At one vertex of a cube there are three squares, and their angles add up. $3 \times 90° = 270°$. There are $90°$ left to complete one turn: this is the defect of the vertex. By Around 1630, Descartes discovered that, in a convex polyhedron, the sum of the defects of all vertices is always 720°.

Polyhedron Angles at a vertex Defect Vertices Sum
Tetrahedron 3 × 60° = 180° 180° 4 720°
Cube 3 × 90° = 270° 90° 8 720°
Octahedron 4 × 60° = 240° 120° 6 720°
Dodecahedron 3 × 108° = 324° 36° 20 720°
Icosahedron 5 × 60° = 300° 60° 12 720°
The faces around a vertex, open in the plane. The red slice is the defect, the angle left to complete 360°.

The manuscript was lost for centuries. It only survived because Leibniz copied the text into 1676, and the A copy was only found in the 19th century. Now comes the surprise: from this theorem comes the formula $V - A + F = 2$, which we saw in the chapter on Euler.

The interior angles of a face with $n$ sides add up to $(n - 2) \cdot 180°$. Adding across all sides, and remembering that each edge is a side of two faces (so the sum of $n$ of all faces is $2A$):

$$\text{soma de todos os ângulos das faces} = (2A - 2F) \cdot 180° = (A - F) \cdot 360°.$$

The total defect is $360°$ per vertex minus this sum:

$$720° = 360° \cdot V - 360° \cdot (A - F) = 360° \cdot (V - A + F) \;\Longrightarrow\; V - A + F = 2.$$

In other words, Descartes had in his hands, more than a century before, a result equivalent to the formula of Euler. He just didn’t write the final bill.

Rotate the dodecahedron: 20 defect vertices of 36° each, 20 × 36° = 720°.

All topics by René Descartes · ← The rule of signs · The kissing circles theorem →

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