In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. (Equivalently: it separates the faces by reducing them, and adds a new face between each two adjacent faces; but it only moves the vertices lower.) For a polyhedron, this operation adds a new hexagonal face in place of each original edge.
In Conway polyhedron notation, chamfering 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 lengths, and in a canonical version where all edges touch the same midsphere. (They look noticeably different only for solids containing triangles.) The shown dual polyhedra are dual to the canonical versions.
Seed Platonic solid | {3,3} | {4,3} | {3,4} | {5,3} | {3,5} |
---|---|---|---|---|---|
Chamfered Platonic solid |
Chamfered tetrahedron | |
---|---|
(equilateral-faced form) | |
Conway notation | cT |
Goldberg polyhedron | GPIII(2,0) = {3+,3}2,0 |
Faces | 4 congruent equilateral triangles 6 congruent hexagons (equilateral for a certain chamfering depth) |
Edges | 24 (2 types: triangle-hexagon, hexagon-hexagon) |
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 (for a certain chamfering depth) |
Net |
The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed:
The chamfered tetrahedron is the Goldberg polyhedron GIII(2,0), containing triangular and hexagonal faces.
chamfered tetrahedron (canonical form) | dual of the tetratetrahedron | chamfered tetrahedron (canonical form) |
alternate-triakis tetratetrahedron | tetratetrahedron | alternate-triakis tetratetrahedron |
Chamfered cube | |
---|---|
(equilateral-faced form) | |
Conway notation | cC = t4daC |
Goldberg polyhedron | GPIV(2,0) = {4+,3}2,0 |
Faces | 6 congruent squares 12 congruent hexagons (equilateral for a certain chamfering depth) |
Edges | 48 (2 types: square-hexagon, hexagon-hexagon) |
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 (for a certain chamfering depth) |
Net (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.) |
The chamfered cube is constructed as a chamfer of a cube: the squares are reduced in size and new faces, hexagons, are added in place of all the original edges. The chamfered cube is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 6 congruent (and regular) squares, and 12 congruent flattened hexagons which are equilateral for a certain depth of chamfering.
The dual of the chamfered cube 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 (6) order-4 vertices of the rhombic dodecahedron are truncated.
The hexagonal faces of the chamfered cube are equilateral but not regular. They are congruent partly truncated rhombi, have 2 internal angles of and 4 internal angles of while a regular hexagon would have all internal angles.
Because all the faces of the chamfered cube have an even number of sides and are centrally symmetric, 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 the eight order-3 vertices of the rhombic dodecahedron are at and its six order-4 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 form) | rhombic dodecahedron | chamfered octahedron (canonical form) |
tetrakis cuboctahedron | cuboctahedron | triakis cuboctahedron |
Chamfered octahedron | |
---|---|
(equilateral-faced form) | |
Conway notation | cO = t3daO |
Faces | 8 congruent equilateral triangles 12 congruent hexagons (equilateral for a certain chamfering depth) |
Edges | 48 (2 types: triangle-hexagon, hexagon-hexagon) |
Vertices | 30 (2 types) |
Vertex configuration | (24) 3.6.6 (6) 6.6.6.6 |
Symmetry | Oh, [4,3], (*432) |
Dual polyhedron | Triakis cuboctahedron |
Properties | convex, equilateral-faced (for a certain chamfering depth) |
In geometry, the chamfered octahedron (or tritruncated rhombic dodecahedron) is a convex polyhedron constructed by truncating the 8 order-3 vertices of the rhombic dodecahedron. These truncated vertices become congruent equilateral triangles, and the original 12 rhombic faces become congruent flattened hexagons.
For a certain depth of truncation, all final edges have same length; then, the hexagons are equilateral, but not regular.
Chamfered dodecahedron | |
---|---|
(equilateral-faced form) | |
Conway notation | cD = t5daD = dk5aD |
Goldberg polyhedron | GV(2,0) = {5+,3}2,0 |
Fullerene | C80 [2] |
Faces | 12 congruent regular pentagons 30 congruent hexagons (equilateral for a certain chamfering depth) |
Edges | 120 (2 types: pentagon-hexagon, hexagon-hexagon) |
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 (for a certain chamfering depth) |
The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons.
It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new faces, flattened hexagons, are added in place of all the original edges. For a certain depth of chamfering, all final edges have same length; then, the hexagons are equilateral, but not regular.
The dual of the chamfered dodecahedron is the pentakis icosidodecahedron.
The chamfered dodecahedron 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 (12) order-5 vertices of the rhombic triacontahedron are truncated.
chamfered dodecahedron (canonical form) | rhombic triacontahedron | chamfered icosahedron (canonical form) |
pentakis icosidodecahedron | icosidodecahedron | triakis icosidodecahedron |
Chamfered icosahedron | |
---|---|
(equilateral-faced form) | |
Conway notation | cI = t3daI |
Faces | 20 congruent equilateral triangles 30 congruent hexagons |
Edges | 120 (2 types) |
Vertices | 72 (2 types) |
Vertex configuration | (24) 3.6.6 (12) 6.6.6.6.6 |
Symmetry | Ih, [5,3], (*532) |
Dual polyhedron | Triakis icosidodecahedron |
Properties | convex |
In geometry, the chamfered icosahedron (or tritruncated rhombic triacontahedron) is a convex polyhedron constructed by truncating the 20 order-3 vertices of the rhombic triacontahedron. The hexagonal faces can be made equilateral, but not regular.
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, a deltahedron is a polyhedron whose faces are all (congruent) equilateral triangles. The name is taken from the Greek upper case delta letter (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra. By the handshaking lemma, each deltahedron has an even number of faces. Only eight deltahedra are strictly convex; these have 4, 6, 8, 10, 12, 14, 16, or 20 faces. The eight convex deltahedra, with their respective numbers of faces, edges, and vertices, are listed below.
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.
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 zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
In geometry, an n-gonaltrapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron, is the dual polyhedron of an n-gonal antiprism. The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its 2n faces are kites.
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 and topology, 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 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 polyhedron.
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".
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 polyhedra, of which all but the smallest one have mostly hexagonal faces.
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.