The angle test becomes a short list
For a regular solid, two choices determine the local shape: how many sides each face has, and how many faces meet at each corner. Call these numbers p and q. A regular p-sided face has interior angles of \((p-2)\times180/p\) degrees. At a corner, that angle is repeated q times, so the corner can fold into a convex solid only when the total is less than 360 degrees.
The smallest possible face is a triangle, and at least three faces must meet at a corner. Trying larger values quickly rules them out: three hexagons already total 360 degrees, while four squares do too. Bigger faces or more faces make the total still larger.
Five pairings, five solids
The combinations that stay below the limit are: triangles meeting in groups of three, four, or five; squares meeting in groups of three; and pentagons meeting in groups of three. Each pairing produces one regular convex solid: the tetrahedron, octahedron, icosahedron, cube, or dodecahedron.
This is more than a count of familiar shapes. The angle test explains why the list stops: beyond those pairings, the faces cannot turn inward enough at a corner to enclose space. The restriction concerns regular convex solids—shapes without inward dents—not every possible polyhedron. Allow irregular faces, or allow a nonconvex form, and the geometric possibilities change.
