Truncated Cuboctahedron - Vertices

Vertices

To derive the number of vertices, we note that each vertex is the meeting point of a square, hexagon, and octagon.

• Each of the 12 squares with their 4 vertices contribute 48 vertices because 12 × 4 = 48.
• Each of the 8 hexagons with their 6 vertices contribute 48 vertices because 8 × 6 = 48.
• Each of the 6 octagons with their 8 vertices contribute 48 vertices because 6 × 8 = 48.

Therefore, there may seem to exist 48 + 48 + 48 = 144 vertices. However, we have over-counted the vertices thrice since a square, hexagon, and octagon meet at each vertex. Consequently, we divide 144 by 3 to correct for our over-counting: 144 / 3 = 48.