User:Tomruen/4D cupola

Cupola (geometry) as regular or quasiregular base and related cells in parallel in parallel.

• Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000 Convex Segmentochora
• Segmentotopes

Polyhedral cupolae

Name	Tetrahedral cupola		Cubic cupola		Octahedral cupola		Dodecahedral cupola		Icosahedral cupola (Hyperbolic)		Hexagonal tiling cupola (Hyperbolic)
Schläfli symbol	{3,3} ∨ rr{3,3}		{4,3} ∨ rr{4,3}		{3,4} ∨ rr{3,4}		{5,3} ∨ rr{5,3}		{3,5} ∨ rr{3,5}		{6,3} ∨ rr{6,3}
Segmentochora index	K4.23		K4.71		K4.107		K4.152
circumradius	1		sqrt((3+sqrt(2))/2) = 1.485634		sqrt(2+sqrt(2)) = 1.847759		3+sqrt(5) = 5.236068
Image
Cap cells
Vertices	16		32		30		80		72		∞
Edges	42		84		84		210		210		∞
Faces	42	24 {3} + 18 {4}	80	32 {3} + 48 {4}	82	40 {3} + 42 {4}	194	80 {3} + 90 {4} + 24 {5}	202	100 {3} + 90 {4} + 12 {5}	∞	∞ {3} + ∞ {4} + ∞ {6}
Cells	16	1 tetrahedron 4 triangular prisms 6 triangular prisms 4 triangular pyramids 1 cuboctahedron	28	1 cube  6 square prisms 12 triangular prisms  8 triangular pyramids  1 rhombicuboctahedron	28	1 octahedron  8 triangular prisms 12 triangular prisms  6 square pyramids 1 rhombicuboctahedron	64	1 dodecahedron 12 pentagonal prisms 30 triangular prisms 20 triangular pyramids  1 rhombicosidodecahedron	64	1 icosahedron 20 triangular prisms 30 triangular prisms 12 pentagonal pyramids  1 rhombicosidodecahedron	∞	1 hexagonal tiling ∞ hexagonal prisms ∞ triangular prisms ∞ triangular pyramids 1 rhombitrihexagonal tiling
Related uniform polychora	runcinated 5-cell		runcinated tesseract		runcinated 24-cell		runcinated 120-cell		runcinated icosahedral honeycomb		runcinated hexagonal tiling honeycomb

Polyhedral recti-cupolae

Schläfli symbol	{3,3} ∨ r{3,3} Rectified 5-cell		{3,4} ∨ r{3,4}		{4,3} ∨ r{4,3}		r{4,3} ∨ rr{4,3}		{5,3} ∨ r{5,3}		{3,5} ∨ r{3,5}		r{5,3} ∨ rr{5,3}
Symmetry	[1,3,3]		[1,4,3]						[1,5,3]
Images
Cap cells
Segmentochora index	K4.5		K4.29		K4.35		K4.61		K4.77		K4.137		K4.131
circumradius	sqrt(3/5) = 0.774597		1		sqrt((16+6*sqrt(2))/23) = 1.031784		(1+sqrt(8))/sqrt(7) = 1.447009		(1+sqrt(5))/2 = 1.618034		sqrt(5+2*sqrt(5)) = 3.077684		sqrt(5+2*sqrt(5)) = 3.077684
Vertices	4+6		6+12		8+12		12+24		20+30		12+30		30+60
Edges	30		60		60		120		150		150		300
Faces	30	30 {3}	58	52 {3}, 6 {4}	56	44 {3}, 12 {4}	112	88 {3}, 24 {4}	134	110 {3}, 24 {5}	142	130 {3}, 12 {5}	274	220 {3}, 30 {4}, 24 {5}
Cells	10	1 tetrahedron 4 tetrahedral 4 octahedra 1 octahedron	16	1 octahedron 8 octahedra 6 square pyramids 1 cuboctahedron	16	1 cube 8 tetrahedra 6 square antiprisms 1 cuboctahedron	28	1 cuboctahedron 8 octahedra 12 square pyramids 6 square antiprisms 1 rhombicuboctahedron	34	1 dodecahedron 20 tetrahedra 12 pentagonal antiprisms 1 icosidodecahedron	34	1 icosahedron 20 octahedra 12 pentagonal pyramids 1 icosidodecahedron	64	1 icosidodecahedron 20 octahedra 12 pentagonal antiprisms 30 square pyramids 1 rhombicosidodecahedron
Related uniform polychora	Rectified 5-cell		half 24-cell								Rectified 600-cell

Polyhedral antiprisms

Schläfli symbol	{3,3} ∨ 2r{3,3}		t{3,3} ∨ 2t{3,3}		{4,3} ∨ 2r{4,3}		t{3,4} ∨ 2t{3,4}		{5,3} ∨ 2r{5,3}		t{5,3} ∨ 2t{5,3}
Symmetry	[2+,4,3]				[1,4,3]				[1,5,3]
Image
Cap cells
Segmentochora index	K4.2		K4.55		K4.15		K4.98		K4.78		K4.151
Vertices	4+4		12+12		6+8		24+24		12+20		60+60
Edges	24		60		48		120		120		300
Faces	32	32 {3}	52	52 {3}	62	56 {3}, 6 {4}	100	88 {3}, 12 {4}	152	140 {3}, 12 {5}	244	220 {3}, 24 {5}
Cells	16	1 tetrahedron 4 tetrahedra 6 tetrahedra 4 tetrahedra 1 tetrahedron	16	1 truncated tetrahedron 4 trigonal cupola 6 tetrahedra 4 trigonal cupola 1 truncated tetrahedron	28	1 octahedron 8 tetrahedra 12 tetrahedra 6 square pyramids 1 cube	28	1 truncated octahedron 8 trigonal cupolae 12 tetrahedral 6 square cupolae 1 truncated cube	64	1 icosahedron 20 tetrahedra 30 tetrahedra 12 pentagonal pyramids 1 dodecahedron	64	1 truncated icosahedron 20 trigonal cupolae 30 tetrahedra 12 pentagonal cupolae 1 truncated dodecahedron
Related uniform polychora	16-cell Tetrahedral antiprism s{2,4,3}		Rectified tesseract