Chamfer (geometry)
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
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]
- 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
 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
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} |
 C |
 cC |
 ccC |
 cccC |
ccccC |
... |
| 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} |
 tΔ |
 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} |
 tkC |
 ctkC |
cctkC |
... |
| GPV {5+,3} |
 tkD |
 ctkD |
cctkD |
... |
| GPVI {6+,3} |
 tkH |
 ctkH |
cctkH |
... |
See also
References
- ^ 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.
- ^ "TwistyPuzzles.com > Museum > Show Museum Item". twistypuzzles.com. Retrieved 2025-02-09.
- ^ Gelişgen & Yavuz 2019b, Chamfered Octahedron Metric and Some Properties.
- ^ Gelişgen & Yavuz 2019a.
Sources
- 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.
- 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..
- Gelişgen, Özcan; Yavuz, Serhat (2019a). "A Note About Isometry Groups of Chamfered Dodecahedron and Chamfered Icosahedron Spaces" (PDF). International Journal of Geometry. 8 (2): 33–45.
- Gelişgen, Özcan; Yavuz, Serhat (2019b). "Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces". Mathematical Sciences and Applications E-Notes. 7 (2): 174–182. doi:10.36753/mathenot.542272.
- Spencer, Leonard James (1911). . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 07 (11th ed.). Cambridge University Press. pp. 569–591.
External links
- 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