folding all the faces, one by one, using the 3DRotate command.

Then, we built a octahedron.

Just like the previous ones, this one is no exception. We have to start it by doing the 2D plan of the solid.

This 2D plan is composed by four equilateral triangles that create between them a square. This square is the base of our first half of the polyhedron.

Using the same method used on the first solid, we draw a vertical line passing on the center of the base and a circunference passing in two points of one of the triangles.

Then, we 3DRotate the circunference and fold the first triangle with the Align command.

After this, we use, again, the 3DArray, to repeat this face to the other sides of the base, but this time, we chose to repeat it 4 times.

At last, we use the 3DMirror command to copy this face, using the square base as a common plan.

The fourth polyhedron drawn was a dodecahedron.

I found this one a bit more complex, since it is composed by regular pentagons.

The first step is to draw 3 pentagons attached to each other.

Then, we have to find an intersection point between two faces, for us to be able to know the height where their sides touch each other. For that, we draw lines perpendicular to each other and circunferences that give us the heigh. Always using the same commands as the other polyhedrons.

After this steps, we use the Align command to fold the first face.

And then, we can use again the 3DArray to repeat this face to the other sides of the base.

Now, because this polyhedron is composed by another half equal to this one, we have to copy this half to the side. After that, we use, again, the Align command, always matching opposite vertices, so that both halfs fit perfectly.