Exploring Some Geometric Themes

Faces, vertices and edges of solids and Euler’s formula, prisms and pyramids, nets, painted cubes, and fractals that repeat a pattern at smaller sizes.

Key ideas

For any polyhedron without holes, V − E + F = 2.
A net is a flat shape that folds into a solid; a cube has 11 different nets.
In a painted cube cut into small cubes, corner cubes have 3 painted faces, edge cubes 2, face cubes 1, and inside cubes none.
Sierpinski triangle: each step removes the middle triangle from every triangle, leaving 3 times as many triangles.
Koch snowflake: each step replaces every side with 4 sides, each a third as long, so the perimeter grows.

Remember

Euler’s formula	V − E + F = 2
Prism on an n-sided base	V = 2n, E = 3n, F = n + 2
Pyramid on an n-sided base	V = n + 1, E = 2n, F = n + 1
Painted n × n × n cube	3 faces: 8; 2: 12(n − 2); 1: 6(n − 2)²; 0: (n − 2)³
Fractal steps	Sierpinski: triangles × 3; Koch: perimeter × 4/3

Watch out

Counting an edge twice when it is shared by two faces.
Forgetting the 8 corner cubes when counting painted cubes.
Mixing up the prism and pyramid counts.

Worked example

A 8 × 5 × 5 block is painted on the outside and cut into 200 unit cubes. How many cubes have paint on exactly 3, 2, 1 and 0 faces?

Working: corners 8; edges 4 × (6 + 3 + 3); faces 2 × (18 + 9 + 18); inside 6 × 3 × 3

Answer: 3 faces: 8; 2 faces: 48; 1 face: 90; none: 54