Slingshot argument

Template:Polyhedron types There are many relations among the uniform polyhedra.

Here they are grouped by the Wythoff symbol.

Key

Image Name Bowers pet name V Number of vertices,E Number of edges,F Number of faces=Face configuration ?=Euler characteristic, group=Symmetry group Wythoff symbol - Vertex figure W - Wenninger number, U - Uniform number, K- Kalido number, C -Coxeter number alternative name second alternative name

The vertex figure can be discovered by considering the Wythoff symbol:

p|q r - 2p edges, alternating q-gons and r-gons. Vertex figure (q.r)^p.
p|q 2 - p edges, q-gons (here r=2 so the r-gons are degenerate lines).
2|q r - 4 edges, alternating q-gons and r-gons
q r|p - 4 edges, 2p-gons, q-gons, 2p-gons r-gons, Vertex figure 2p.q.2p.r.
q 2|p - 3 edges, 2p-gons, q-gons, 2p-gons, Vertex figure 2p.q.2p.
p q r|- 3 edges, 2p-gons, 2q-gons, 2r-gons, vertex figure 2p.2q.2r

Convex

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{3} \frac{π}{3} \frac{π}{2}$	Template:Reg polyhedra db		Octahedron	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,		Cuboctahedron	Truncated octahedron	Icosahedron
$\frac{π}{4} \frac{π}{3} \frac{π}{2}$	Template:Reg polyhedra db	Template:Reg polyhedra db	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,
$\frac{π}{5} \frac{π}{3} \frac{π}{2}$	Template:Reg polyhedra db	Template:Reg polyhedra db	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,	Wilber Berryhill is what his wife loves to call him and he completely enjoys this title. The preferred pastime for him and his kids is to perform lacross and he would by no means give it up. Distributing production real psychics has been his profession for online reader; 1.234.36.240, some time. Alaska is exactly where he's always been residing. Feel free to surf to my site - online psychic; mouse click for source,

Non-convex

a b 2

3 3 2

$\frac{a π}{3} \frac{b π}{3} \frac{c π}{2}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{3} \frac{π}{2} \frac{2 π}{3}$					Template:Uniform polyhedra db

4 3 2

$\frac{a π}{4} \frac{b π}{3} \frac{c π}{2}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	q r\|p	p q\|r	p q r\|
$\frac{π}{4} \frac{2 π}{3} \frac{π}{2}$	octahedron	cube	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{3 π}{4} \frac{π}{3} \frac{π}{2}$					Template:Uniform polyhedra db
$\frac{3 π}{4} \frac{2 π}{3} \frac{π}{2}$					Template:Uniform polyhedra db

5 3 2

$\frac{a π}{5} \frac{b π}{3} \frac{c π}{2}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
$\frac{2 π}{5} \frac{π}{3} \frac{π}{2}$	Template:Reg polyhedra db	Template:Reg polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
	p q r\|	p q r\|	p q r\|	\|p q r
$\frac{3 π}{5} \frac{π}{3} \frac{π}{2}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db

5 5 2

$\frac{a π}{5} \frac{b π}{5} \frac{c π}{2}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
$\frac{π}{5} \frac{2 π}{5} \frac{π}{2}$	Template:Reg polyhedra db	Template:Reg polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
	p q r\|	p q r\|	\|p q r
$\frac{π}{5} \frac{3 π}{5} \frac{π}{2}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db

a b 3

3 3 3

$\frac{a π}{3} \frac{b π}{3} \frac{c π}{3}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{3} \frac{π}{3} \frac{2 π}{3}$				Template:Uniform polyhedra db

4 3 3

$\frac{a π}{4} \frac{b π}{3} \frac{c π}{3}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r

5 3 3

$\frac{a π}{5} \frac{b π}{3} \frac{c π}{3}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r
$\frac{3 π}{5} \frac{π}{3} \frac{π}{3}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db		Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
	p q r\|	p q r\|	\|p q r
$\frac{π}{5} \frac{2 π}{3} \frac{π}{3}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db

4 4 3

$\frac{a π}{4} \frac{b π}{4} \frac{c π}{3}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{π}{4} \frac{π}{3} \frac{3 π}{4}$					Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{π}{4} \frac{π}{4} \frac{2 π}{3}$				Template:Uniform polyhedra db

5 5 3

$\frac{a π}{5} \frac{b π}{5} \frac{c π}{3}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	q\|p r	q r\|p	p r\|q	p q\|r
$\frac{π}{3} \frac{2 π}{5} \frac{3 π}{5}$		Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{π}{3} \frac{π}{5} \frac{4 π}{5}$		Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{π}{5} \frac{π}{5} \frac{2 π}{3}$		Template:Uniform polyhedra db	Template:Uniform polyhedra db
$\frac{π}{5} \frac{π}{3} \frac{3 π}{5}$	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db	Template:Uniform polyhedra db

a b 5

5 5 5

$\frac{a π}{5} \frac{b π}{5} \frac{c π}{5}$ Group

Spherical triangle $\frac{π}{p} \frac{π}{q} \frac{π}{r}$	p\|q r	q\|p r	r\|p q	q r\|p	p r\|q	p q\|r	p q r\|	\|p q r
$\frac{2 π}{5} \frac{3 π}{5} \frac{3 π}{5}$					Template:Uniform polyhedra db

Slingshot argument

Contents

Key

Convex

Non-convex

a b 2

3 3 2

4 3 2

5 3 2

5 5 2

a b 3

3 3 3

4 3 3

5 3 3

4 4 3

5 5 3

a b 5

5 5 5

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Slingshot argument

Key

Convex

Non-convex

a b 2

3 3 2

4 3 2

5 3 2

5 5 2

a b 3

3 3 3

4 3 3

5 3 3

4 4 3

5 5 3

a b 5

5 5 5

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