Truncated Rhombic Dodecahedron
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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a chamfer or edge-truncation is a topological operator that modifies one
polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
into another. It separates the
faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...
by reducing them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, similar to
expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansi ...
, 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
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...
al face in place of each original
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
. In
Conway polyhedron notation In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the i ...
, ''chamfering'' is represented by the letter "c". A polyhedron with edges will have a chamfered form containing new vertices, new edges, and new hexagonal faces.


Chamfered Platonic solids

Chamfers of five
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s are described in detail below. Each is shown in an
equilateral An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
version where all edges have the same length, and in a canonical version where all edges touch the same
midsphere In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every Edge (geometry), edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedron, uniform polyhedra, including the reg ...
. The shown
dual polyhedra In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...
are dual to the canonical versions.


Chamfered tetrahedron

The chamfered tetrahedron or alternate truncated cube is a
convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
constructed: * by chamfering a
regular tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
: replacing its 6 edges with
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
flattened hexagons; * or by alternately truncating a (regular)
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
: 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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, but not
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
. The dual of the ''chamfered tetrahedron'' is the alternate-triakis tetratetrahedron. The cT is the
Goldberg polyhedron In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990 ...
GPIII(2,0) or 2,0, containing triangular and hexagonal faces.


Chamfered cube

The chamfered cube is constructed as a chamfer of a
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
: 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 In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
with 32 vertices, 48 edges, and 18 faces: 6
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
(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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, but not
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
. They are congruent alternately truncated
rhombi In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
, have 2
internal angle In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
s of \cos^(-\frac) \approx 109.47 ^\circ and 4 internal angles of \pi - \frac \cos^(-\frac) \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 In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for ''truncated cuboctahedral rhombus'', w ...
. The cC can more accurately be called a tetratruncated rhombic dodecahedron, because only the (6) order-4 vertices of the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...
are truncated. The dual of the ''chamfered cube'' is the
tetrakis cuboctahedron In geometry, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron. Its name comes from a topological construction from the cuboctahedron wi ...
. Because all the faces of the cC have an even number of sides and are
centrally symmetric In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...
, it is a
zonohedron In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...
: GPIV(2,0) or 2,0, containing square and hexagonal faces. 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 Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
equivalent to the ''chamfered cube'', but with
pyritohedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...
and rectangular faces, can be constructed by chamfering the axial edges of a
pyritohedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular st ...
. This occurs in
pyrite The mineral pyrite ( ), or iron pyrite, also known as fool's gold, is an iron sulfide with the chemical formula Fe S2 (iron (II) disulfide). Pyrite is the most abundant sulfide mineral. Pyrite's metallic luster and pale brass-yellow hue ...
crystals.


Uses

The DaYan Gem 7 is a
twisty puzzle Twist may refer to: In arts and entertainment Film, television, and stage * ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist'' * ''Twist'' (2021 film), a 2021 modern rendition of ''Olive ...
in the shape of a chamfered cube.


Chamfered octahedron

The chamfered octahedron is a
convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
constructed by truncating the 8 order-3 vertices of the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...
. These truncated vertices become congruent equilateral triangles, and the original 12 rhombic faces become
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
flattened hexagons.
For a certain depth of truncation, all (final) edges of the cO have the same length; then, the hexagons are
equilateral An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, but not
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
. The ''chamfered octahedron'' can also be called a tritruncated rhombic dodecahedron. The dual of the cO is the triakis cuboctahedron.


Chamfered dodecahedron

The chamfered dodecahedron is a
convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
with 80 vertices, 120 edges, and 42 faces: 12 congruent
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
pentagons and 30
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
flattened hexagons.
It is constructed as a chamfer of a
regular dodecahedron A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...
. 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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, but not regular. The cD is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a
rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has a total of 62 faces: 20 regular triangular faces, 30 square f ...
. 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 In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990 ...
GPV(2,0) or 2,0, containing pentagonal and hexagonal faces.


Chamfered icosahedron

The chamfered icosahedron is a
convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
constructed by truncating the 20 order-3 vertices of the
rhombic triacontahedron The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex ...
. The hexagonal faces of the cI can be made
equilateral An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, but not
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
, 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


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 In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three prop ...
sequence: GP(1,0), GP(2,0), GP(4,0), GP(8,0), GP(16,0)... The
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
or
truncated icosahedron In geometry, the truncated icosahedron is a polyhedron that can be constructed by Truncation (geometry), truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as Ball (association football), footballs (or soccer ...
, GP(1,1), creates a Goldberg sequence: GP(1,1), GP(2,2), GP(4,4), GP(8,8)... A truncated
tetrakis hexahedron In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be called a disdyakis hexahedron or hexaki ...
or
pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is a polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. Specifically, the term typically refers to ...
, GP(3,0), creates a Goldberg sequence: GP(3,0), GP(6,0), GP(12,0)...


Chamfered polytopes and honeycombs

Like the
expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansi ...
operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. Example: : 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. omething may be wrong in this passage/span>


See also

*
Conway polyhedron notation In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the i ...
*
Near-miss Johnson solid In geometry, a near-miss Johnson solid is a strictly convex set, convex polyhedron whose face (geometry), faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a John ...
*
Cantellation (geometry) In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...


References


Sources

* * Joseph D. Clinton, ''Clinton’s Equal Central Angle Conjecture'

* * * Antoine Deza, Michel Deza, Viatcheslav Grishukhin, ''Fullerenes and coordination polyhedra versus half-cube embeddings'', 1998
PDF Portable document format (PDF), standardized as ISO 32000, is a file format developed by Adobe Inc., Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, computer hardware, ...
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(p. 72 Fig. 26. Chamfered tetrahedron) * . * {{cite EB1911, wstitle= Crystallography , volume= 07 , pages= 569–591 , last1= Spencer , first1= Leonard James , author-link= Leonard James Spencer


External links


Chamfered Tetrahedron




Livio Zefiro * ttp://www.georgehart.com/virtual-polyhedra/conway_notation.html VRML polyhedral generator(
Conway polyhedron notation In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the i ...
) **
VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...
mode
Chamfered cube


,6fullerene * Fullerene C80 *

(Number 7 -Ih) *


How to make a chamfered cube
Goldberg polyhedra Polyhedra Mathematical notation