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 chamfered dodecahedron is a

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

with 80 vertices, 120

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

, and 42

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

: 30

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 and 12

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

s. It is constructed as a

chamfer A chamfer or is a transitional edge between two faces of an object. Sometimes defined as a form of bevel, it is often created at a 45° angle between two adjoining right-angled faces. Chamfers are frequently used in machining, carpentry, fu ...

(edge-truncation) of a

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...

. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the

pentakis icosidodecahedron In geometry, the pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the ''truncated rhombic triacontahedron'' (chamfered dodecahedron). Constructi ...

. It is also called a truncated rhombic triacontahedron, constructed as a

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

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 ...

. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated.

Structure

These 12 order-5 vertices can be truncated such that all edges are equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons. The hexagon faces can be

equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

but not regular with D symmetry. The angles at the two vertices with

vertex configuration In geometry, a vertex configurationCrystallography ...

are

\arccos\left(\frac\right) = 116.565^

and at the remaining four vertices with , they are each. It is the

Goldberg polyhedron In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three pro ...

, containing pentagonal and hexagonal faces. It also represents the exterior envelope of a cell-centered

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

of the

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...

, one of six ( convex regular 4-polytopes).

Chemistry

This is the shape of the

fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...

; sometimes this shape is denoted to describe its icosahedral symmetry and distinguish it from other less-symmetric 80-vertex fullerenes. It is one of only four fullerenes found by to have a

skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...

that can be isometrically embeddable into an L space. Chamfered dodecahedron

Related polyhedra

This polyhedron looks very similar to the uniform

truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. ...

which has 12 pentagons, but only 20 hexagons. Image:Truncated rhombic triacontahedron.png, Truncated rhombic triacontahedron
G(2,0) Image:Truncated icosahedron.png,

Truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. ...

G(1,1) File:Ortho solid 120-cell.png, cell-centered

of the

The chamfered dodecahedron creates more polyhedra by basic

Conway polyhedron notation In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using op ...

. The zip chamfered dodecahedron makes a chamfered truncated icosahedron, and Goldberg (2,2).

Chamfered truncated icosahedron

, the chamfered truncated icosahedron is a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

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 ...

with 240 vertices, 360 edges, and 122 faces, 110 hexagons and 12 pentagons. It is constructed by a chamfer operation to the

, adding new hexagons in place of original edges. It can also be constructed as a zip (= dk = dual of kis of) operation from the ''chamfered dodecahedron''. In other words, raising pentagonal and hexagonal pyramids on a chamfered dodecahedron (kis operation) will yield the (2,2)

geodesic polyhedron A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhed ...

. Taking the dual of that yields the (2,2)

, which is the chamfered truncated icosahedron, and is also

Fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...

C₂₄₀.

Dual

Its dual, the hexapentakis chamfered dodecahedron has 240 triangle faces (grouped as 60 (blue), 60 (red) around 12 5-fold symmetry vertices and 120 around 20 6-fold symmetry vertices), 360 edges, and 122 vertices. Conway_polyhedron_kcD

Hexapentakis chamfered dodecahedron

References

Bibliography

* * * *{{cite web , title=Mathematical Impressions: Goldberg Polyhedra , first=George , last=Hart , authorlink = George W. Hart , date=June 18, 2013 , url=https://www.simonsfoundation.org/multimedia/mathematical-impressions-goldberg-polyhedra/ , publisher= Simons Science News

External links

Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra
Livio Zefiro

(

) Goldberg polyhedra Polyhedra Mathematical notation