Chamfer (geometry)

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Unchamfered, slightly chamfered, and chamfered cube

Historical crystal models of slightly chamfered Platonic solids

Transformation of a cube (left) and a regular dodecahedron (right) into their chamfered forms, with equilateral (but not equiangular) hexagons

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

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 GP_III(2,0) or {3+,3}_2,0, containing triangular and hexagonal faces. Its dual is the alternate-triakis tetratetrahedron.
• chamfered cube: from a cube, the resulting polyhedron has twelve hexagonal and six square centrally symmetric faces, a zonohedron. This is also an example of the Goldberg polyhedron GP_IV(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.
• chamfered octahedron or tritruncated rhombic dodecahedron: from a regular octahedron by chamfering, 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 GP_V(2,0) or {5+,3}_2,0. Its dual is triakis cuboctahedron.

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. GP_V(2,0) = {5+,3}_2,0. The structure resembles C₆₀ fullerene. Its dual is the pentakis icosidodecahedron.
• 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.

Regular tilings

Chamfered regular and quasiregular 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

• Cantellation (geometry)
• Conway polyhedron notation
• Near-miss Johnson solid
• Uniform 4-polytope
• Uniform polyhedron

References

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
• 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). "Crystallography" . 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
  1. (Number 7 -Ih)
• How to make a chamfered cube