In
Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a
polytope by marking the midpoints of all its
edges, and cutting off its
vertices at those points.
The resulting polytope will be bounded by
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
facets and the rectified facets of the original polytope.
A rectification operator is sometimes denoted by the letter with a
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
. For example, is the rectified
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
, also called a
cuboctahedron, and also represented as
. And a rectified cuboctahedron is a
rhombicuboctahedron, and also represented as
.
Conway polyhedron notation uses for ambo as this operator. In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
this operation creates a
medial graph.
The rectification of any regular
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
polyhedron or tiling will result in another regular polyhedron or tiling with a
tiling order
Tiling may refer to:
*The physical act of laying tiles
*Tessellations
Computing
*The compiler optimization of loop tiling
*Tiled rendering, the process of subdividing an image by regular grid
*Tiling window manager
People
*Heinrich Sylvester Th ...
of 4, for example the
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
becoming an
octahedron As a special case, a
square tiling will turn into another square tiling under a rectification operation.
Example of rectification as a final truncation to an edge
Rectification is the final point of a truncation process. For example, on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:
Higher degree rectifications
Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the
dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.
Example of birectification as a final truncation to a face
This sequence shows a ''birectified cube'' as the final sequence from a cube to the dual where the original faces are truncated down to a single point:
:
In polygons
The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
In polyhedra and plane tilings
Each
platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
and its
dual have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
# The rectified
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
, whose dual is the tetrahedron, is the ''tetratetrahedron'', better known as the
octahedron.
# The rectified
octahedron, whose dual is the
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
, is the
cuboctahedron.
# The rectified
icosahedron, whose dual is the
dodecahedron, is the
icosidodecahedron.
# A rectified
square tiling is a
square tiling.
# A rectified
triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
or
hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathemati ...
is a
trihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...
.
Examples
In nonregular polyhedra
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a
polyhedral graph as its
1-skeleton, and from that graph one may form the
medial graph by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by
Steinitz's theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar gra ...
it can be represented as a polyhedron.
The
Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, (rectifying a rectification) is Conway's
expand operation, e, which is the same as Johnson's
cantellation operation, t
0,2 generated from regular polyhedral and tilings.
In 4-polytopes and 3D honeycomb tessellations
Each
Convex regular 4-polytope has a rectified form as a
uniform 4-polytope.
A regular 4-polytope has cells . Its rectification will have two cell types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex.
A rectified is not the same as a rectified , however. A further truncation, called
bitruncation
In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves.
Bitruncated regular pol ...
, is symmetric between a 4-polytope and its dual. See
Uniform 4-polytope#Geometric derivations.
Examples
Degrees of rectification
A first rectification truncates edges down to points. If a polytope is
regular, this form is represented by an extended
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
notation t
1 or r.
A second rectification, or birectification, truncates
faces down to points. If regular it has notation t
2 or 2r. For
polyhedra, a birectification creates a
dual polyhedron
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 oth ...
.
Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates ''n-faces'' to points.
If an n-polytope is (n-1)-rectified, its
facets are reduced to points and the polytope becomes its
dual.
Notations and facets
There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of
facets for each.
Regular
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s
Facets are edges, represented as .
Regular polyhedra and tilings
Facets are regular polygons.
Regular Uniform 4-polytopes and honeycombs
Facets are regular or rectified polyhedra.
Regular 5-polytopes and 4-space honeycombs
Facets are regular or rectified 4-polytopes.
See also
*
Dual polytope
*
Quasiregular polyhedron
*
List of regular polytopes
*
Truncation (geometry)
*
Conway polyhedron notation
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation)
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
**
N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...
, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26)
External links
*
{{Polyhedron_operators
Polytopes