In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.
In Conway polyhedron notation it is represented by the letter c. A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and e new hexagonal faces.
In the chapters below the chamfers of the five Platonic solids 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. (They only look noticeably different for solids containing triangles.) The shown duals are dual to the canonical versions.
Seed | {3,3} | {4,3} | {3,4} | {5,3} | {3,5} |
---|---|---|---|---|---|
Chamfered |
Chamfered tetrahedron | |
---|---|
(with equal edge length) | |
Conway notation | cT |
Goldberg polyhedron | GPIII(2,0) = {3+,3}2,0 |
Faces | 4 triangles 6 hexagons |
Edges | 24 (2 types) |
Vertices | 16 (2 types) |
Vertex configuration | (12) 3.6.6 (4) 6.6.6 |
Symmetry group | Tetrahedral (Td) |
Dual polyhedron | Alternate-triakis tetratetrahedron |
Properties | convex, equilateral-faced |
net |
The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons.
It is the Goldberg polyhedron GIII(2,0), containing triangular and hexagonal faces.
chamfered tetrahedron (canonical) | dual of the tetratetrahedron | chamfered tetrahedron (canonical) |
alternate-triakis tetratetrahedron | tetratetrahedron | alternate-triakis tetratetrahedron |
Chamfered cube | |
---|---|
(with equal edge length) | |
Conway notation | cC = t4daC |
Goldberg polyhedron | GPIV(2,0) = {4+,3}2,0 |
Faces | 6 squares 12 hexagons |
Edges | 48 (2 types) |
Vertices | 32 (2 types) |
Vertex configuration | (24) 4.6.6 (8) 6.6.6 |
Symmetry | Oh, [4,3], (*432) Th, [4,3+], (3*2) |
Dual polyhedron | Tetrakis cuboctahedron |
Properties | convex, equilateral-faced |
net |
The chamfered cube is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 12 hexagons and 6 squares. It is constructed as a chamfer of a cube. 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. It can more accurately be called a tetratruncated rhombic dodecahedron because only the order-4 vertices are truncated.
The hexagonal faces are equilateral but not regular. They are formed by a truncated rhombus, have 2 internal angles of about 109.47°, or , 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. It is also the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0, containing square and hexagonal faces.
The chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at and its six vertices are at the permutations of .
A topological equivalent with pyritohedral symmetry and rectangular faces can be constructed by chamfering the axial edges of a pyritohedron. This occurs in pyrite crystals.
chamfered cube (canonical) | rhombic dodecahedron | chamfered octahedron (canonical) |
tetrakis cuboctahedron | cuboctahedron | triakis cuboctahedron |
Chamfered octahedron | |
---|---|
(with equal edge length) | |
Conway notation | cO = t3daO |
Faces | 8 triangles 12 hexagons |
Edges | 48 (2 types) |
Vertices | 30 (2 types) |
Vertex configuration | (24) 3.6.6 (6) 6.6.6 |
Symmetry | Oh, [4,3], (*432) |
Dual polyhedron | Triakis cuboctahedron |
Properties | convex |
In geometry, the chamfered octahedron is a convex polyhedron constructed from the rhombic dodecahedron 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 rhombic dodecahedron.
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 equilateral but not regular.
Chamfered dodecahedron | |
---|---|
(with equal edge length) | |
Conway notation | cD] = t5daD = dk5aD |
Goldberg polyhedron | GV(2,0) = {5+,3}2,0 |
Fullerene | C80 [1] |
Faces | 12 pentagons 30 hexagons |
Edges | 120 (2 types) |
Vertices | 80 (2 types) |
Vertex configuration | (60) 5.6.6 (20) 6.6.6 |
Symmetry group | Icosahedral (Ih) |
Dual polyhedron | Pentakis icosidodecahedron |
Properties | convex, equilateral-faced |
The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer of a regular dodecahedron. 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.
It is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a rhombicosidodecahedron. It can more accurately be called a pentatruncated rhombic triacontahedron because only the order-5 vertices are truncated.
chamfered dodecahedron (canonical) | rhombic triacontahedron | chamfered icosahedron (canonical) |
pentakis icosidodecahedron | icosidodecahedron | triakis icosidodecahedron |
Chamfered icosahedron | |
---|---|
(with equal edge length) | |
Conway notation | cI = t3daI |
Faces | 20 triangles 30 hexagons |
Edges | 120 (2 types) |
Vertices | 72 (2 types) |
Vertex configuration | (24) 3.6.6 (12) 6.6.6 |
Symmetry | Ih, [5,3], (*532) |
Dual polyhedron | triakis icosidodecahedron |
Properties | convex |
In geometry, the chamfered icosahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the 20 order-3 vertices. The hexagonal faces can be made equilateral but not regular.
It can also be called a tritruncated rhombic triacontahedron, a truncation of the order-3 vertices of the rhombic triacontahedron.
Square tiling, Q {4,4} | Triangular tiling, Δ {3,6} | Hexagonal tiling, H {6,3} | Rhombille, daH dr{6,3} |
cQ | cΔ | cH | cdaH |
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 sequence: GP(1,0), GP(2,0), GP(4,0), GP(8,0), GP(16,0)...
GP(1,0) | GP(2,0) | GP(4,0) | GP(8,0) | GP(16,0)... | |
---|---|---|---|---|---|
GPIV {4+,3} | C | cC | ccC | cccC | |
GPV {5+,3} | D | cD | ccD | cccD | ccccD |
GPVI {6+,3} | H | cH | ccH | cccH | ccccH |
The truncated octahedron or truncated icosahedron, GP(1,1) creates a Goldberg sequence: GP(1,1), GP(2,2), GP(4,4), GP(8,8)....
GP(1,1) | GP(2,2) | GP(4,4)... | |
---|---|---|---|
GPIV {4+,3} | tO | ctO | cctO |
GPV {5+,3} | tI | ctI | cctI |
GPVI {6+,3} | tH | ctH | cctH |
A truncated tetrakis hexahedron or pentakis dodecahedron, GP(3,0), creates a Goldberg sequence: GP(3,0), GP(6,0), GP(12,0)...
GP(3,0) | GP(6,0) | GP(12,0)... | |
---|---|---|---|
GPIV {4+,3} | tkC | ctkC | cctkC |
GPV {5+,3} | tkD | ctkD | cctkD |
GPVI {6+,3} | tkH | ctkH | cctkH |
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.
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
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.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
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.
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 Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,5⁄2}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).
In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. 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, 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.
In geometry, a polytope 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 edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
In geometry, a d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets). The vertex figure of a simple d-polytope is a (d – 1)-simplex.
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.
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a shallow pyramid. Kleetopes are named after Victor Klee.
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 properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic sphere.
In geometry, the rectified truncated icosahedron is a convex polyhedron. It has 92 faces: 60 isosceles triangles, 12 regular pentagons, and 20 regular hexagons. It is constructed as a rectified, truncated icosahedron, rectification truncating vertices down to mid-edges.
The truncated pentakis dodecahedron is a convex polyhedron constructed as a truncation of the pentakis dodecahedron. It is Goldberg polyhedron GV(3,0), with pentagonal faces separated by an edge-direct distance of 3 steps.
In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty', and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".
The order-5 truncated pentagonal hexecontahedron is a convex polyhedron with 72 faces: 60 hexagons and 12 pentagons triangular, with 210 edges, and 140 vertices. Its dual is the pentakis snub dodecahedron.