View source for Truncation (geometry)

{{Short description|Operation that cuts polytope vertices, creating a new facet in place of each vertex}}
{| class=wikitable align=right width=360
|- valign=top
|[[File:Regular polygon truncation 4 1.svg|130px]]<BR>Truncated square is a regular octagon:<BR>t{4} = {8}<BR>{{CDD|node_1|4|node_1}} = {{CDD|node_1|8|node}}
|[[Image:Truncated hexahedron.png|130px]]<BR>[[Truncated cube]]<BR>t{4,3} or {{CDD|node_1|4|node_1|3|node}}
|[[File:Truncated cubic honeycomb1.jpg|130px]]<BR>[[Truncated cubic honeycomb]]<BR>t{4,3,4} or {{CDD|node_1|4|node_1|3|node|4|node}}
|}
In [[Geometry|geometry]], a '''truncation''' is an operation in any dimension that cuts [[Polytope|polytope]] vertices, creating a new [[Facet (geometry)|facet]] in place of each vertex. The term originates from Kepler's names for the [[Archimedean solid]]s.

== Uniform truncation ==
In general any [[Polyhedron|polyhedron]] (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in [[Conway polyhedron notation]] truncation operation.

A special kind of truncation, usually implied, is a '''uniform truncation''', a truncation operator applied to a [[Regular polyhedron|regular polyhedron]] (or [[Regular polytope|regular polytope]]) which creates a resulting [[Uniform polyhedron|uniform polyhedron]] (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra.

In general all single ringed uniform polytopes have a uniform truncation. For example, the [[Icosidodecahedron|icosidodecahedron]], represented as [[Schläfli symbol]]s r{5,3} or <math>\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}</math>, and Coxeter-Dynkin diagram {{CDD|node|5|node_1|3|node}} or {{CDD|node_1|split1-53|nodes}} has a uniform truncation, the [[Truncated icosidodecahedron|truncated icosidodecahedron]], represented as tr{5,3} or <math>t\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}</math>, {{CDD|node_1|split1-53|nodes_11}}. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the nodes adjacent to the ringed node.

A uniform truncation performed on the regular [[Triangular tiling|triangular tiling]] {3,6} results in the regular [[Hexagonal tiling|hexagonal tiling]] {6,3}.

== Truncation of polygons ==

A truncated n-sided [[Polygon|polygon]] will have 2n sides (edges). A regular polygon uniformly truncated will become another regular polygon: t{n} is {2n}. A complete truncation (or [[Rectification (geometry)|rectification]]), r{3}, is another regular polygon in its [[Dual polygon|dual]] position.

A regular polygon can also be represented by its Coxeter-Dynkin diagram, {{CDD|node_1|n|node}}, and its uniform truncation {{CDD|node_1|n|node_1}}, and its complete truncation {{CDD|node|n|node_1}}. The graph {{CDD|node|n|node}} represents [[Coxeter group]] I<sub>2</sub>(n), with each node representing a mirror, and the edge representing the angle &pi;/''n'' between the mirrors, and a circle is given around one or both mirrors to show which ones are active.
{| class=wikitable
|+ Parametric truncations of a triangle
|- align=center valign=top
|[[File:Regular_truncation_3_0.0.svg|120px]]<BR>{3}<BR>{{CDD|node_1|3|node}}
|[[File:Regular_truncation_3_0.2.svg|120px]]
|[[File:Regular_truncation_3_0.333.svg|120px]]<BR>t{3} = {6}<BR>{{CDD|node_1|3|node_1}}
|[[File:Regular_truncation_3_0.45.svg|120px]]
|[[File:Regular_truncation_3_0.5.svg|120px]]<BR>r{3} = {3}<BR>{{CDD|node|3|node_1}}
|}

[[Star polygon]]s can also be truncated. A truncated [[Pentagram|pentagram]] {5/2} will look like a [[Pentagon|pentagon]], but is actually a double-covered (degenerate) [[Decagon|decagon]] ({10/2}) with two sets of overlapping vertices and edges. A truncated great [[Heptagram|heptagram]] {7/3} gives a [[Tetradecagon|tetradecagram]] {14/3}.

== Uniform truncation in regular polyhedra and tilings and higher ==
[[File:Birectified cube sequence.png|thumb|Truncations of the cube beyond rectification]]
When "truncation" applies to [[Platonic solid|platonic solid]]s or regular tilings, usually "uniform truncation" is implied, which means truncating until the original faces become regular polygons with twice as many sides as the original form.

[[Image:Cube truncation sequence.svg|320px]]

This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full [[Cube|cube]] and a [[Rectification (geometry)|rectified]] cube. The final polyhedron is a [[Cuboctahedron|cuboctahedron]]. The middle image is the uniform [[Truncated cube|truncated cube]]; it is represented by a [[Schläfli symbol]] '''''t'''''{''p'',''q'',...}.

A '''[[Bitruncation|bitruncation]]''' is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. Example: a [[Truncated octahedron|truncated octahedron]] is a bitruncated cube: t{3,4} = 2t{4,3}.

A complete bitruncation, called a '''birectification''', reduces original faces to points. For polyhedra, this becomes the [[Dual polyhedron|dual polyhedron]]. Example: an [[Octahedron|octahedron]] is a birectification of a [[Cube|cube]]: {3,4} = 2r{4,3}.

Another type of truncation, cantellation, cuts edges and vertices, removing the original edges, replacing them with rectangles, removing the original vertices, and replacing them with the faces of the dual of the original regular polyhedra or tiling.

Higher dimensional polytopes have higher truncations. [[Runcination]] cuts faces, edges, and vertices. In 5 dimensions, sterication cuts cells, faces, and edges.

== Edge-truncation==
{{multiple image
 | align = right  | total_width = 400
 | image1 = Polyhedron 6 unchamfered.png |width1=1|height1=1
 | image2 = Polyhedron 6 slightly chamfered.png |width2=1|height2=1
 | image3 = Polyhedron chamfered 6 edeq.png |width3=1|height3=1
 | footer = Truncating the edges of a cube, creating a chamfered cube
}}
'''Edge-truncation''' is a beveling, or [[Chamfer (geometry)|chamfer]] for polyhedra, similar to cantellation, but retaining the original vertices, and replacing edges by hexagons. In 4-polytopes, edge-truncation replaces edges with [[Elongated bipyramid|elongated bipyramid]] cells.
{{-}}

== Alternation or partial truncation ==
{{multiple image
 | align = right  | total_width = 400
 | image1 = Polyhedron great rhombi 6-8 subsolid snub left maxmatch.png
 | image2 = Polyhedron great rhombi 6-8 max.png
 | image3 = Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png
 | footer = A uniform alternation of a [[Truncated cuboctahedron|truncated cuboctahedron]] gives a nonuniform [[Snub cube|snub cube]].
}}

Alternation or partial truncation removes only some of the original vertices.

In '''partial truncation''', or [[Alternation (geometry)|'''alternation''']], half of the vertices and connecting edges are completely removed. The operation applies only to polytopes with even-sided faces. Faces are reduced to half as many sides, and square faces degenerate into edges. For example, the [[Tetrahedron|tetrahedron]] is an alternated cube, h{4,3}.

'''Diminishment''' is a more general term used in reference to [[Johnson solid]]s for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices. For example, the [[Tridiminished icosahedron|tridiminished icosahedron]] starts with a regular [[Icosahedron|icosahedron]] with 3 vertices removed.

Other partial truncations are symmetry-based; for example, the [[Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]].

== Generalized truncations ==
[[File:Types_of_truncation_on_edge.png|400px|thumb|Types of truncations shown on an [[Edge (geometry)|edge]] isolated from a larger polygon or polyhedron with red and blue vertices. The edge reverses direction after complete truncation.]]

The linear truncation process can be generalized by allowing parametric truncations that are negative, or that go beyond the midpoint of the edges, causing self-intersecting star polyhedra, and can parametrically relate to some of the regular star polygons and [[Uniform star polyhedron|uniform star polyhedra]].

* '''Shallow truncation''' - Edges are reduced in length, faces are truncated to have twice as many sides, while new [[Facet (geometry)|facets]] are formed, centered at the old vertices. 
* '''Uniform truncation''' are a special case of this with equal edge lengths. The [[Truncated cube|truncated cube]], t{4,3}, with square faces becoming octagons, with new triangular faces are the vertices. 
* '''Antitruncation''' A reverse ''shallow truncation'', truncated outwards off the original edges, rather than inward. This results in a polytope which looks like the original, but has parts of the dual dangling off its corners, instead of the dual cutting into its own corners.
* '''Complete truncation''' or [[Rectification (geometry)|'''rectification''']] - The limit of a shallow truncation, where edges are reduced to points. The [[Cuboctahedron|cuboctahedron]], r{4,3}, is an example.
* '''Hypertruncation''' A form of truncation that goes past the rectification, inverting the original edges, and causing self-intersections to appear.
* '''Quasitruncation''' A form of truncation that goes even farther than hypertruncation where the inverted edge becomes longer than the original edge. It can be generated from the original polytope by treating all the faces as retrograde, i.e. going backwards round the vertex. For example, quasitruncating the [[Square|square]] gives a regular [[Octagram|octagram]] (t{4,3}={8/3}), and quasitruncating the [[Cube|cube]] gives the uniform [[Stellated truncated hexahedron|stellated truncated hexahedron]], t{4/3,3}.
{| class=wikitable width=640
|+ Truncations on a square
|[[File:Types_of_truncation_on_square4.png|640px]]<BR>Types of truncation on a square, {4}, showing red original edges, and new truncated edges in cyan. A uniform truncated square is a regular octagon, t{4}={8}. A complete truncated square becomes a new square, with a diagonal orientation. Vertices are sequenced around counterclockwise, 1-4, with truncated pairs of vertices as ''a'' and ''b''.
|}
{| class=wikitable width=840
|+ Truncations of the cube
|- align=center valign=top
|bgcolor="#f0d0d0"|[[File:Cube_truncation_3.75.png|120px]]<BR><big><big><big>⇨</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-4)dC t<sub>a</sub>C]
|[[File:Cube_truncation_0.00.png|120px]]<BR>Cube<BR>[[Cube|{4,3}]] [https://levskaya.github.io/polyhedronisme/?recipe=C C]
|bgcolor="#d0d0f0"|[[File:Cube_truncation_0.25.png|120px]]<BR><big><big><big>⇨</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C4)dC tC]
|bgcolor="#d0d0f0"|[[File:Cube_truncation_0.50.png|120px]]<BR>Truncation<BR>[[Truncated cube|t{4,3}]] [https://levskaya.github.io/polyhedronisme/?recipe=tC tC]
|bgcolor="#d0d0f0"|[[File:Cube_truncation_0.75.png|120px]]<BR><big><big><big>⇨</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-.1)dC tC]
|[[File:Cube_truncation_1.00.png|120px]]<BR>Complete truncation<BR>[[Cuboctahedron|r{4,3}]] [https://levskaya.github.io/polyhedronisme/?recipe=aC aC]
|bgcolor="#d0f0f0"|[[File:Cube_truncation_1.25.png|120px]]<BR><big><big><big>⇩</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-.6)dC t<sub>h</sub>C]
|- align=center valign=top
|bgcolor="#f0d0d0"|[[File:Cube_truncation_3.50.png|120px]]<BR>Antitruncation [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-3)dC t<sub>a</sub>C]
|colspan=5|
|bgcolor="#d0f0f0"|[[File:Cube_truncation_1.50.png|120px]]<BR>Hypertruncation [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-1)dC t<sub>h</sub>C]
|- align=center valign=top
|bgcolor="#f0d0d0"|[[File:Cube_truncation_3.25.png|120px]]<BR><big><big><big>⇧</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-1.8)dC t<sub>a</sub>C]
|[[File:Cube_truncation_3.00.png|120px]]<BR>Complete quasitruncation<BR>[https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-1.65)dC a<sub>q</sub>C]
|bgcolor="#d0f0d0"|[[File:Cube_truncation_2.75.png|120px]]<BR><big><big><big>⇦</big></big></big>  
|bgcolor="#d0f0d0"|[[File:Cube_truncation_2.50.png|120px]]<BR>Quasitruncation<BR>t{4/3,3} [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-1.5)dC t<sub>q</sub>C]
|bgcolor="#d0f0d0"|[[File:Cube_truncation_2.25.png|120px]]<BR><big><big><big>⇦</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-1.5)dC t<sub>q</sub>C]
|[[File:Cube_truncation_2.00.png|120px]]<BR>Complete hypertruncation [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-1)dC a<sub>h</sub>C]
|bgcolor="#d0f0f0"|[[File:Cube_truncation_1.75.png|120px]]<BR><big><big><big>⇦</big></big></big> [https://levskaya.github.io/polyhedronisme/?recipe=dk(0%2C-.8)dC t<sub>h</sub>C]
|}

== See also ==
* [[Uniform polyhedron]]
* Uniform 4-polytope
* Bitruncation (geometry)
* [[Rectification (geometry)]]
* [[Alternation (geometry)]]
* [[Conway polyhedron notation]]
* Truncated cone

== References ==
* Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} (pp.&nbsp;145–154 Chapter 8: Truncation)
* [[Biography:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** [[Biography:Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966

== External links ==
* {{mathworld | urlname = Truncation | title = Truncation}}
*{{GlossaryForHyperspace | anchor=Truncation | title=Truncation }}
* [http://www.georgehart.com/virtual-polyhedra/naming.html Polyhedra Names, truncation]


[[Category:Polytopes]]
[[Category:Truncated tilings| ]]

{{Sourceattribution|Truncation (geometry)}}