Triakis tetrahedron, pentagonal icositetrahedron and disdyakis triacontahedron.

The solids above (dark) shown together with their duals (light). The visible parts of the Catalan solids are regular pyramids.

In mathematics, a Catalan solid, or Archimedean dual, is a polyhedron that is dual to an Archimedean solid. There are 13 Catalan solids. They are named after the Belgian mathematician Eugène Catalan, who first described them in 1865.

The Catalan solids are all convex. They are face-transitive but not vertex-transitive. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons. However, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra.

Additionally, two of the Catalan solids are edge-transitive: the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids.

Just as prisms and antiprisms are generally not considered Archimedean solids, bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive.

Two of the Catalan solids are chiral: the pentagonal icositetrahedron and the pentagonal hexecontahedron, dual to the chiral snub cube and snub dodecahedron. These each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids.

Eleven of the 13 Catalan solids have the Rupert property: a copy of the solid, of the same or larger shape, can be passed through a hole in the solid. ^[1]

List of Catalan solids and their duals

Conway name	Archimedean dual	Face polygon	Orthogonal wireframes	Pictures	Face angles (°)	Dihedral angle (°)	Midradius[2]	Faces	Edges	Vert	Sym.
triakis tetrahedron "kT"	truncated tetrahedron	Isosceles V3.6.6			112.885 33.557 33.557	129.521	1.0607	12	18	8	Td
rhombic dodecahedron "jC"	cuboctahedron	Rhombus V3.4.3.4			70.529 109.471 70.529 109.471	120	0.8660	12	24	14	Oh
triakis octahedron "kO"	truncated cube	Isosceles V3.8.8			117.201 31.400 31.400	147.350	1.7071	24	36	14	Oh
tetrakis hexahedron "kC"	truncated octahedron	Isosceles V4.6.6			83.621 48.190 48.190	143.130	1.5000	24	36	14	Oh
deltoidal icositetrahedron "oC"	rhombicuboctahedron	Kite V3.4.4.4			81.579 81.579 81.579 115.263	138.118	1.3066	24	48	26	Oh
disdyakis dodecahedron "mC"	truncated cuboctahedron	Scalene V4.6.8			87.202 55.025 37.773	155.082	2.2630	48	72	26	Oh
pentagonal icositetrahedron "gC"	snub cube	Pentagon V3.3.3.3.4			114.812 114.812 114.812 114.812 80.752	136.309	1.2472	24	60	38	O
rhombic triacontahedron "jD"	icosidodecahedron	Rhombus V3.5.3.5			63.435 116.565 63.435 116.565	144	1.5388	30	60	32	Ih
triakis icosahedron "kI"	truncated dodecahedron	Isosceles V3.10.10			119.039 30.480 30.480	160.613	2.9271	60	90	32	Ih
pentakis dodecahedron "kD"	truncated icosahedron	Isosceles V5.6.6			68.619 55.691 55.691	156.719	2.4271	60	90	32	Ih
deltoidal hexecontahedron "oD"	rhombicosidodecahedron	Kite V3.4.5.4			86.974 67.783 86.974 118.269	154.121	2.1763	60	120	62	Ih
disdyakis triacontahedron "mD"	truncated icosidodecahedron	Scalene V4.6.10			88.992 58.238 32.770	164.888	3.7694	120	180	62	Ih
pentagonal hexecontahedron "gD"	snub dodecahedron	Pentagon V3.3.3.3.5			118.137 118.137 118.137 118.137 67.454	153.179	2.0971	60	150	92	I

Symmetry

The Catalan solids, along with their dual Archimedean solids, can be grouped in those with tetrahedral, octahedral and icosahedral symmetry. For both octahedral and icosahedral symmetry there are six forms. The only Catalan solid with genuine tetrahedral symmetry is the triakis tetrahedron (dual of the truncated tetrahedron). The rhombic dodecahedron and tetrakis hexahedron have octahedral symmetry, but they can be colored to have only tetrahedral symmetry. Rectification and snub also exist with tetrahedral symmetry, but they are Platonic instead of Archimedean, so their duals are Platonic instead of Catalan. (They are shown with brown background in the table below.)

Tetrahedral symmetry

Archimedean (Platonic)
Catalan (Platonic)

Octahedral symmetry

Archimedean
Catalan

Icosahedral symmetry

Archimedean
Catalan

Geometry

All dihedral angles of a Catalan solid are equal. Denoting their value by ${\displaystyle \theta }$ , and denoting the face angle at the vertices where ${\displaystyle p}$ faces meet by ${\displaystyle \alpha _{p}}$, we have

${\displaystyle \sin(\theta /2)=\cos(\pi /p)/\cos(\alpha _{p}/2)}$.

This can be used to compute ${\displaystyle \theta }$ and ${\displaystyle \alpha _{p}}$, ${\displaystyle \alpha _{q}}$, ... , from ${\displaystyle p}$, ${\displaystyle q}$ ... only.

Triangular faces

Of the 13 Catalan solids, 7 have triangular faces. These are of the form Vp.q.r, where p, q and r take their values among 3, 4, 5, 6, 8 and 10. The angles ${\displaystyle \alpha _{p}}$, ${\displaystyle \alpha _{q}}$ and ${\displaystyle \alpha _{r}}$ can be computed in the following way. Put ${\displaystyle a=4\cos ^{2}(\pi /p)}$, ${\displaystyle b=}$Zdroj:https://en.wikipedia.org?pojem=Catalan_solid
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