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 ...
, an antiprism or is 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 polyhedron is the convex hull of finitely many points, not all o ...
composed of two
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
direct
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), ...
copies (not mirror images) of an
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 ...
, connected by an alternating band of
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s. They are represented by the
Conway notation .
Antiprisms are a subclass of
prismatoid
In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or ...
s, and are a (degenerate) type of
snub polyhedron.
Antiprisms are similar to
prisms
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are triangles, rather than
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
s.
The
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 ...
of an -gonal antiprism is an -gonal
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 hi ...
.
History
At the intersection of modern-day
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 ...
and
coding theory, the
triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle me ...
of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Points ...
have interested mathematicians since
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
, who fruitlessly sought a
mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...
of the
kissing number problem
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...
in 1694.
The existence of antiprisms was discussed, and their name was coined by
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
, though it is possible that they were previously known to
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
, as they satisfy the same conditions on faces and on vertices as the
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 compose ...
s. According to Ericson and Zinoviev,
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
wrote at length on the topic,
[ and was among the first to apply the mathematics of Victor Schlegel to this field.
Knowledge in this field is "quite incomplete" and "was obtained fairly recently", i.e. in the 20th century. For example, as of 2001 it had been proven for only a limited number of non-trivial cases that the -gonal antiprism is the mathematically optimal arrangement of points in the sense of maximizing the minimum ]Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
between any two points on the set: in 1943 by László Fejes Tóth
László Fejes Tóth ( hu, Fejes Tóth László, 12 March 1915 – 17 March 2005) was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on ...
for 4 and 6 points (digonal and trigonal antiprisms, which are Platonic solids); in 1951 by Kurt Schütte and Bartel Leendert van der Waerden for 8 points (tetragonal antiprism, which is not a cube).[
The chemical structure of binary compounds has been remarked to be in the family of antiprisms;] especially those of the family of boron hydrides (in 1975) and carboranes because they are isoelectronic
Isoelectronicity is a phenomenon observed when two or more molecules have the same structure (positions and connectivities among atoms) and the same electronic configurations, but differ by what specific elements are at certain locations in th ...
. This is a mathematically real conclusion reached by studies of X-ray diffraction
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
patterns,[“Boron Hydride Chemistry” (E. L. Muetterties, ed.), Academic Press, New York] and stems from the 1971 work of Kenneth Wade, the nominative source for Wade's rules
In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by ...
of polyhedral skeletal electron pair theory.
Rare-earth metals such as the lanthanides form antiprismatic compounds with some of the halides
In chemistry, a halide (rarely halogenide) is a binary chemical compound, of which one part is a halogen atom and the other part is an element or radical that is less electronegative (or more electropositive) than the halogen, to make a fluor ...
or some of the iodides
An iodide ion is the ion I−. Compounds with iodine in formal oxidation state −1 are called iodides. In everyday life, iodide is most commonly encountered as a component of iodized salt, which many governments mandate. Worldwide, iodine de ...
. The study of 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 ...
is useful here. Some lanthanides, when arranged in peculiar antiprismatic structures with chlorine
Chlorine is a chemical element with the symbol Cl and atomic number 17. The second-lightest of the halogens, it appears between fluorine and bromine in the periodic table and its properties are mostly intermediate between them. Chlorine i ...
and water, can form molecule-based magnets.
Right antiprism
For an antiprism with regular -gon bases, one usually considers the case where these two copies are twisted by an angle of degrees.
The axis of a regular polygon is the line perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to the polygon plane and lying in the polygon centre.
For an antiprism with congruent ''regular'' -gon bases, twisted by an angle of degrees, more regularity is obtained if the bases have the same axis: are ''coaxial
In geometry, coaxial means that several three-dimensional linear or planar forms share a common axis. The two-dimensional analog is ''concentric''.
Common examples:
A coaxial cable is a three-dimensional linear structure. It has a wire condu ...
''; i.e. (for non- coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its side faces are ''isosceles'' triangles.
Uniform antiprism
A uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
-antiprism has two congruent ''regular'' -gons as base faces, and ''equilateral'' triangles as side faces.
Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For , we have the regular 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 ...
as a ''digonal antiprism'' (degenerate antiprism); for , the 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 ...
as a ''triangular antiprism'' (non-degenerate antiprism).
Schlegel diagrams
Cartesian coordinates
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 ...
for the vertices of a right
Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical ...
-antiprism (i.e. with regular -gon bases and isosceles triangle side faces) are:
:
where ;
if the -antiprism is uniform (i.e. if the triangles are equilateral), then:
:
Volume and surface area
Let be the edge-length of a uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
-gonal antiprism; then the volume is:
:
and the surface area is:
:
Related polyhedra
There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are 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 each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...
s, and the ''snub triangular antiprism'' is a lower symmetry form of the regular icosahedron.
Symmetry
The symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of a right
Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical ...
-antiprism (i.e. with regular bases and isosceles side faces) is of order , except in the cases of:
*: the regular 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 ...
, which has the larger symmetry group of order , which has three versions of as subgroups;
*: the 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 ...
, which has the larger symmetry group of order , which has four versions of as subgroups.
The symmetry group contains inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
is odd.
The rotation group is of order , except in the cases of:
*: the regular tetrahedron, which has the larger rotation group of order , which has three versions of as subgroups;
*: the regular octahedron, which has the larger rotation group of order , which has four versions of as subgroups.
Note: The right -antiprisms have congruent regular -gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform -antiprism, for .
Star antiprism
Uniform star antiprisms are named by their star polygon
In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operatio ...
bases, , and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting 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 ...
s, and are denoted by "inverted" fractions: ''p''/(''p'' – ''q'') instead of ''p''/''q''; example: 5/3 instead of 5/2.
A right star antiprism has two congruent coaxial
In geometry, coaxial means that several three-dimensional linear or planar forms share a common axis. The two-dimensional analog is ''concentric''.
Common examples:
A coaxial cable is a three-dimensional linear structure. It has a wire condu ...
regular ''convex'' or ''star'' polygon base faces, and 2''n'' isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
side faces.
Any star antiprism with ''regular'' convex or star polygon bases can be made a ''right'' star antiprism (by translating and/or twisting one of its bases, if necessary).
In the retrograde forms but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:
*Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron.
*Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.
Also, star antiprism compounds with regular star ''p''/''q''-gon bases can be constructed if ''p'' and ''q'' have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.
See also
* Apeirogonal antiprism
* Grand antiprism – a four-dimensional polytope
*One World Trade Center
One World Trade Center (also known as One World Trade, One WTC, and formerly Freedom Tower) is the main building of the rebuilt World Trade Center complex in Lower Manhattan, New York City. Designed by David Childs of Skidmore, Owings & Mer ...
, a building consisting primarily of an elongated square antiprism
*Skew polygon
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do ...
References
Bibliography
* Chapter 2: Archimedean polyhedra, prisms and antiprisms
External links
*
*
Nonconvex Prisms and Antiprisms
Paper models of prisms and antiprisms
{{Polyhedron navigator
Uniform polyhedra
Prismatoid polyhedra
Topological graph theory
Graph drawing
Coxeter groups
Elementary geometry
Polyhedra
Polytopes
Triangulation (geometry)
Knot invariants