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 ...
, 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 on ...
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 of ...
direct 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 to ...
, 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 trap ...
s, and are a (degenerate) type of
snub polyhedron
In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub poly ...
.
Antiprisms are similar to
prisms, 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 other. ...
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
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...
, the
triangulation of a
set of
points have interested mathematicians since
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
, 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 proo ...
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 of ...
in 1694.
The existence of antiprisms was discussed, and their name was coined by
Johannes Kepler, though it is possible that they were previously known to
Archimedes, as they satisfy the same conditions on faces and on vertices as the
Archimedean solids. 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
Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams.
In the nineteenth centur ...
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, therefor ...
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 th ...
for 4 and 6 points (digonal and trigonal antiprisms, which are 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 ...
); in 1951 by Kurt Schütte
Kurt Schütte (14 October 1909, Salzwedel – 18 August 1998, Munich) was a German mathematician who worked on proof theory and ordinal analysis. The Feferman–Schütte ordinal, which he showed to be the precise ordinal bound for predicativi ...
and Bartel Leendert van der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics.
Biography
Education and early career
Van der Waerden learned advanced mathematics at the University of Amster ...
for 8 points (tetragonal antiprism, which is not a cube).[
The chemical structure of ]binary compounds
In materials chemistry, a binary phase or binary compound is a chemical compound containing two different elements. Some binary phase compounds are molecular, e.g. carbon tetrachloride (CCl4). More typically binary phase refers to extended soli ...
has been remarked to be in the family of antiprisms; especially those of the family of boron hydrides
Boranes is the name given to compounds with the formula BxHy and related anions. Many such boranes are known. Most common are those with 1 to 12 boron atoms. Although they have few practical applications, the boranes exhibit structures and bond ...
(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 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 of polyhedral skeletal electron pair theory
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 ...
.
Rare-earth metals
The rare-earth elements (REE), also called the rare-earth metals or (in context) rare-earth oxides or sometimes the lanthanides (yttrium and scandium are usually included as rare earths), are a set of 17 nearly-indistinguishable lustrous silve ...
such as the lanthanides
The lanthanide () or lanthanoid () series of chemical elements comprises the 15 metallic chemical elements with atomic numbers 57–71, from lanthanum through lutetium. These elements, along with the chemically similar elements scandium and ytt ...
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. The study of crystallography 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 ca ...
to the polygon plane and lying in the polygon centre.
For an antiprism with 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 ...
''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
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...
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
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 ...
''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 th ...
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 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
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...
(truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the ''snub triangular antiprism'' is a lower symmetry form of the regular icosahedron
In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
It has five equilateral triangular faces meeting at each vertex. It ...
.
Symmetry
The symmetry group 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 th ...
, 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 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 b ...
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 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
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 ...
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 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
In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane.
If the sides are equilateral triangles, ...
*Grand antiprism
In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope ...
– 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 not ...
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