![Hyperbolic Octahedron](https://upload.wikimedia.org/wikipedia/commons/8/85/Hyperbolic_Octahedron.jpg)
In three-dimensional
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
, an ideal polyhedron is a
convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
all of whose
vertices are
ideal point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.
Given a line ''l'' and a point ''P'' not on ''l'', right- and left- limiting parallels to ''l'' through ''P' ...
s, points "at infinity" rather than interior to three-dimensional
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...
. It can be defined as the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of a finite set of ideal points. An ideal polyhedron has ideal polygons as its
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 ...
, meeting along lines of the hyperbolic space.
The
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 have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform
hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra – a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a
circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
. Using
linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
, it is possible to test whether a given polyhedron has an ideal version, in
polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
.
Every two ideal polyhedra with the same number of vertices have the same surface area, and it is possible to calculate the volume of an ideal polyhedron using the
Lobachevsky function. The surface of an ideal polyhedron forms a
hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, res ...
, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique ideal polyhedron.
Examples and counterexamples
An ideal polyhedron can be constructed as the convex hull of a finite set of ideal points of hyperbolic space, whenever the points do not all lie on a single plane. The resulting shape is the intersection of all closed
half-spaces that have the given ideal points as limit points. Alternatively, any Euclidean convex polyhedron that has a
circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
can be reinterpreted as an ideal polyhedron by interpreting the interior of the sphere as a
Klein model
Klein may refer to:
People
*Klein (surname)
*Klein (musician)
Places
*Klein (crater), a lunar feature
*Klein, Montana, United States
*Klein, Texas, United States
*Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm
*Klein River, a river ...
for hyperbolic space. In the Klein model, every Euclidean polyhedron enclosed by the sphere represents a hyperbolic polyhedron, and every Euclidean polyhedron with its vertices on the sphere represents an ideal hyperbolic polyhedron.
Every
isogonal convex polyhedron (one with symmetries taking every vertex to every other vertex) can be represented as an ideal polyhedron, in a way that respects its symmetries, because it has a circumscribed sphere centered at the center of symmetry of the polyhedron. In particular, this implies that the
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 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 composed ...
s all have ideal forms. However, another highly symmetric class of polyhedra, the
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 ...
s, do not all have ideal forms. The Catalan solids are the dual polyhedra to the Archimedean solids, and have symmetries taking any face to any other face. Catalan solids that cannot be ideal include 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
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 see ...
.
Removing certain triples of vertices from the triakis tetrahedron separates the remaining vertices into multiple connected components. When no such three-vertex separation exists, a polyhedron is said to be
4-connected. Every 4-connected polyhedron has a representation as an ideal polyhedron; for instance this is true of the
tetrakis hexahedron, another Catalan solid.
Truncating a single vertex from a cube produces a
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
polyhedron (one with three edges per vertex) that cannot be realized as an ideal polyhedron: by
Miquel's six circles theorem, if seven of the eight vertices of a cube are ideal, the eighth vertex is also ideal, and so the vertices created by truncating it cannot be ideal. There also exist polyhedra with four edges per vertex that cannot be realized as ideal polyhedra. If a
simplicial
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
polyhedron (one with all faces triangles) has all vertex degrees between four and six (inclusive) then it has an ideal representation, but the triakis tetrahedron is simplicial and non-ideal, and the 4-regular non-ideal example above shows that for non-simplicial polyhedra, having all degrees in this range does not guarantee an ideal realization.
Properties
Measurements
Every ideal polyhedron with
vertices has a surface that can be subdivided into
ideal triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometime ...
s, each with area
. Therefore, the surface area is exactly
.
In an ideal polyhedron, all face angles and all solid angles at vertices are zero. However, 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 ...
s on the edges of an ideal polyhedron are nonzero. At each vertex, the
supplementary angles of the dihedral angles incident to that vertex sum to exactly
. This fact can be used to calculate the dihedral angles themselves for a regular or
edge-symmetric ideal polyhedron (in which all these angles are equal), by counting how many edges meet at each vertex: an ideal regular tetrahedron, cube or dodecahedron, with three edges per vertex, has dihedral angles
, an ideal regular octahedron or
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, with four edges per vertex, has dihedral angles
, and an ideal regular icosahedron, with five edges per vertex, has dihedral angles
.
The volume of an ideal
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 the o ...
can be expressed in terms of the
Clausen function
In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimatel ...
or
Lobachevsky function of its dihedral angles, and the volume of an arbitrary ideal polyhedron can then be found by partitioning it into tetrahedra and summing the volumes of the tetrahedra.
The
Dehn invariant
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissection problem, dissected") into another, and whether a polyhedron or its dissections can Honeycomb (geometry), tile s ...
of a polyhedron is normally found by combining the edge lengths and dihedral angles of the polyhedron, but in the case of an ideal polyhedron the edge lengths are infinite. This difficulty can be avoided by using a
horosphere
In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of ...
to
truncate
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
each vertex, leaving a finite length along each edge. The resulting shape is not itself a polyhedron because the truncated faces are not flat, but it has finite edge lengths, and its Dehn invariant can be calculated in the normal way, ignoring the new edges where the truncated faces meet the original faces of the polyhedron. Because of the way the Dehn invariant is defined, and the constraints on the dihedral angles meeting at a single vertex of an ideal polyhedron, the result of this calculation does not depend on the choice of horospheres used to truncate the vertices.
Combinatorial structure
As proved, the
maximum independent set
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the tw ...
of any ideal polyhedron (the largest possible subset of non-adjacent vertices) must have at most half of the vertices of the polyhedron. It can have exactly half only when the vertices can be partitioned into two equal-size independent sets, so that the graph of the polyhedron is a balanced
bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...
, as it is for an ideal cube.
[; .] More strongly, the graph of any ideal polyhedron is
1-tough, meaning that, for any
, removing
vertices from the graph leaves at most
connected components. For example, 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 ...
is bipartite, but has an independent set with more than half of its vertices, and 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 see ...
has an independent set of exactly half the vertices but is not bipartite, so neither can be realized as an ideal polyhedron.
[
]
Characterization and recognition
Not all convex polyhedra are combinatorially equivalent to ideal polyhedra. The geometric characterization of inscribed polyhedra was attempted, unsuccessfully, by René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
in his c.1630 manuscript ''De solidorum elementis''. The question of finding a combinatorial characterization of the ideal polyhedra, analogous to Steinitz's theorem characterizing the Euclidean convex polyhedra, was raised by ; a numerical (rather than combinatorial) characterization was provided by . Their characterization is based on the fact that 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 ...
s of an ideal polyhedron, incident to a single ideal vertex, must have supplementary angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles are ...
that sum to exactly , while the supplementary angles crossed by any Jordan curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
on the surface of the polyhedron that has more than one vertex on both of its sides must be larger. For instance, for the ideal cube, the dihedral angles are and their supplements are . The three supplementary angles at a single vertex sum to but the four angles crossed by a curve midway between two opposite faces sum to , and other curves cross even more of these angles with even larger sums. show that a convex polyhedron is equivalent to an ideal polyhedron if and only if it is possible to assign numbers to its edges with the same properties: these numbers all lie between and , they add up to at each vertex, and they add up to more than on each non-facial cycle of the dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop ...
. When such an assignment exists, there is a unique ideal polyhedron whose dihedral angles are supplementary to these numbers. As a consequence of this characterization, realizability as an ideal polyhedron can be expressed as a linear program
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming i ...
with exponentially many constraints (one for each non-facial cycle), and tested in polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
using the ellipsoid algorithm.
A more combinatorial characterization was provided by for the special case of simple polyhedra, polyhedra with only three faces and three edges meeting at each (ideal) vertex. According to their characterization, a simple polyhedron is ideal or inscribable if and only if one of two conditions is met: either the graph of the polyhedron is a bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...
and its dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop ...
is 4-connected, or it is a 1-supertough graph. In this condition, being 1-supertough is a variation of graph toughness
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
; it means that, for every set of more than one vertex of the graph, the removal of from the graph leaves a number of connected components that is strictly smaller than . Based on this characterization they found a linear time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
combinatorial algorithm for testing realizability of simple polyhedra as ideal polyhedra.
Honeycombs
Because the ideal regular tetrahedron, cube, octahedron, and dodecahedron all have dihedral angles that are integer fractions of , they can all tile hyperbolic space, forming a regular honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen.
beekeeping, Beekee ...
.[ In this they differ from the Euclidean regular solids, among which only the cube can tile space.][ The ideal tetrahedron, cube, octahedron, and dodecahedron form respectively the order-6 tetrahedral honeycomb, order-6 cubic honeycomb, order-4 octahedral honeycomb, and order-6 dodecahedral honeycomb; here the order refers to the number of cells meeting at each edge. However, the ideal icosahedron does not tile space in the same way.][; ; .]
The Epstein–Penner decomposition, a construction of , can be used to decompose any cusped hyperbolic 3-manifold into ideal polyhedra, and to represent the manifold as the result of gluing together these ideal polyhedra. Each manifold that can be represented in this way has a finite number of representations. The universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of the manifold inherits the same decomposition, which forms a honeycomb of ideal polyhedra. Examples of cusped manifolds, leading to honeycombs in this way, arise naturally as the knot complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
s of hyperbolic link
In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.
As a ...
s, which have a cusp for each component of the link. For example, the complement of the figure-eight knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...
is associated in this way with the order-6 tetrahedral honeycomb, and the complement of the Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the t ...
is associated in the same way with the order-4 octahedral honeycomb. These two honeycombs, and three others using the ideal cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...
, and 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 ...
, arise in the study of the Bianchi group
In mathematics, a Bianchi group is a group of the form
:PSL_2(\mathcal_d)
where ''d'' is a positive square-free integer. Here, PSL denotes the projective special linear group and \mathcal_d is the ring of integers of the imaginary quadratic fiel ...
s, and come from cusped manifolds formed as quotients of hyperbolic space by subgroups of Bianchi groups. The same manifolds can also be interpreted as link complements.
Surface manifold
The surface of an ideal polyhedron (not including its vertices) forms a manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, topologically equivalent to a punctured sphere, with a uniform two-dimensional hyperbolic geometry; the folds of the surface in its embedding into hyperbolic space are not detectable as folds in the intrinsic geometry of the surface. Because this surface can be partitioned into ideal triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometime ...
s, its total area is finite. Conversely, and analogously to Alexandrov's uniqueness theorem
The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each othe ...
, every two-dimensional manifold with uniform hyperbolic geometry and finite area, combinatorially equivalent to a finitely-punctured sphere, can be realized as the surface of an ideal polyhedron. (As with Alexandrov's theorem, such surfaces must be allowed to include ideal dihedra
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
.)[; .] From this point of view, the theory of ideal polyhedra has close connections with discrete approximations to conformal map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s.
Surfaces of ideal polyhedra may also be considered more abstractly as topological spaces formed by gluing together ideal triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometime ...
s by isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
along their edges. For every such surface, and every closed curve which does not merely wrap around a single vertex of the polyhedron (one or more times) without separating any others, there is a unique geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
on the surface that is homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to the given curve. In this respect, ideal polyhedra are different from Euclidean polyhedra (and from their Euclidean Klein models): for instance, on a Euclidean cube, any geodesic can cross at most two edges incident to a single vertex consecutively, before crossing a non-incident edge, but geodesics on the ideal cube are not limited in this way.
See also
*Canonical polyhedron
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex po ...
, a polyhedron in which each edge is tangent to a common sphere
Notes
References
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{{refend
Polyhedra
Spheres
Hyperbolic geometry