Medial rhombic triacontahedron
Encyclopedia
In geometry
, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron
. It is the dual
of the dodecadodecahedron. It has 30 intersecting rhombic faces.
It can also be called the small stellated triacontahedron.
It is topologically equivalent to the hyperbolic
order-5 square tiling, by distorting the rhombi into squares
. As such, it is topologically a regular polyhedron
of index two:
Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, which is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
. It is the dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
of the dodecadodecahedron. It has 30 intersecting rhombic faces.
It can also be called the small stellated triacontahedron.
It is topologically equivalent to the hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
order-5 square tiling, by distorting the rhombi into squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
. As such, it is topologically a regular polyhedron
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...
of index two:
Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, which is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.