In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
The Catalan sol ...
. Its 24 faces are
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 mod ...
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. ...
. The deltoidal icositetrahedron, whose
dual is the (uniform)
rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at eac ...
, is tightly related to the
pseudo-deltoidal icositetrahedron
The pseudo-deltoidal icositetrahedron is a convex polyhedron with congruent kites as its faces. It is the dual of the elongated square gyrobicupola, also known as the pseudorhombicuboctahedron.
As the pseudorhombicuboctahedron is tightly relat ...
, whose dual is the
pseudorhombicuboctahedron
In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same patt ...
; but the actual and pseudo-d.i. are not to be confused with each other.
Cartesian coordinates
In the image above, the long body diagonals are those between opposite red vertices and between opposite blue vertices, and the short body diagonals are those between opposite yellow vertices.
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of the deltoidal icositetrahedron centered at the origin and with long body diagonal length
are:
*red vertices (lying in
-fold symmetry axes):
:
*blue vertices (lying in
-fold symmetry axes):
:
*yellow vertices (lying in
-fold symmetry axes):
:
For example, the point with coordinates
is the intersection of the plane with equation
and of the line with system of equations
A deltoidal icositetrahedron has three regular-octagon equators, lying in three orthogonal planes.
Dimensions and angles
Dimensions
The deltoidal icositetrahedron with long body diagonal length
has:
*short body diagonal length:
::
*long edge length:
In this MathWorld entry, the small rhombicuboctahedron has edge length so this s.r.c.o.h. has circumradius and midradius so this s.r.c.o.h.'s dual with respect to their common midsphere is the deltoidal icositetrahedron with inradius ××
::
*short edge length:
::
*inradius:
::
is the distance from the center to any face plane; it may be calculated by
normalizing the equation of plane
above, replacing
with
and taking the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the result.
A deltoidal icositetrahedron has its long and short edges in the ratio:
:
The deltoidal icositetrahedron with short edge length
has:
*area:
::
*volume:
::
Angles
For a deltoidal icositetrahedron, each kite face has:
*three equal acute angles, with value:
::
*one obtuse angle (between the short edges), with value:
::
Occurrences in nature and culture
The deltoidal icositetrahedron is a
crystal habit
In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. The habit of a crystal is dependent on its crystallographic form and growth conditions, which generally creates irregularities due to l ...
often formed by the mineral
analcime
Analcime (; ) or analcite is a white, gray, or colorless tectosilicate mineral. Analcime consists of hydrated sodium aluminium silicate in cubic crystalline form. Its chemical formula is Na Al Si2 O6· H2O. Minor amounts of potassium and calciu ...
and occasionally
garnet
Garnets () are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives.
All species of garnets possess similar physical properties and crystal forms, but differ in chemical composition. The different s ...
. The shape is often called a trapezohedron in mineral contexts, although in
solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid fig ...
the name
trapezohedron
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 high ...
has another meaning.
Orthogonal projections
The ''deltoidal icositetrahedron'' has three symmetry positions, all centered on vertices:
Related polyhedra
The deltoidal icositetrahedron's projection onto a
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 r ...
divides its squares into quadrants. The projection onto a regular
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
divides its equilateral triangles into kite faces. In
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 op ...
this represents an ''ortho'' operation to a cube or octahedron.
The deltoidal icositetrahedron
(dual of the small rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square (geometry), square, and twelve rectangle, rectangular faces. There are 24 identical vertices, with one triangle, one square, and ...
) is tightly related to the
disdyakis dodecahedron
In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is fa ...
(dual of the great rhombicuboctahedron). The main difference is that the latter also has edges between the vertices on 3- and 4-fold symmetry axes
(between yellow and red vertices in the images below).
Dyakis dodecahedron
A variant 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 ...
is called a dyakis dodecahedron or diploid. It is common in
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
.
A dyakis dodecahedron can be created by enlarging 24 of the 48 faces of a disdyakis dodecahedron. A
tetartoid
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron.
[Both is indicated in the two crystal models in the top right corner of this photo. A visual demonstration can be seen ]here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
and here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
.
Stellation
The
great triakis octahedron is a stellation of the deltoidal icositetrahedron.
Related polyhedra and tilings
The deltoidal icositetrahedron is a member of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
When projected onto a sphere (see right), it can be seen that the edges make up
the edges of a cube and regular octahedron arranged in their dual positions. It can also be seen that the 3- and 4-fold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rhombicuboctahedron for a dual, since the centers of the square and triangle faces of a rhombicuboctahedron are at different distances from its center.
This polyhedron is a term of a sequence of topologically related deltoidal polyhedra with face configuration V3.4.''n''.4; this sequence continues with tilings of the
Euclidean and
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
planes. These
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 congruent ...
figures have (*''n''32) reflectional
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
.
See also
*
Deltoidal hexecontahedron
In geometry, a deltoidal hexecontahedron (also sometimes called a ''trapezoidal hexecontahedron'', a ''strombic hexecontahedron'', or a ''tetragonal hexacontahedron'') is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, ...
*
Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube.
* "
The Haunter of the Dark
"The Haunter of the Dark" is a horror short story by American author H. P. Lovecraft, written between 5–9 November 1935 and published in the December 1936 edition of ''Weird Tales'' (Vol. 28, No. 5, p. 538–53). It was the last written ...
", a story by H.P. Lovecraft, whose plot involves this figure.
*
Pseudo-deltoidal icositetrahedron
The pseudo-deltoidal icositetrahedron is a convex polyhedron with congruent kites as its faces. It is the dual of the elongated square gyrobicupola, also known as the pseudorhombicuboctahedron.
As the pseudorhombicuboctahedron is tightly relat ...
References
* (Section 3-9)
* (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron)
*''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)
External links
*
Deltoidal (Trapezoidal) Icositetrahedron– Interactive Polyhedron model
{{Polyhedron navigator
Catalan solids