Catalan Solids

Starting with

Taking the convex hull of

End up with

a regular tetrahedron

four adjacent tetrahedra

Triakis Tetrahedron

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Starting with

Taking the convex hull of

End up with

a regular dodecahedron

an icosahedron and a dodecahedron

Triakis Icosahedron

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Starting with

Taking the convex hull of

End up with

A cube

a cube and an octahedron

Tetrakis Hexahedron

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Starting with

Taking the convex hull of

End up with

a cube

a cube and an octahedron

@

Rhombic Dodecahedron

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Starting with

Taking the convex hull of

End up with

an icosidodecahedron

an icosidodecahedron, a dodecahedron and an icosahedron

Deltoidal Hexecontahedron

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Starting with

Taking the convex hull of

End up with

a regular octahedron

three octagons and a cube

Deltoidal Icositetrahedron

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Starting with

Taking the convex hull of

End up with

a cuboctahedron

a cuboctahedron, an octahedron and a cube

Disdyakis Dodecahedron

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Starting with

Taking the convex hull of

End up with

an icosidodecahedron

an icosidodecahedron, an icosahedron and a dodecahedron

Disdyakis Triacontahedron

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Starting with

Taking the convex hull of

End up with

an octahedron

an octahedron and a cube

Small Triakis Octahedron

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Starting with

Taking the convex hull of

End up with

a dodecahedron

an icosahedron and a dodecahedron

Rhombic Triacontahedron

@

Starting with

Taking the convex hull of

End up with

a dodecahedron

an icosahedron and a dodecahedron

Pentakis Dodecahedron

References: Eugene Charles Catalan  Rhombic Triacontahedron