No simply connected polyhedron since it does not meet the requirements of the Euler characteristic, which states that F + V = E + 2
F = number of faces
V = number of vertices
E = number of edges
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A decagonal based pyramid will have 11 faces, 11 vertices and 20 edges
Not a polyhedral shape. You need at least three vertices to define a plane shape. With 5 vertices, the maximum number of triplets you can get is 10 so there cannot be 11 faces.
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Here is some information about the 13 Archimedean solids:Tetrahedron-related:1) The truncated tetrahedron has 12 vertices, 18 edges, & 8 faces (4 triangles & 4 hexagons).Cube-related:2) The truncated cube has 24 vertices, 36 edges, & 14 faces (8 triangles & 6 octagons). Also known as a truncated hexahedron.3) The truncated octahedron has 24 vertices, 36 edges, & 14 faces (6 squares & 8 hexagons).4) The cuboctahedron has 12 verticies, 24 edges, & 14 faces (8 triangles & 6 squares).5) The small rhombicuboctahedron has 24 vertices, 48 edges, & 26 faces (8 triangles & 18 squares). Also known simply as a rhombicuboctahedron.6) The great rhombicuboctahedron has 48 vertices, 72 edges, & 26 faces (12 squares, 8 hexagons, & 6 octagons). Also known as a truncated cuboctahedron.7) The snub cube has 24 vertices, 60 edges, & 38 faces (32 triangles & 6 squares). Also known variously as a snub hexahedron, snub octahedron, or snub cuboctahedron. This shape, along with the snub dodecahedron, has 2 chiral forms.Icosahedron-related:8) The truncated icosahedron has 60 vertices, 90 edges, & 32 faces (12 pentagons & 20 hexagons). Also known as a buckyball or a football/soccer ball.9) The truncated dodecahedron has 60 vertices, 90 edges, & 32 faces (20 triangles & 12 decagons).10) The icosidodecahedron has 30 vertices, 60 edges, & 32 faces (20 triangles & 12 pentagons).11) The small rhombicosidodecahedron has 60 vertices, 120 edges, & 62 faces (20 triangles, 30 squares, & 12 pentagons). Also known simply as a rhombicosidodecahedron.12) The great rhombicosidodecahedron has 120 vertices, 180 edges, & 62 faces (30 squares, 20 hexagons, & 12 decagons). Also known as a truncated icosidodecahedron.13) The snub dodecahedron has 60 vertices, 150 edges, & 92 faces (80 triangles & 12 pentagons). Also known as a snub icosahedron or a snub icosidodecahedron. This shape, along with the snub cube, has 2 chiral forms.
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