A Non-Convex Polyhedron with Sixty Non-Convex Pentagonal Faces

Sixty nonconvex pentagons as faces of a nonconvex polyhedron

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On Classification of Concave Polygons By Number of Concavities

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On Classification of Concave Polygons By Number of Concavities

Concave triangles do not exist, so concavity does not appear in the examination of polygons by ascending side length until the quadrilateral. A quadrilateral may only have one concavity, as shown in the red figure. Any polygon with exactly one concavity is called a uniconcave polygon.

Beginning with pentagons, the potential for two concavities appears. A polygon with two concavities, such as the yellow pentagon shown here, is a biconcave polygon.

Triconcave polygons, such as the blue hexagon here, have exactly three concavities. It is not possible for a triconcave polygon to have fewer than six sides.

For a tetraconcave polygon, with four concavities, at least eight sides are needed. The example shown here is the green octagon.

For higher number of concavities, simply double the number of sides to find the minimum number of sides for such a polygon. This pattern begins on the bottom row in the diagram here, but does not apply to the polygons shown in the top row.