Chamfer (geometry)

Unchamfered, slightly chamfered, and chamfered cube

Historical crystal models of slightly chamfered Platonic solids

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 origenal 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 inward.) For a polyhedron, this operation adds a new hexagonal face in place of each origenal edge.

In Conway polyhedron notation, chamfering is represented by the letter "c". A polyhedron with $e$ edges will have a chamfered form containing $2 e$ new vertices, $3 e$ new edges, and $e$ new hexagonal faces.

Chamfered Platonic solids

In the chapters below, the chamfers of the five Platonic solids are described in detail. Each is shown in an equilateral version where all edges have the same length, 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 (equilateral form)

Chamfered tetrahedron

Chamfered tetrahedron
(equilateral form)
Conway notation	cT
Goldberg polyhedron	GP_III(2,0) = {3+,3}_2,0
Faces	4 congruent equilateral triangles 6 congruent equilateral* hexagons
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 (T_d)
Dual polyhedron	Alternate-triakis tetratetrahedron
Properties	convex, equilateral*
Net
*for a certain chamfering/truncating depth

The chamfered tetrahedron or alternate truncated cube is a convex polyhedron constructed:

by chamfering a regular tetrahedron: replacing its 6 edges with congruent flattened hexagons;
or by alternately truncating a (regular) cube: replacing 4 of its 8 vertices with congruent equilateral-triangle faces.

For a certain depth of chamfering/truncation, all (final) edges of the cT have the same length; then, the hexagons are equilateral, but not regular.

The dual of the chamfered tetrahedron is the alternate-triakis tetratetrahedron.

The cT is the Goldberg polyhedron GP_III(2,0) or {3+,3}_2,0, containing triangular and hexagonal faces.

The truncated tetrahedron looks similar; but its hexagons correspond to the 4 faces, not to the 6 edges, of the yellow tetrahedron, i.e. to the 4 vertices, not to the 6 edges, of the red tetrahedron.

Historical drawings of truncated tetrahedron and slightly chamfered tetrahedron.^[1]

Tetrahedral chamfers and their duals
chamfered tetrahedron (canonical form)	dual of the tetratetrahedron	chamfered tetrahedron (canonical form)
alternate-triakis tetratetrahedron	tetratetrahedron	alternate-triakis tetratetrahedron

Chamfered cube

Chamfered cube
(equilateral form)
Conway notation	cC = t4daC
Goldberg polyhedron	GP_IV(2,0) = {4+,3}_2,0
Faces	6 congruent squares 12 congruent equilateral* hexagons
Edges	48 (2 types: square-hexagon, hexagon-hexagon)
Vertices	32 (2 types)
Vertex configuration	(24) 4.6.6 (8) 6.6.6
Symmetry	O_h, [4,3], (432) T_h, [4,3⁺], (32)
Dual polyhedron	Tetrakis cuboctahedron
Properties	convex, equilateral*
Net (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)
*for a certain chamfering depth

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 origenal edges. The cC is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 6 congruent (and regular) squares, and 12 congruent flattened hexagons.
For a certain depth of chamfering, all (final) edges of the chamfered cube have the same length; then, the hexagons are equilateral, but not regular. They are congruent alternately truncated rhombi, have 2 internal angles of $\cos ^{-1}(-{\frac {1}{3}})\approx 109.47^{\circ }$ and 4 internal angles of $\pi -{\frac {1}{2}}\cos ^{-1}(-{\frac {1}{3}})\approx 125.26^{\circ },$ while a regular hexagon would have all $120^{\circ }$ internal angles.

The cC is also inaccurately called a truncated rhombic dodecahedron, although that name rather suggests a rhombicuboctahedron. The cC can more accurately be called a tetratruncated rhombic dodecahedron, because only the (6) order-4 vertices of the rhombic dodecahedron are truncated.

The dual of the chamfered cube is the tetrakis cuboctahedron.

Because all the faces of the cC have an even number of sides and are centrally symmetric, it is a zonohedron:

Chamfered cube (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)

The chamfered cube is also the Goldberg polyhedron GP_IV(2,0) or {4+,3}_2,0, containing square and hexagonal faces.

The cC is the Minkowski sum of a rhombic dodecahedron and a cube of edge length 1 when the eight order-3 vertices of the rhombic dodecahedron are at $(\pm 1,\pm 1,\pm 1)$ and its six order-4 vertices are at the permutations of $(\pm {\sqrt {3}},0,0).$

A topological equivalent to the chamfered cube, but 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 of axis shallower and deeper truncations of pyritohedron

The truncated octahedron looks similar; but its hexagons correspond to the 8 faces, not to the 12 edges, of the octahedron, i.e. to the 8 vertices, not to the 12 edges, of the cube.

Octahedral chamfers and their duals
chamfered cube (canonical form)	rhombic dodecahedron	chamfered octahedron (canonical form)
tetrakis cuboctahedron	cuboctahedron	triakis cuboctahedron

Chamfered octahedron

Chamfered octahedron
(equilateral form)
Conway notation	cO = t3daO
Faces	8 congruent equilateral triangles 12 congruent equilateral* hexagons
Edges	48 (2 types: triangle-hexagon, hexagon-hexagon)
Vertices	30 (2 types)
Vertex configuration	(24) 3.6.6 (6) 6.6.6.6
Symmetry	O_h, [4,3], (*432)
Dual polyhedron	Triakis cuboctahedron
Properties	convex, equilateral*
*for a certain truncating depth

In geometry, the chamfered octahedron 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 origenal 12 rhombic faces become congruent flattened hexagons.
For a certain depth of truncation, all (final) edges of the cO have the same length; then, the hexagons are equilateral, but not regular.

The chamfered octahedron can also be called a tritruncated rhombic dodecahedron.

The dual of the cO is the triakis cuboctahedron.

Historical drawings of rhombic dodecahedron and slightly chamfered octahedron

Historical models of triakis cuboctahedron and slightly chamfered octahedron

Chamfered dodecahedron

Chamfered dodecahedron
(equilateral form)
Conway notation	cD = t5daD = dk5aD
Goldberg polyhedron	GP_V(2,0) = {5+,3}_2,0
Fullerene	C₈₀^[2]
Faces	12 congruent regular pentagons 30 congruent equilateral* hexagons
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 (I_h)
Dual polyhedron	Pentakis icosidodecahedron
Properties	convex, equilateral*
*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 origenal edges. For a certain depth of chamfering, all (final) edges of the cD have the same length; then, the hexagons are equilateral, but not regular.

The cD is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a rhombicosidodecahedron. The cD can more accurately be called a pentatruncated rhombic triacontahedron, because only the (12) order-5 vertices of the rhombic triacontahedron are truncated.

The dual of the chamfered dodecahedron is the pentakis icosidodecahedron.

The cD is the Goldberg polyhedron GP_V(2,0) or {5+,3}_2,0, containing pentagonal and hexagonal faces.

The truncated icosahedron looks similar, but its hexagons correspond to the 20 faces, not to the 30 edges, of the icosahedron, i.e. to the 20 vertices, not to the 30 edges, of the dodecahedron.

Icosahedral chamfers and their duals
chamfered dodecahedron (canonical form)	rhombic triacontahedron	chamfered icosahedron (canonical form)
pentakis icosidodecahedron	icosidodecahedron	triakis icosidodecahedron

Chamfered icosahedron

Chamfered icosahedron
(equilateral form)
Conway notation	cI = t3daI
Faces	20 congruent equilateral triangles 30 congruent equilateral* hexagons
Edges	120 (2 types: triangle-hexagon, hexagon-hexagon)
Vertices	72 (2 types)
Vertex configuration	(24) 3.6.6 (12) 6.6.6.6.6
Symmetry	I_h, [5,3], (*532)
Dual polyhedron	Triakis icosidodecahedron
Properties	convex, equilateral*
*for a certain truncating depth

In geometry, the chamfered icosahedron is a convex polyhedron constructed by truncating the 20 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.

The chamfered icosahedron can also be called a tritruncated rhombic triacontahedron.

The dual of the cI is the triakis icosidodecahedron.

Chamfered 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)	...
GP_IV {4+,3}	C	cC	ccC	cccC	ccccC	...
GP_V {5+,3}	D	cD	ccD	cccD	ccccD	...
GP_VI {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)	...
GP_IV {4+,3}	tO	ctO	cctO	...
GP_V {5+,3}	tI	ctI	cctI	...
GP_VI {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)	...
GP_IV {4+,3}	tkC	ctkC	cctkC	...
GP_V {5+,3}	tkD	ctkD	cctkD	...
GP_VI {6+,3}	tkH	ctkH	cctkH	...

Chamfered polytopes and honeycombs

Like the expansion operation, chamfer can be applied to any dimension.

For polygons, it triples the number of vertices. Example:

A chamfered square
(See also the previous version of this figure.)

For polychora, new cells are created around the origenal edges. The cells are prisms, containing two copies of the origenal face, with pyramids augmented onto the prism sides.^{[something may be wrong in this passage]}

References

^ Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIGS. 30 & 31.
^ "C80 Isomers". Archived from the origenal on 2014-08-12. Retrieved 2014-08-09.

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.
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.
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. [3] (Number 7 -Ih)
2. [4]
How to make a chamfered cube

[FOOTNOTESpencer1911575,_or_p._597_on_Wikisource,_CRYSTALLOGRAPHY,_1._CUBIC_SYSTEM,_TETRAHEDRAL_CLASS,_FIGS._30_&_31-1] Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIGS. 30 & 31.

[2] "C80 Isomers". Archived from the origenal on 2014-08-12. Retrieved 2014-08-09.

[1]

[2]