Chamfer (geometry)#Chamfered regular tilings

Chamfers of five Platonic solids are described in detail below. 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. The shown dual polyhedra are dual to the canonical versions.

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.

File:Polyhedron truncated 4b max.png 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.]]

File:EB1911 Crystallography Figs. 30 & 31.jpg

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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 original 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 $\backslash cos^\{-1\}(-\backslash frac\{1\}\{3\})\; \backslash approx\; 109.47\; ^\backslash circ$ and 4 internal angles of $\backslash pi\; -\; \backslash frac\{1\}\{2\}\; \backslash cos^\{-1\}(-\backslash frac\{1\}\{3\})\; \backslash approx\; 125.26\; ^\backslash circ,$ while a regular hexagon would have all $120\; ^\backslash circ$ 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:

File:Truncated rhombic dodecahedron.png

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

{{nowrap|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 $(\backslash pm\; 1,\; \backslash pm\; 1,\; \backslash pm\; 1)$ and its six order-4 vertices are at the permutations of $(\backslash pm\; \backslash sqrt\; 3,\; 0,\; 0).$

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.

The DaYan Gem 7 is a twisty puzzle in the shape of a chamfered cube. {{Cite web |title=TwistyPuzzles.com > Museum > Show Museum Item |url=https://twistypuzzles.com/app/museum/museum_showitem.php?pkey=4308 |access-date=2025-02-09 |website=twistypuzzles.com}}

File:Polyhedron truncated 8 max.png 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.]]

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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 original 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.

File:Brockhaus and Efron Encyclopedic Dictionary b48 862-4.jpg

{{multiple image

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| image1 = Modell, Kristallform Würfel-Deltoidikositetraeder -Krantz 426-.jpg

| image2 = Modell, Kristallform Oktaeder-Rhombendodekaeder -Krantz 432-.jpg

| footer = Historical models of triakis cuboctahedron and slightly chamfered octahedron

}}

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{{main|Chamfered dodecahedron}}

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 original 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.

File:Polyhedron truncated 20 max.png 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.]]

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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.

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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)...

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)...

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

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

:File:Chamfered square.png

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.^{[something may be wrong in this passage]}

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

{{reflist}}

• {{cite journal | title=A class of multi-symmetric polyhedra | first=Michael | last=Goldberg | journal=Tohoku Mathematical Journal | year=1937 | volume=43 | pages=104–108 | url=https://www.jstage.jst.go.jp/article/tmj1911/43/0/43_0_104/_article}}
• Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture [http://www.domerama.com/wp-content/uploads/2012/08/ClintonEqualEdge.pdf]
• {{cite book | first=George | last=Hart | authorlink=George W. Hart | chapter=Goldberg Polyhedra | title=Shaping Space | url=https://archive.org/details/shapingspaceexpl00sene | url-access=limited | edition=2nd | editor-first=Marjorie | editor-last=Senechal | editor-link=Marjorie Senechal | pages=[https://archive.org/details/shapingspaceexpl00sene/page/n126 125]–138 | publisher=Springer | year=2012 | doi=10.1007/978-0-387-92714-5_9 | isbn=978-0-387-92713-8 }}
• {{cite web | title=Mathematical Impressions: Goldberg Polyhedra | first=George | last=Hart | authorlink=George W. Hart | date=June 18, 2013 | url=https://www.simonsfoundation.org/multimedia/mathematical-impressions-goldberg-polyhedra/ | publisher=Simons Science News }}
• Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF [http://www.cas.mcmaster.ca/~deza/dm1998.pdf] (p. 72 Fig. 26. Chamfered tetrahedron)
• {{citation

|last1=Deza

|first1=A.

|last2=Deza

|first2=M.

|author2-link=Michel Deza

|last3=Grishukhin

|first3=V.

|title=Fullerenes and coordination polyhedra versus half-cube embeddings

|journal=Discrete Mathematics

|volume=192

|issue=1

|year=1998

|pages=41–80

|doi=10.1016/S0012-365X(98)00065-X

|doi-access=

}}.

• {{cite EB1911|wstitle= Crystallography |volume= 07 |pages= 569–591 |last1= Spencer |first1= Leonard James |author-link= Leonard James Spencer}}

• [http://dmccooey.com/polyhedra/ChamferedTetrahedron1.html Chamfered Tetrahedron]
• [http://dmccooey.com/polyhedra/Chamfer.html Chamfered Solids]
• [http://www.mi.sanu.ac.rs/vismath/zefirosept2011/_truncation_Archimedean_polyhedra.htm Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra] Livio Zefiro
• [http://www.georgehart.com/virtual-polyhedra/conway_notation.html VRML polyhedral generator] (Conway polyhedron notation)
• VRML model [http://www.georgehart.com/virtual-polyhedra/vrml/zono-7_from_cube.wrl Chamfered cube]
• [http://www.chem.qmul.ac.uk/iupac/fullerene2/327.html 3.2.7. Systematic numbering for (C80-Ih)] [5,6] fullerene
• Fullerene C80
• # [https://web.archive.org/web/20110718135554/http://www.cochem2.tutkie.tut.ac.jp/Fuller/fsl/c80.html] (Number 7 -Ih)
• # [http://www.chem.qmul.ac.uk/iupac/fullerene2/327.html]
• [http://video.fc2.com/content/20141016xYf9JQFy How to make a chamfered cube]

Category:Goldberg polyhedra

Category:Polyhedra

Category:Mathematical notation