Chamfer (geometry)
(Redirected from Alternate truncated cube)
This article needs additional citations for verification. (March 2025) |
Unchamfered, slightly chamfered, and chamfered cube
Historical crystal models of slightly chamfered Platonic solids
In geometry, a chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the faces by reducing them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, 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.
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.
Platonic solids
[edit | edit source]
Left to right: chamfered tetrahedron, cube, octahedron, dodecahedron, and icosahedron
Chamfers of five Platonic solids are described in detail below.
- chamfered tetrahedron or alternated truncated cube: from a regular tetrahedron, this replaces its six edges with congruent flattened hexagons; or alternately truncating a cube, replacing four of its eight vertices with congruent equilateral-triangle faces. This is an example of Goldberg polyhedron GPIII(2,0) or {3+,3}2,0, containing triangular and hexagonal faces. Its dual is the alternate-triakis tetratetrahedron.[2]
- chamfered cube: from a cube, the resulting polyhedron has twelve hexagonal and six square centrally symmetric faces, a zonohedron.[3] This is also an example of the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0. Its dual is the tetrakis cuboctahedron. A twisty puzzle of the DaYan Gem 7 is the shape of a chamfered cube.[4]
- chamfered octahedron or tritruncated rhombic dodecahedron: from a regular octahedron by chamfering,[5] or by truncating the eight order-3 vertices of the rhombic dodecahedron, which become congruent equilateral triangles, and the original twelve rhombic faces become congruent flattened hexagons. It is a Goldberg polyhedron GPV(2,0) or {5+,3}2,0. Its dual is triakis cuboctahedron.[2]
pentakis icosidodecahedron and triakis icosidodecahedron
- chamfered dodecahedron: by chamfering a regular dodecahedron, the resulting polyhedron has 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons. GPV(2,0) = {5+,3}2,0. The structure resembles C60 fullerene.[6] Its dual is the pentakis icosidodecahedron.[2]
- chamfered icosahedron or tritruncated rhombic triacontahedron: by chamfering a regular icosahedron, or truncating the twenty order-3 vertices of the rhombic triacontahedron. The hexagonal faces of the cI can be made equilateral, but not regular, with a certain depth of truncation. Its dual is the triakis icosidodecahedron.[2]
Regular tilings
[edit | edit source]| File:Tiling 4a simple.svg Square tiling, Q {4,4} |
File:Tiling 3 simple.svg Triangular tiling, Δ {3,6} |
File:Tiling 6 simple.svg Hexagonal tiling, H {6,3} |
File:Tiling 3-6 dual.svg Rhombille, daH dr{6,3} |
| File:Chamfer square tiling.svg | File:Chamfer triangular tiling.svg | File:Chamfer hexagonal tiling.svg | File:Chamfered rhombille tiling.svg |
| cQ | cΔ | cH | cdaH |
Relation to Goldberg polyhedra
[edit | edit source]The chamfer operation applied in series creates progressively larger polyhedra with new faces, hexagonal, replacing the edges of the current one. The chamfer operator transforms GP(m,n) to GP(2m,2n).
A regular polyhedron, GP(1,0), creates 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} |
File:Uniform polyhedron-43-t0.svg C |
File:Truncated rhombic dodecahedron2.png cC |
File:Octahedral goldberg polyhedron 04 00.svg ccC |
File:Octahedral goldberg polyhedron 08 00.svg cccC |
ccccC |
... |
| GPV {5+,3} |
File:Uniform polyhedron-53-t0.svg D |
File:Truncated rhombic triacontahedron.png cD |
File:Chamfered chamfered dodecahedron.png ccD |
File:Chamfered chamfered chamfered dodecahedron.png cccD |
File:Chamfered chamfered chamfered chamfered dodecahedron.png ccccD |
... |
| GPVI {6+,3} |
File:Tiling 6 simple.svg H |
File:Truncated rhombille tiling.svg cH |
File:Chamfered chamfered hexagonal tiling.png 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} |
File:Uniform polyhedron-43-t12.svg tO |
File:Chamfered truncated octahedron.png ctO |
File:Chamfered chamfered truncated octahedron.png cctO |
... |
| GPV {5+,3} |
File:Uniform polyhedron-53-t12.svg tI |
File:Chamfered truncated icosahedron.png ctI |
File:Chamfered chamfered truncated icosahedron.png cctI |
... |
| GPVI {6+,3} |
File:Uniform tiling 63-t12.svg tΔ |
File:Chamfered truncated triangular tiling.svg ctΔ |
cctΔ |
... |
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} |
File:Octahedral goldberg polyhedron 03 00.svg tkC |
File:Octahedral goldberg polyhedron 06 00.svg ctkC |
cctkC |
... |
| GPV {5+,3} |
File:Conway polyhedron Dk6k5tI.png tkD |
File:Chamfered truncated pentakis dodecahedron.png ctkD |
cctkD |
... |
| GPVI {6+,3} |
File:Truncated hexakis hexagonal tiling.png tkH |
File:Chamfered truncated hexakis hexagonal tiling.svg ctkH |
cctkH |
... |
See also
[edit | edit source]- Cantellation (geometry)
- Conway polyhedron notation
- Near-miss Johnson solid
- Uniform 4-polytope
- Uniform polyhedron
References
[edit | edit source]- ^ Spencer 1911, p. 575, or p. 597 on Wikisource, Crystallography, 1. Cubic System, Tetrahedral Class, Figs. 30 & 31.
- ^ a b c d Deza, Deza & Grishukhin 1998, 3.4.3. Edge truncations.
- ^ Gelişgen & Yavuz 2019b, Chamfered Cube Metric and Some Properties.
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Gelişgen & Yavuz 2019b, Chamfered Octahedron Metric and Some Properties.
- ^ Gelişgen & Yavuz 2019a.
Sources
[edit | edit source]- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture [1]
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)..
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
External links
[edit | edit source]- 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
- How to make a chamfered cube