Platonic solids.

Step 1 of 9

A gap at the corner

A Platonic solid is built from one regular polygon, over and over, and all its corners are alike. Everything is decided at the corner: at least three faces meet there, and to fold up into a point their angles must add up to less than $360^\circ$⁠.

The gap left when the faces at a corner are pressed flat is called the angle defect. Predict how big it is for the cube, then look.

Step 2 of 9

The icosahedron

In every Platonic solid the defect is the same at every corner, and it equals $360^\circ$⁠ minus the face angles at the corner. In the Icosahedron five equilateral triangles meet at each corner. The Solid and Unfold are fixed now.

Step 3 of 9

Only five

Each angle of an equilateral triangle is $60^\circ$⁠, of a square $90^\circ$⁠, and of a regular pentagon $108^\circ$⁠. Try every possibility: three, four or five triangles at a corner, three squares, three pentagons. Each leaves a gap, and gives one solid. Choose a Solid and check the gap for each.

Step 4 of 9

The net and Euler’s formula

At the Count stage the corners ($V$⁠), edges ($E$⁠) and faces ($F$⁠) are counted, and for every solid $V - E + F = 2$⁠. The net explains why. The folds hold all the faces in one piece without a loop, so there are $F - 1$⁠ of them. The cuts reach every corner, and they never close a loop either. Every edge is a fold or a cut. The Solid and Unfold are fixed now.

Step 5 of 9

Faces from Euler

At Count you see all three numbers. Knowing two of them, Euler’s formula gives the third.

Step 6 of 9

Faces and corners

Each regular solid is fixed by two numbers: how many sides each face has, and how many faces meet at each corner. At Corner you see one corner up close.

Step 7 of 9

A pentagon

To know how many faces fit at a corner, you need to know how big each of their angles is.

Step 8 of 9

All the gaps together

At every corner something is missing from a flat plane, and those gaps are what close the solid up. How much is missing at all the corners together?

Step 9 of 9

Between two faces

A solid has one more angle: the angle at which two faces meet along an edge, like the pages of an open book.