In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Catalan solid, or Archimedean dual, is 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 other. ...
to an
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
. There are 13 Catalan solids. They are named for the
Belgian
Belgian may refer to:
* Something of, or related to, Belgium
* Belgians, people from Belgium or of Belgian descent
* Languages of Belgium, languages spoken in Belgium, such as Dutch, French, and German
*Ancient Belgian language, an extinct languag ...
mathematician
Eugène Catalan, who first described them in 1865.
The Catalan solids are all convex. They are
face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...
but not
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike
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 edges c ...
s and
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s, the faces of Catalan solids are ''not''
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s. However, the
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s of Catalan solids are regular, and they have constant
dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
s. Being face-transitive, Catalan solids are
isohedra.
Additionally, two of the Catalan solids are
edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...
: the
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
and the
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Ca ...
. These are the
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...
of the two
quasi-regular Archimedean solids.
Just as
prisms and
antiprisms
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass ...
are generally not considered Archimedean solids, so
bipyramids and
trapezohedra
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
are generally not considered Catalan solids, despite being face-transitive.
Two of the Catalan solids are
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
: the
pentagonal icositetrahedron and the
pentagonal hexecontahedron
In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catala ...
, dual to the chiral
snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...
and
snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most ...
. These each come in two
enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids.
List of Catalan solids and their duals
All Catalan solids to scale with
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
edges superimposed. Sorted by
midradius in descending order:
All
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
of Catalan solids, same scale as above:
Symmetry
The Catalan solids, along with their dual
Archimedean solids
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are compose ...
, can be grouped in those with tetrahedral, octahedral and icosahedral symmetry.
For both octahedral and icosahedral symmetry there are six forms. The only Catalan solid with genuine tetrahedral symmetry is the
triakis tetrahedron
In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.
The triakis tetrahedron can be se ...
(dual of the
truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncation (geometry), truncating all 4 vertices of ...
). The
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
and
tetrakis hexahedron have octahedral symmetry, but they can be colored to have only tetrahedral symmetry.
Rectification and snub also exist with tetrahedral symmetry, but they are
Platonic
Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply acceptance of that philosophy as a whole. It ...
instead of Archimedean, so their duals are Platonic instead of Catalan. (They are shown with brown background in the table below.)
Geometry
All
dihedral angles
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
of a Catalan solid are equal. Denoting their value by
, and denoting the face angle at the vertices where
faces meet by
, we have
:
.
This can be used to compute
and
,
, ... , from
,
... only.
Triangular faces
Of the 13 Catalan solids, 7 have triangular faces. These are of the form Vp.q.r, where p, q and r take their values among 3, 4, 5, 6, 8 and 10. The angles
,
and
can be computed in the following way. Put
,
,
and put
:
.
Then
:
,
:
.
For
and
the expressions are similar of course. The
dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
can be computed from
:
.
Applying this, for example, to the
disdyakis triacontahedron (
,
and
, hence
,
and
, where
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
) gives
and
.
Quadrilateral faces
Of the 13 Catalan solids, 4 have quadrilateral faces. These are of the form Vp.q.p.r, where p, q and r take their values among 3, 4, and 5. The angle
can be computed by the following formula:
:
.
From this,
,
and the dihedral angle can be easily computed. Alternatively, put
,
,
. Then
and
can be found by applying the formulas for the triangular case. The angle
can be computed similarly of course.
The faces are
kites
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
, or, if
,
rhombi
In plane Euclidean geometry, a rhombus (plural 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 ...
.
Applying this, for example, to the
deltoidal icositetrahedron
In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...
(
,
and
), we get
.
Pentagonal faces
Of the 13 Catalan solids, 2 have pentagonal faces. These are of the form Vp.p.p.p.q, where p=3, and q=4 or 5. The angle
can be computed by solving a degree three equation:
:
.
Metric properties
For a Catalan solid
let
be the dual with respect to the
midsphere
In geometry, the midsphere or intersphere of a polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex poly ...
of
. Then
is an Archimedean solid with the same midsphere. Denote the length of the edges of
by
. Let
be the
inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
of the faces of
,
the midradius of
and
,
the inradius of
, and
the circumradius of
. Then these quantities can be expressed in
and the dihedral angle
as follows:
:
,
:
,
:
,
:
.
These quantities are related by
,
and
.
As an example, let
be a cuboctahedron with edge length
. Then
is a rhombic dodecahedron. Applying the formula for quadrilateral faces with
and
gives
, hence
,
,
,
.
All vertices of
of type
lie on a sphere with radius
given by
:
,
and similarly for
.
Dually, there is a sphere which touches all faces of
which are regular
-gons (and similarly for
) in their center. The radius
of this sphere is given by
:
.
These two radii are related by
. Continuing the above example:
and
, which gives
,
,
and
.
If
is a vertex of
of type
,
an edge of
starting at
, and
the point where the edge
touches the midsphere of
, denote the distance
by
. Then the edges of
joining vertices of type
and type
have length
. These quantities can be computed by
:
,
and similarly for
. Continuing the above example:
,
,
,
, so the edges of the rhombic dodecahedron have length
.
The dihedral angles
between
-gonal and
-gonal faces of
satisfy
:
.
Finishing the rhombic dodecahedron example, the dihedral angle
of the cuboctahedron is given by
.
Construction
The face of any Catalan polyhedron may be obtained from the
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
of the dual
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
using the
Dorman Luke construction
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. ...
.
[, p. 117; , p. 30.]
Application to other solids
All of the formulae of this section apply to the
Platonic solids
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 edges c ...
, and
bipyramids
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices.
The "-gonal" in the name of a bipyramid does no ...
and
trapezohedra
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
with equal dihedral angles as well, because they can be derived from the constant dihedral angle property only. For the
pentagonal trapezohedron
In geometry, a pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites.
It can ...
, for example, with faces V3.3.5.3, we get
, or
. This is not surprising: it is possible to cut off both apexes in such a way as to obtain a
regular dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
.
See also
*
List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...
Shows dual uniform polygonal tilings similar to the Catalan solids
*
Conway polyhedron notation
In geometry, 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 idea of using o ...
A notational construction process
*
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
*
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
Notes
References
*
Eugène Catalan ''Mémoire sur la Théorie des Polyèdres.'' J. l'École Polytechnique (Paris) 41, 1-71, 1865.
*.
*.
*
Alan Holden ''Shapes, Space, and Symmetry''. New York: Dover, 1991.
* (The thirteen semiregular convex polyhedra and their duals)
* (Section 3-9)
* Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
External links
*
*
Catalan Solids– at Visual Polyhedra
– at Virtual Reality Polyhedra
in Java
Download link for Catalan's original 1865 publication– with beautiful figures, PDF format
{{DEFAULTSORT:Catalan Solid
*
Polyhedra