A kaleidocyle is a rotating ring of triangular pyramids.

Hexagonal Kaleidocycle

Number of faces: 24

Number of edges: 30

Number of vertices: 12

Octagonal Kaleidocycle

Number of faces: 32

Number of edges: 40

Number of vertices: 16

Decagonal Kaleidocycle

Number of faces: 40

Number of edges: 50

Number of vertices: 20

Closed Octagonal Kaleidocycle

Number of faces: 32

Number of edges: 40

Number of vertices: 16

Closed Decagonal Kaleidocycle

Number of faces: 40

Number of edges: 50

Number of vertices: 20

Closed Dodecagonal Kaleidocycle

Number of faces: 48

Number of edges: 60

Number of vertices: 24

Dodecagonal Kaleidocycle

Number of faces: 48

Number of edges: 60

Number of vertices: 24

Half Closed Hexagonal Kaleidocycle

Number of faces: 24

Number of edges: 30

Number of vertices: 12

Half Closed Octagonal Kaleidocycle

Number of faces: 32

Number of edges: 40

Number of vertices: 16

Tetrakaidecagonal Kaleidocycle

Number of faces: 56

Number of edges: 70

Number of vertices: 28

Closed Tetrakaidecagonal Kaleidocycle

Number of faces: 56

Number of edges: 70

Number of vertices: 28

Half Closed Decagonal Kaleidocycle

Number of faces: 40

Number of edges: 50

Number of vertices: 20

Half Closed Dodecagonal Kaleidocycle

Number of faces: 48

Number of edges: 60

Number of vertices: 24

Half Closed Tetrakaidecagonal Kaleidocycle

Number of faces: 56

Number of edges: 70

Number of vertices: 28

Open Hexagonal Kaleidocycle

Number of faces: 24

Number of edges: 30

Number of vertices: 12

Quarter Closed Hexagonal Kaleidocycle

(Invertible cube or Schatz cube)

Number of faces: 24

Number of edges: 30

Number of vertices: 12

Seven Twelfths Closed Hexagonal Kaleidocycle

Number of faces: 24

Number of edges: 30

Number of vertices: 12