__NOTOC__

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

. It is a

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...

of the

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...

, and can also be called small stellated triacontahedron. Its dual is the

dodecadodecahedron In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The e ...

. Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

). This means that on each axis there is an inner and an outer vertex. The ratio of outer to inner vertex radius is

\varphi \approx 1.618

, the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. It has 30 intersecting rhombic faces, which correspond to the faces of the convex

. The diagonals in the rhombs of the convex solid have a ratio of 1 to

\varphi

. The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to

\varphi^3 \approx 4.236

. So the ratio of rhomb diagonals in the medial solid is 1 to

\varphi^2 \approx 2.618

. This solid is to the

compound of small stellated dodecahedron and great dodecahedron The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron. This can be seen as one of the two three-dimensional equivalen ...

what the convex one is to the

compound of dodecahedron and icosahedron In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound. As a compound It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Ke ...

: The crossing edges in the

dual compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected ...

are the diagonals of the rhombs. The faces have two angles of

\arccos(\frac\sqrt)\approx 41.810\,314\,895\,78^

, and two of

\arccos(-\frac\sqrt)\approx 138.189\,685\,104\,22^

. Its

dihedral angles A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

equal

\arccos(-\frac)= 120^

. Part of each rhomb lies inside the solid, hence is invisible in solid models.

Related hyperbolic tiling

It is topologically equivalent to a quotient space of the

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

order-5 square tiling In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Related polyhedra and tiling This tiling is topologically related as a part of sequence of regular polyhedra and tilings with ver ...

, by distorting the rhombi into

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

. As such, it is topologically a

regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

of index two:The Regular Polyhedra (of index two)
David A. Richter Uniform tiling 45-t0

Note that the order-5 square tiling is dual to the

order-4 pentagonal tiling In geometry, the order-4 pentagonal tiling is a List_of_regular_polytopes#Hyperbolic_tilings, regular tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of . It can also be called a pentapentagonal tiling in a bicolored q ...

, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the

References

External links

* * David I. McCooey
animation and measurements

Uniform polyhedra and duals
Dual uniform polyhedra {{Polyhedron-stub

Related hyperbolic tiling

See also

References

External links