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 ...

, chamfering or edge-truncation is a topological operator that modifies one

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 ...

into another. It is similar to

expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...

, moving

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 ...

apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original

edge 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 ...

. In

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 ...

it is represented by the letter . A polyhedron with edges will have a chamfered form containing new vertices, new edges, and new hexagonal faces.

Chamfered Platonic solids

In the chapters below the chamfers of the five

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s are described in detail. Each is shown in a version with edges of equal length and in a canonical version where all edges touch the same

midsphere In geometry, the midsphere or intersphere of a 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 poly ...

. (They only look noticeably different for solids containing triangles.) The shown

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...

are dual to the canonical versions.

Chamfered tetrahedron

The

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 ...

ed tetrahedron (or alternate truncated cube) 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 ...

constructed as an alternately

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons. 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 ...

G_III(2,0), containing triangular and hexagonal faces. Polyhedron truncated 4b max

Chamfered cube

The chamfered cube is a

with 32 vertices, 48 edges, and 18 faces: 12 hexagons and 6 squares. It is constructed as a chamfer of a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

. The squares are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the tetrakis cuboctahedron. It is also inaccurately called a truncated rhombic dodecahedron, although that name rather suggests a

rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at eac ...

. It can more accurately be called a tetratruncated rhombic dodecahedron because only the order-4 vertices are truncated. The hexagonal faces are

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. They are formed by a truncated rhombus, have 2 internal angles of about 109.47°

\cos^(-\frac)

and 4 internal angles of about 125.26°, while a regular hexagon would have all 120° angles. Because all its faces have an even number of sides with 180° rotation symmetry, it is a

zonohedron In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...

. It is also the

GP_IV(2,0) or _2,0, containing square and hexagonal faces. The ''chamfered cube'' is the

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at

(\pm 1, \pm 1, \pm 1)

and its six vertices are at the permutations of

(\pm \sqrt 3, 0, 0)

. A

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

equivalent with

pyritohedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...

and rectangular faces can be constructed by chamfering the axial edges of a

pyritohedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

. This occurs in

pyrite The mineral pyrite (), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Iron, FeSulfur, S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral. Pyrite's metallic Luster (mineralogy), lust ...

crystals.

Chamfered octahedron

, the chamfered octahedron is a

constructed from the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

by truncating the 8 (order 3) vertices. It can also be called a tritruncated rhombic dodecahedron, a truncation of the order-3 vertices of the

. The 8 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles. The hexagonal faces are

but not regular. Brockhaus and Efron Encyclopedic Dictionary b48 862-4

Chamfered dodecahedron

The chamfered dodecahedron is a

with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer 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). Construc ...

. It is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a

rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square (geometry), square face ...

. It can more accurately be called a pentatruncated rhombic triacontahedron because only the order-5 vertices are truncated. Polyhedron truncated 20 max

Chamfered icosahedron

, the chamfered icosahedron is a

constructed from 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 ...

by truncating the 20 order-3 vertices. The hexagonal faces can be made

but not regular. It can also be called a tritruncated rhombic triacontahedron, a truncation of the order-3 vertices of the

Chamfered regular tilings

Relation to Goldberg polyhedra

The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. The chamfer operator transforms GP(m,n) to GP(2m,2n). A regular polyhedron, GP(1,0), create a

Goldberg polyhedra 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 (mathematician), Michael Goldberg (1902–1990 ...

sequence: GP(1,0), GP(2,0), GP(4,0), GP(8,0), GP(16,0)... The

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

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. ...

, GP(1,1) creates a Goldberg sequence: GP(1,1), GP(2,2), GP(4,4), GP(8,8).... A truncated tetrakis hexahedron or

pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that i ...

, GP(3,0), creates a Goldberg sequence: GP(3,0), GP(6,0), GP(12,0)...

Chamfered polytopes and honeycombs

Like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. For polychora, new cells are created around the original edges. The cells are prisms, containing two copies of the original face, with pyramids augmented onto the prism sides.

References

* * Joseph D. Clinton, ''Clinton’s Equal Central Angle Conjecture'

* * * Antoine Deza, Michel Deza, Viatcheslav Grishukhin, ''Fullerenes and coordination polyhedra versus half-cube embeddings'', 1998

PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

br>
(p. 72 Fig. 26. Chamfered tetrahedron) *{{citation , last1=Deza , first1=A. , last2=Deza , first2=M. , author2-link=Michel Deza , last3=Grishukhin , first3=V. , title=Fullerenes and coordination polyhedra versus half-cube embeddings , journal= Discrete Mathematics (journal), Discrete Mathematics , volume=192 , issue=1 , year=1998 , pages=41–80 , url=http://www.ehess.fr/centres/cams/papers/144.ps.gz , doi=10.1016/S0012-365X(98)00065-X , url-status=dead , archiveurl=https://web.archive.org/web/20070206064633/http://www.ehess.fr/centres/cams/papers/144.ps.gz , archivedate=2007-02-06 , doi-access=free .

External links

Chamfered Tetrahedron

Livio Zefiro * ttp://www.georgehart.com/virtual-polyhedra/conway_notation.html VRML polyhedral generator(

) **

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...

mode
Chamfered cube

,6fullerene * Fullerene C80 *

(Number 7 -Ih) *

How to make a chamfered cube
Goldberg polyhedra Polyhedra Mathematical notation