Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane This article is about the Perimetry concept For other uses of the word see Meridian. Due to conformal property of Stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line).

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it A 3-sphere is an example of a 3-manifold. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions

A 3-sphere is also called a hypersphere, although the term hypersphere can in general describe any n-sphere for n ≥ 3. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension.

Definition

In coordinates, a 3-sphere with center (C₀, C₁, C₂, C₃) and radius r is the set of all points (x₀, x₁, x₂, x₃) in real, 4-dimensional space (R⁴) such that

The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S³:

It is often convenient to regard R⁴ as the space with 2 complex dimensions (C²) or the quaternions (H). In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician The unit 3-sphere is then given by

or

The last description is often the most useful. It describes the 3-sphere as the set of all unit quaternions—quaternions with absolute value equal to unity. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. In Mathematics, a unit circle is In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

Properties

Elementary properties

The 3-dimensional volume (or hyperarea) of a 3-sphere of radius r is

while the 4-dimensional hypervolume (the volume of the 4-dimensional region bounded by the 3-sphere) is

Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). A hyperplane is a concept in Geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere which reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

Topological properties

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be It is also simply-connected. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be What this means, loosely speaking, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture proposes that the 3-sphere is the only three dimensional manifold with these properties (up to homeomorphism). In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among Topological equivalence redirects here see also Topological equivalence (dynamical systems. This conjecture was proved in 2003 by Grigori Perelman. Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман born 13 June 1966 in Leningrad, USSR (now St

The 3-sphere is homeomorphic to the one-point compactification of . Generally, any topological space which is homeomorphic to the 3-sphere is called a topological 3-sphere. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

The homology groups of the 3-sphere are as follows: H₀(S³,Z) and H₃(S³,Z) are both infinite cyclic, while H_i(S³,Z) = {0} for all other indices i. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Any topological space with these homology groups is known as a homology 3-sphere. In Algebraic topology, a homology sphere is an n -manifold X having the Homology groups of an n - Sphere, for some integer Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S³, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Algebraic topology, a homology sphere is an n -manifold X having the Homology groups of an n - Sphere, for some integer Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the three-sphere gives a homology sphere; typically these are not homeomorphic to the three-sphere. In Mathematics, hyperbolic Dehn surgery refers to an operation by which one can obtain further Hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces

As to the homotopy groups, we have π₁(S³) = π₂(S³) = {0} and π₃(S³) is infinite cyclic. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional The higher homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernible pattern. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the For more discussion see homotopy groups of spheres. In the mathematical field of Algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other

Geometric properties

The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R⁴. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, a submanifold of a Manifold M is a Subset S which itself has the structure of a manifold and for which the Inclusion The Euclidean metric on R⁴ induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1/r² where r is the radius. In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group). In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex

Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the One can even find three linearly-independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie This implies that the 3-sphere is parallelizable. In Mathematics, a parallelizable manifold M is a Smooth manifold of dimension n having Vector fields V 1 It follows that the tangent bundle of the 3-sphere is trivial. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. For a general discussion of the number of linear independent vector fields on a n-sphere see the article vector fields on spheres. In Mathematics, the discussion of vector fields on spheres was a classical problem of Differential topology, beginning with the Hairy ball theorem, and

There is an interesting action of the circle group T on S³ giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex In Mathematics, a circle bundle is a Fiber bundle where the fiber is the Circle \mathbf{S}^1 or more precisely a principal ''U''(1-bundle In the mathematical field of Topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a Hypersphere If one thinks of S³ as a subset of C², the action is given by

.

The orbit space of this action is homeomorphic to the two-sphere S². In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Since S³ is not homeomorphic to S²×S¹, the Hopf bundle is nontrivial.

Topological construction

Two convenient constructions for the topologist are the reverse of "slicing in half" and "puncturing".

Unslicing

A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other.

The interiors of the 3-balls do not match: only their boundaries. In fact, the fourth dimension can be thought of as a continuous scalar field, a function of the 3-dimensional coordinates of the 3-ball, similar to "temperature". In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point Let this "temperature" be zero at the 2-spherical boundary, but let one of the 3-balls be "hot" (have positive values of its scalar field) and let the other 3-ball be "cold" (have negative values of its scalar field). The "hot" 3-ball could be thought of as the "hot hemi-3-sphere" and the "cold" 3-ball could be thought of as the "cold hemi-3-sphere". The temperature is highest at the hot 3-ball's very center and lowest at the cold 3-ball's center.

This construction is analogous to a construction of a 2-sphere, performed by joining the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter; superpose them so that their circular boundaries match, then let corresponding points on the circular boundaries become equivalent identically to each other. The boundaries are now glued together. Now "inflate" the disks. One disk inflates upwards and becomes the Northern hemisphere and the other inflates downwards and becomes the Southern hemisphere.

It is possible for a point traveling on the 3-sphere to move from one hemi-3-sphere to the other hemi-3-sphere by crossing the 2-spherical boundary, which could be thought of as a "3-quator" — analogous to an equator on a 2-sphere. The point would seem to be bouncing off the 3-quator and reversing direction of motion in 3-D, but also its "temperature" would become reversed, e. g. from positive on the "hot hemi-3-sphere" to zero on the 3-quator to negative on the "cold hemi-3-sphere".

Unpuncturing

Consider a topological 2-sphere to be a seamless balloon. When punctured and flattened, the missing point becomes a circle (a 1-sphere) and the remaining balloon surface becomes a disk (a 2-ball) inside the circle. In the same way, a 3-ball is a punctured and flattened 3-sphere. To recreate the 3-sphere, merge all points on the 3-ball boundary (a 2-sphere) into a single point.

Another view of puncturing is stereographic projection. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane Rest the South Pole of a 2-sphere on an infinite plane, and draw lines from the North Pole through the sphere to intersect the plane. Each sphere point corresponds to a unique plane point, and vice versa, excepting the North Pole itself. The balloon has been stretched to infinity. Stereographic projection of a 3-sphere (except for the projection point) fills all of 3-space in the same manner. A benefit of this correspondence is that geometric spheres in 3-space map to geometric spheres of the 3-sphere, and planes in 3-space map to spheres containing the Pole.

Another view is a "shooting map". Place a marble at the South Pole and give it a flick of a measured strength in a chosen direction. Assuming the marble stays on the sphere and rolls without friction, its position after a fixed time interval (say, 1 second) will be some definite point of the sphere. Plotting direction in the plane and strength as radius, the North Pole is equally far away in every direction; this is the equivalent of the punctured balloon. Performing the same shooting experiment on the 3-sphere gives a map on the 3-ball. When the 3-sphere is considered a Lie group, the marble paths are one-parameter subgroups, the 3-ball is the tangent space at the identity (taken to be the South Pole), and the mapping to the 3-sphere is the exponential map. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine

Coordinate systems on the 3-sphere

The four Euclidean coordinates for S³ are redundant since they are subject to the condition that . As a 3-dimensional manifold one should be able to parameterize S³ by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement Due to the nontrivial topology of S³ it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use at least two coordinate charts. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how Some different choices of coordinates are given below.

Hyperspherical coordinates

It is convenient to have some sort of hyperspherical coordinates on S³ in analogy to the usual spherical coordinates on S². In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial One such choice—by no means unique—is to use (ψ, θ, φ) where

where ψ and θ runs over the range 0 to π, and φ runs over 0 to 2π. Note that for any fixed value of ψ, θ and φ parameterize a 2-sphere of radius sin(ψ), except for the degenerate cases, when ψ equals 0 or π, in which case they describe a point.

The round metric on the 3-sphere in these coordinates is given by

and the volume form by

These coordinates have an elegant description in terms of quaternions. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Any unit quaternion q can be written in the form:

q = e^τψ = cos ψ + τ sin ψ

where τ is a unit imaginary quaternion—that is, any quaternion which satisfies τ² = −1. This is the quaternionic analogue of Euler's formula. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Now the unit imaginary quaternions all lie on the unit 2-sphere in Im H so any such τ can be written:

τ = cos φ sin θ i + sin φ sin θ j + cos θ k

With τ in this form, the unit quaternion q is given by

q = e^τψ = x₀ + x₁ i + x₂ j + x₃ k

where the x’s are as above.

When q is used to describe spatial rotations (cf. quaternions and spatial rotations) it describes a rotation about τ through an angle of 2ψ. Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions

Hopf coordinates

Another choice of hyperspherical coordinates, (η, ξ₁, ξ₂), makes use of the embedding of S³ in C². In complex coordinates (z₁, z₂) ∈ C² we write

Here η runs over the range 0 to π/2, and ξ₁ and ξ₂ can take any values between 0 and 2π. These coordinates are useful in the description of the 3-sphere as the Hopf bundle

For any fixed value of η between 0 and π/2, the coordinates (ξ₁, ξ₂) parameterize a 2-dimensional torus. In the mathematical field of Topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a Hypersphere In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In the degenerate cases, when η equals 0 or π/2, these coordinates describe a circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

The round metric on the 3-sphere in these coordinates is given by

and the volume form by

Stereographic coordinates

Another convenient set of coordinates can be obtained via stereographic projection of S³ onto a tangent R³ hyperplane. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane A hyperplane is a concept in Geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a For example, if we project onto the plane tangent to the point (1, 0, 0, 0) we can write a point p in S³ as

where u = (u₁, u₂, u₃) is a vector in R³ and ||u||² = u₁² + u₂² + u₃². In the second equality above we have identified p with a unit quaternion and u = u₁ i + u₂ j + u₃ k with a pure quaternion. (Note that the division here is well-defined even though quaternionic multiplication is generally noncommutative). The inverse of this map takes p = (x₀, x₁, x₂, x₃) in S³ to

We could just have well have projected onto the plane tangent to the point (−1, 0, 0, 0) in which case the point p is given by

where v = (v₁, v₂, v₃) is a vector in the second R³. The inverse of this map takes p to

Note that the u coordinates are defined everywhere but (−1, 0, 0, 0) and the v coordinates everywhere but (1, 0, 0, 0). Both patches together cover all of S³. This defines an atlas on S³ consisting of two coordinate charts. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how Note that the transition function between these two charts on their overlap is given by

and vice-versa.

Group structure

When considered as the set of unit quaternions, S³ inherits an important structure, namely that of quaternionic multiplication. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Because the set of unit quaternions is closed under multiplication, S³ takes on the structure of a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Moreover, since quaternionic multiplication is smooth, S³ can be regarded as a real Lie group. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group It is a nonabelian, compact Lie group of dimension 3. In Mathematics, a nonabelian group, also sometimes called a non-commutative group, is a group ( G, *) such that there are at least two elements When thought of as a Lie group S³ is often denoted Sp(1) or U(1, H). In Mathematics, the name symplectic group can refer to two different but closely related types of mathematical groups.

It turns out that the only spheres which admit a Lie group structure are S¹, thought of as the set of unit complex numbers, and S³, the set of unit quaternions. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a unit circle is Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician One might think that S⁷, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Mathematics, associativity is a property that a Binary operation can have The octonionic structure does give S⁷ one important property: parallelizability. In Mathematics, a parallelizable manifold M is a Smooth manifold of dimension n having Vector fields V 1 It turns out that the only spheres which are parallelizable are S¹, S³, and S⁷.

By using a matrix representation of the quaternions, H, one obtains a matrix representation of S³. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally One convenient choice is given by the Pauli matrices:

This map gives an injective algebra homomorphism from H to the set of 2×2 complex matrices. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. A homomorphism between two algebras over a field K, A and B, is a map FA\rightarrow B such that for all k It has the property that the absolute value of a quaternion q is equal to the square root of the determinant of the matrix image of q. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group SU(2). Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Thus, S³ as a Lie group is isomorphic to SU(2). In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Using our hyperspherical coordinates (η, ξ₁, ξ₂) we can then write any element of SU(2) in the form

In literature

In Edwin Abbott Abbott's Flatland, published in 1884, and in Sphereland, a 1965 sequel to Flatland by Dionys Burger, the 3-sphere is referred to as an oversphere, and a 4-sphere is referred to as a hypersphere. Edwin Abbott Abbott ( December 20, 1838 &ndash October 12, 1926) English Schoolmaster and theologian, For other uses see Flatland (disambiguation Flatland A Romance of Many Dimensions is an 1884 Science fiction Year 1884 ( MDCCCLXXXIV) was a Leap year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Leap year Sphereland is a 1965 Novel by Dionys Burger, and is a sequel to Flatland, a novel by "A Square" (a pen name of Edwin Abbott Abbott) Year 1965 ( MCMLXV) was a Common year starting on Friday (link will display full calendar of the 1965 Gregorian calendar. Dionys Burger ( 10 July 1892, Amsterdam - 19 April 1987) was a Dutch mathematician (in original Dutch: Dionijs

Writing in the American Journal of Physics^[1], Mark A. The American Journal of Physics is a Peer-reviewed Scientific journal published by the American Association of Physics Teachers devoted to the educational Peterson describes three different ways of visualizing 3-spheres and points out language in The Divine Comedy that suggests Dante viewed the Universe in the same way. The Divine Comedy

See also

References

• David W. The American Journal of Physics is a Peer-reviewed Scientific journal published by the American Association of Physics Teachers devoted to the educational Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition, 2001, [1] (Chapter 20: 3-spheres and hyperbolic 3-spaces. )
• Jeffrey R. Weeks, The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds, 1985, ([2]) (Chapter 14: The Hypersphere) (Says: A Warning on terminology: Our two-sphere is defined in three-dimensional space, where it is the boundary of a three-dimensional ball. Jeffrey Renwick Weeks is an American Mathematician. He became a MacArthur Fellow in 1999 This terminology is standard among mathematicians, but not among physicists. So don't be surprised if you find people calling the two-sphere a three-sphere. )

External links

• Eric W. Weisstein, Hypersphere at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W Note: This article uses the alternate naming scheme for spheres in which a sphere in n-dimensional space is termed an n-sphere.