Projective 3-sphere
In mechanics and geometry, the 3D rotation group is the group of all rotations about the origin (mathematics) of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (such as it is an isometry), and orientation (i.e. handedness of space). A distance-preserving transformation which reverses orientation is an improper rotation, that is a reflection or, in the general position, a rotoreflection. The origin in Euclidean space establishes certain one-to-one correspondence between points and their coordinate vectors. Rotations about the origin can be thought of as magnitude-preserving linear transformations of Euclidean 3-dimensional vectors (whose vector space is also denoted as R3).
Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties (along with the associative property, which rotations obey), the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. The rotation group is often denoted SO(3) (or, less ambiguously, SO(3, R)) for reasons explained below.

This is an excerpt from the article Projective 3-sphere from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
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Projective space - Wikipedia, the free encyclopedia
1 Introduction; 2 Definition of projective space; 3 Projective space as a manifold ... Every such line meets the sphere of radius one centered in the origin exactly ...
en.wikipedia.org/wiki/Projective_space
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HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES ...
in spheres and complex projective spaces. In this paper, we shall classify con- formal non-superminimal harmonic two-tori in a 2- or 3-dimensional quater- ...
www.med.nihon-u.ac.jp/home/sudagawa/ProcAMS.pdf
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HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES ...
in spheres and complex projective spaces. In this paper, we shall classify con- formal non-superminimal harmonic two-tori in a 2- or 3-dimensional quater- ...
www.ams.org/proc/1997-125-01/S0002-9939-97-03638-1/S0002-9939-97-03638-1.pdf
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Spheres and Projective Spaces - University of Waterloo
Jul 7, 2010 ... Complex Projective Spaces. Quaternionic Projective Spaces. Octonionic Projective Spaces. The Hopf Fibrations. 3 Spheres and Almost ...
www.math.uwaterloo.ca/~karigian/talks/CUMC-2010.pdf
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Möbius and projective 3-manifolds - MathOverflow
May 5, 2010 ... A M\"obius 3-manifold is defined similarly, with the projective space replaced with the standard 3-sphere and projective transformations ...
mathoverflow.net/questions/23567/mobius-and-projective-3-manifolds
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PICTURES OF THE PROJECTIVE PLANE
The Montana Mathematics Enthusiast, ISSN 1551-3440, Monograph 3, pp.3-16 ... antipodal points of the sphere, the model gives us a picture of the projective ...
www.math.umt.edu/tmme/Monograph3/Artmann_Monograph3_pp.3_16.pdf
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Projective 3-sphere | Facebook
Projective 3-sphere. 2 likes · 1 talking about this. ... Projective 3-sphere. Projective 3-space was merged with this page · Interest ...
www.facebook.com/pages/Projective-3-sphere/334419543317024?rf=399680730086477
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Are there 3D tilings of a 3D projective hyperplane or 3-sphere?
I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings ...
math.stackexchange.com/questions/337271/are-there-3d-tilings-of-a-3d-projective-hyperplane-or-3-sphere
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Projective modules over the real algebraic sphere - EPFL
Using this, we show that all the finitely generated projective modules over the algebraic real 3-sphere are free. Contents. 1. Introduction. 1. 1.1. Conventions. 3 ...
alg-geo.epfl.ch/~jfasel/Cancel_rev.pdf
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Trends in Mathematical Physics: Proceedings of the Conference on ... - Google Books Result
books.google.com/books?id=n9RqblJkVpIC&pg=PA323&lpg=PA323&dq=Projective+3-sphere&source=bl&ots=ot-15sa_gO&sig=xvn_PNLVOp_LU1l7CzdlxiUD3mg&hl=en&sa=X&ei=XSK2UZfjKsPvOaKbgOAN&ved=0CDkQ6AEwCg
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Projective 3-sphere in science
Riemann sphere - Wikipedia, the free encyclopedia
In projective geometry, the sphere can be thought of as the complex projective ... 2 As a complex manifold; 3 As the complex projective line; 4 As a sphere; 5 Metric ..... by Douglas N. Arnold and Jonathan Rogness (a video by two University of ...
Projective space - Wikipedia, the free encyclopedia
1 Introduction; 2 Definition of projective space; 3 Projective space as a ... Every such line meets the sphere of radius one centered in the origin exactly twice, say in ..... Cambridge Studies in Advanced Mathematics, Cambridge University Press, ...
[PDF]Spheres and Projective Spaces - University of Waterloo
Jul 7, 2010 ... Complex Projective Spaces. Quaternionic Projective Spaces. Octonionic Projective Spaces. The Hopf Fibrations. 3 Spheres and Almost ...
Dr J Bolton - Durham University
Durham University ... 'From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space'. 'Bull. ... Linearly full harmonic 2-spheres in S^4 of area 20\pi.
[PDF]IMMERSIONS OF THE PROJECTIVE PLANE WITH ONE TRIPLE ...
UNIVERSITY OF SOUTH CAROLINA, COLUMBIA. Abstract. We consider C∞ generic immersions of the projective plane into the. 3-sphere. Pinkall has shown ...
[PDF]Invariant differential operators on the sphere - Australian National ...
Third CoE Talk at the University of Tokyo – p. 1/21 ... The projective circle: {(x, y) ∼ λ(x, y) for λ > 0}. E. T .... The contact Projective 3-sphere = Sp(4,R)/P. • The CR ...
W2012 - University of Chicago Geometry/Topology Seminar
Thursdays (and sometimes Tuesdays) 3-4pm, in Eckhart 308 ... Rice University; Polynomial Pick forms for affine spheres and real projective polygons; Abstract: ...
[PDF]Introduction to Topology - The Department of Mathematics at ...
4.2.2 The Projective Plane as a Quotient Space of the Sphere . . . . . . . . . . 65. 4.2.3 The Projective Plane as an identification space of a disc . . . . . . . . . . 66 ...
Understanding Projective Geometry - University of Toronto
Sep 10, 1996 ... Okay, I'm just wondering about the applicability of projective and affine .... Now, if you put a sphere with its centre at the origin in 3-d space, then ...
[PDF]HARMONIC TORI IN QUATERNIONIC PROJECTIVE 3-SPACES ...
formal non-superminimal harmonic two-tori in a 2- or 3-dimensional quater- ... The harmonic two-spheres in n-dimensional quaternionic projective space HPn ...... Department of Mathematics, School of Medicine, Nihon University, Itabashi, ...
Books on the term Projective 3-sphere
Trends in Mathematical Physics: Proceedings of the Conference on ...
Trends in Mathematical Physics: Proceedings of the Conference on ...
Vasilios Alexiades, George Siopsis, 1999
The labels on the 3-orbifolds of Figure 4 denote the characteristic invariant lattice complexes (letters) and singular set ... 3-disk, 3-sphere, real projective 3- plane, real projective 2-plane plus a 3-disk half boundary, and double real projective ...
Differential Topology: First Steps (Dover Books on Mathematics)
Differential Topology: First Steps (Dover Books on Mathematics)
Andrew H. Wallace, 2006
Keeping mathematical prerequisites to a minimum, this undergraduate-level text stimulates students' intuitive understanding of topology while avoiding the more difficult subtleties and technicalities. Its focus is the method of spherical modifications and the study of critical points of functions on manifolds.No previous knowledge of topology ...
Seifert and Threlfall, A textbook of topology
Seifert and Threlfall, A textbook of topology
Herbert Seifert, 1980
In fact, one cannot even completely classify the topologically distinct 3- dimensional spaces. The higher ... These are projective 3-space and the 3-sphere. Projeclive 3-space arises from the requirement that one completes Euclidean 3- ...
How Surfaces Intersect in Space: An Introduction to Topology (2nd Edition) (K & E Series on Knots and Everything...
How Surfaces Intersect in Space: An Introduction to Topology (2nd Edition) (K & E Series on Knots and Everything...
J. Scott Carter, 1995
"This book should be widely read, particularly by undergraduates trying to get a feel for what topology is about." --This text refers to an out of print or unavailable edition of this title.
Handbook of Geometric Topology
Handbook of Geometric Topology
R.B. Sher, R.J. Daverman, 2001
Having analyzed essential 2-spheres, we now turn our attention to essential projective planes. By the previous step, we can restrict the analysis to 3- manifolds that contain no essential 2-spheres. It turns out that we then have a much stronger ...
The Chemistry of Matter Waves
The Chemistry of Matter Waves
2013
The quantum and relativity theories of physics are considered to underpin all of science in an absolute sense. This monograph argues against this proposition primarily on the basis of the two theories' incompatibility and of some untenable philosophical implications of the quantum model. Elementary matter is assumed in both theories to occur a...
Algorithmic Foundations of Robotics VIII: Selected Contributions ...
Algorithmic Foundations of Robotics VIII: Selected Contributions ...
Gregory S. Chirikjian, Howie Choset, Marco Morales, 2010
that the 3-sphere is a double covering of RP3, since one point of the projective space has two corresponding points on the 3-sphere. • Haar Measure on SO(3). Up to a scalar multiple, there exists a unique measure on SO(3) that is invariant ...
How surfaces intersect in space: an introduction to topology
How surfaces intersect in space: an introduction to topology
J. Scott Carter, 1995
The projective plane is substantial in projective space because the core disk of the 2-handle is pierced by a transverse arc ... Identifying Points on the 3-sphere Consider the possibility of identifying diametrically opposite points on the 3-sphere.
An Introduction to Homotopy Theory (Cambridge Tracts in Mathematics)
An Introduction to Homotopy Theory (Cambridge Tracts in Mathematics)
1953
Since the introduction of homotopy groups by Hurewicz in 1935, homotopy theory has occupied a prominent place in the development of algebraic topology. This monograph provides an account of the subject which bridges the gap between the fundamental concepts of topology and the more complex treatment to be found in original papers.
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Blog posts on the term
Projective 3-sphere
Visualizing High Dimensional Data: Azhimutal projection for RPM on projective spaces
Among those manifolds used by relational perspective map (RPM), the 2-dimensional real projective plane (P2) has been one of my favorite space. As a visualization space for high dimensional data, P2 works similarly as the 2-dimensional flat torus (T2) used initially by RPM.
jamesxli.blogspot.com/2013/04/azhimutal-projection-for-rpm-on.html
Real & Complex Projective Space (Part 3) | isaacsolomonmath
General mathematical distractions for people with too much free time
isaacsolomonmath.wordpress.com/2011/12/29/real-complex-projective-part-3/
The Universe | Complex Projective 4-Space
Where exciting things happen
cp4space.wordpress.com/2013/05/26/the-universe/
Interesection Numbers in RP^3
Interesection Numbers in RP^3 in Differential Geometry is being discussed at Physics Forums
www.physicsforums.com/showthread.php?t=473333
Rolling Circles and Balls (Part 5) | Azimuth
Last time I promised to show you how the problem of a little ball rolling on a big stationary ball can be described using an 8-dimensional number system called the split octonions… if the big ball has a radius that’s 3 times the radius of the little one! So, let’s get started. First, I must admit that I lied.
johncarlosbaez.wordpress.com/2013/01/02/rolling-circles-and-balls-part-5/
S0(3)/SO(2) is isomorphic to the projective plane
S0(3)/SO(2) is isomorphic to the projective plane in Linear & Abstract Algebra is being discussed at Physics Forums
www.physicsforums.com/showthread.php?t=161908
Altitudes of a triangle and the Jacobi identity | Mathematics under the Microscope
Atomic objects, structures and concepts of mathematics
micromath.wordpress.com/2008/04/07/altitudes-of-a-triangle-and-the-jacobi-identity/
11011110: Skew lines
www.livejournal.com/go.bml?journal=11011110&itemid=190579&dir=next
On the Topology of Isolated Singularities in Analytic Spaces ebook @ evenmoreo2的部落格 :: 痞客邦 PIXNET ::
Download On the Topology of Isolated Singularities in Analytic Spaces Author: Jos? Seade Type: eBoo
evenmoreo2.pixnet.net/blog/post/55479712-on-the-topology-of-isolated-singularities-in-analytic-spaces
Syntropy: CJ Fearnley’s Blog » Models of Projective Geometry
blog.cjfearnley.com/2012/07/24/models-of-projective-geometry-2/
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