Chamfer (geometry)
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.
Chamfered Platonic solids edit
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.
Chamfered tetrahedron edit
| 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 edit
| 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.
| Pyritohedron and its axis truncation | Historical crystallographic models |
| chamfered cube (canonical) | rhombic dodecahedron | chamfered octahedron (canonical) |
| tetrakis cuboctahedron | cuboctahedron | triakis cuboctahedron |
Chamfered octahedron edit
| 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 edit
| 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 edit
| 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.
Chamfered regular tilings edit
| Square tiling, Q {4,4} | Triangular tiling, Δ {3,6} | Hexagonal tiling, H {6,3} | Rhombille, daH dr{6,3} |
| cQ | cΔ | cH | cdaH |
Relation to Goldberg polyhedra edit
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)....
A truncated tetrakis hexahedron or pentakis dodecahedron, GP(3,0), creates a Goldberg sequence: GP(3,0), GP(6,0), GP(12,0)...
Chamfered polytopes and honeycombs edit
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.
See also edit
References edit
- ^ "C80 Isomers". Archived from the original on 2014-08-12. Retrieved 2014-08-09.
- Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal. 43: 104–108.
- Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture [1]
- Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.
- Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.
- Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF [2] (p. 72 Fig. 26. Chamfered tetrahedron)
- Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus half-cube embeddings", Discrete Mathematics, 192 (1): 41–80, doi:10.1016/S0012-365X(98)00065-X.
External links edit
- Chamfered Tetrahedron
- Chamfered Solids
- Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro
- VRML polyhedral generator (Conway polyhedron notation)
- VRML model Chamfered cube
- 3.2.7. Systematic numbering for (C80-Ih) [5,6] fullerene
- Fullerene C80
- (Number 7 -Ih)
- [4]
- How to make a chamfered cube