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 great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

. The exterior surface also represents the De₂f₂

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

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 ...

. These figures can be differentiated by marking which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the

even–odd rule The even–odd rule is an algorithm implemented in vector-based graphic software, like the PostScript language and Scalable Vector Graphics (SVG), which determines how a graphical shape with more than one closed outline will be filled. Unlike the ...

, the internal structure of both shapes will differ. The 12 vertices of the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

matches the

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

of an

Great triambic icosahedron

The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron, U47. It has 20 inverted-hexagonal (

triambus In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular polygon, equiangular (have all angles equal), but if it does then it ...

) faces, shaped like a three-bladed

propeller A propeller (colloquially often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon ...

. It has 32 vertices: 12 exterior points, and 20 hidden inside. It has 60 edges. The faces have alternating angles of

\arccos(\frac)-60^\approx 15.522\,487\,814\,07^

and

\arccos(-\frac)\approx 104.477\,512\,185\,93^

. The sum of the six angles is

360^

, and not

720^

as might be expected for a hexagon, because the polygon turns around its center twice. The

dihedral angle 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 ...

equals

\arccos(-\frac)\approx 109.471\,220\,634\,49

Medial triambic icosahedron

The medial triambic icosahedron is the dual of the

ditrigonal dodecadodecahedron In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol ...

, U41. It has 20 faces, each being simple concave

isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

s or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges. The faces have alternating angles of

\arccos(\frac)-60^\approx 15.522\,487\,814\,07^

and

\arccos(-\frac)+120^\approx 224.477\,512\,185\,93^

. The

equals

\arccos(-\frac)\approx 109.471\,220\,634\,49

. Unlike the great triambic icosahedron, the medial triambic icosahedron 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 By distorting the triambi into regular

s, one obtains 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 hexagonal tiling In geometry, the order-5 hexagonal 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 tilings with order-5 vertices ...

As a stellation

It is Wenninger's 34th model as his 9th stellation of the icosahedron

References

* * *

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...

'', (3rd edition, 1973), Dover edition, , 3.6 6.2 ''Stellating the Platonic solids'', pp.96-104

External links

* * * gratrix.ne
Uniform polyhedra and duals
* bulatov.org

{{Icosahedron stellations Polyhedra Polyhedral stellation Dual uniform polyhedra

Great triambic icosahedron

Medial triambic icosahedron

As a stellation

See also

References

External links