Archimedean solids are highly symmetric three-dimensional shapes made from regular polygon faces. Unlike the Platonic solids, they use two or more different types of regular polygons, such as triangles and squares or pentagons and hexagons. They matter because they connect pure geometry to real objects, including soccer balls, crystals, molecules, architecture, and computer graphics.
Their patterns show how local rules at each corner can determine a whole solid shape.
The key idea is that every vertex of an Archimedean solid has the same arrangement of polygons around it. This arrangement is called a vertex configuration, such as 5.6.6 for a truncated icosahedron, meaning one pentagon and two hexagons meet at each vertex. There are 13 Archimedean solids if mirror-image snub forms are counted as the same type.
Each one is convex, meaning it has no dents, and each satisfies Euler's formula V - E + F = 2.
Key Facts
- An Archimedean solid is a convex polyhedron with two or more types of regular polygon faces and the same vertex arrangement everywhere.
- Euler's formula for any convex polyhedron is V - E + F = 2.
- The truncated icosahedron has 12 pentagons, 20 hexagons, 60 vertices, and 90 edges.
- The truncated icosahedron vertex configuration is 5.6.6, meaning one pentagon and two hexagons meet at each vertex.
- For a regular n-gon, the interior angle is A = (n - 2)180°/n.
- At each vertex of a convex polyhedron, the angles meeting there must add to less than 360°.
Vocabulary
- Archimedean solid
- A convex polyhedron made from two or more kinds of regular polygon faces with the same pattern of faces at every vertex.
- Regular polygon
- A flat polygon whose sides are all equal in length and whose angles are all equal in measure.
- Vertex configuration
- A notation that lists the regular polygons meeting at each vertex in order, such as 3.4.3.4 or 5.6.6.
- Convex polyhedron
- A solid with flat polygon faces in which every line segment connecting two points inside the solid stays inside or on the solid.
- Truncated icosahedron
- An Archimedean solid with 12 regular pentagons and 20 regular hexagons, commonly recognized as the soccer ball shape.
Common Mistakes to Avoid
- Calling every many-faced solid an Archimedean solid is wrong because the faces must be regular polygons and the same face pattern must occur at every vertex.
- Confusing Platonic solids with Archimedean solids is wrong because Platonic solids use only one type of regular polygon face, while Archimedean solids use two or more.
- Ignoring the order in a vertex configuration is wrong because 3.4.5 and 3.5.4 may describe different cyclic arrangements around a vertex.
- Letting the angles around a vertex add to 360° or more is wrong for a convex polyhedron because the faces would lie flat or overlap instead of forming a corner.
Practice Questions
- 1 A truncated icosahedron has 12 pentagons and 20 hexagons. How many total faces does it have, and using V = 60 and Euler's formula V - E + F = 2, how many edges does it have?
- 2 Find the sum of the face angles at one vertex of a solid with vertex configuration 5.6.6. Use pentagon angle 108° and hexagon angle 120°. Is the angle sum less than 360°?
- 3 Explain why a solid made only of regular hexagons cannot form a convex Archimedean solid, even though regular hexagons tile the plane.