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 enneahedron (or nonahedron) 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 th ...

with nine

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

. There are 2606 types of

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.

Examples

The most familiar enneahedra are the

octagonal In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

and the

heptagonal prism In geometry, the heptagonal prism is a prism with heptagonal base. This polyhedron has 9 faces, 21 edges, and 14 vertices.. Area The area of a right heptagonal prism with height h and with a side length of L and apothem a_p is given by: :A = 7 ...

. The heptagonal prism is a

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

, with two regular heptagon faces and seven square faces. The octagonal pyramid has eight isosceles triangular faces around a regular octagonal base. Two more enneahedra are also found among the

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 ...

s: the

elongated square pyramid In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically ( ...

and the

elongated triangular bipyramid In geometry, the elongated triangular bipyramid (or dipyramid) or triakis triangular prism is one of the Johnson solids (), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangu ...

. The three-dimensional

associahedron In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of ...

, with six pentagonal faces and three quadrilateral faces, is an enneahedron. Five Johnson solids have enneahedral duals: the

triangular cupola In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron. Formulae The following formulae for the volume (V), the surface area (A) and the height (H) can be used if all faces are regular, ...

gyroelongated square pyramid In geometry, the gyroelongated square pyramid is one of the Johnson solids (). As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base. ...

, self-dual

triaugmented triangular prism The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same ...

(whose dual is the associahedron), and

tridiminished icosahedron In geometry, the tridiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing three pentagonal pyramids () from a regular icosahedron, which replaces three sets of five triangular faces fro ...

. Another enneahedron is the

diminished trapezohedron In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle ...

with a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

base, and 4

kite 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. ...

and 4

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...

faces. The

Herschel graph In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Ha ...

represents the vertices and edges of the Herschel enneahedron above, with all of its faces quadrilaterals. It is the simplest polyhedron without a

Hamiltonian cycle In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

, the only enneahedron in which all faces have the same number of edges, and one of only three bipartite enneahedra. Isospectral enneahedra

The smallest pair of

isospectral In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectra ...

polyhedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-con ...

s are enneahedra with eight vertices each.

Space-filling enneahedra

Slicing a

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 ...

in half through the long diagonals of four of its faces results in a self-dual enneahedron, the square

, with one large square face, four rhombus faces, and four isosceles triangle faces. Like the rhombic dodecahedron itself, this shape can be used to

tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

three-dimensional space. An elongated form of this shape that still tiles space can be seen atop the rear side towers of the 12th-century Romanesque

Basilica of Our Lady (Maastricht) The Basilica of Our Lady ( nl, Basiliek van Onze-Lieve-Vrouw; li, Slevrouwe ) is a Romanesque church in the historic center of Maastricht, Netherlands. The church is dedicated to Our Lady of the Assumption ( nl, Onze-Lieve-Vrouw-Tenhemelopnemin ...

. The towers themselves, with their four pentagonal sides, four roof facets, and square base, form another space-filling enneahedron. More generally, found at least 40 topologically distinct space-filling enneahedra.

Topologically distinct enneahedra

There are 2606 topologically distinct ''convex'' enneahedra, excluding mirror images. These can be divided into subsets of 8, 74, 296, 633, 768, 558, 219, 50, with 7 to 14 vertices respectively. A table of these numbers, together with a detailed description of the nine-vertex enneahedra, was first published in the 1870s by

Thomas Kirkman Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, a ...

References

External links

Enumeration of Polyhedra
by Steven Dutch * {{Polyhedra Polyhedra