Euclidean
Geometry| Euclidean
Geometry |
Euclidean Geometry / Space En > s.a.
[differential geometry]; trigonometry.
* Idea: The space Rn with
an affine structure; Choosing an origin and a positive definite quadratic function
: Rn → R (which
defines an inner product) makes it into a Euclidean vector space.
* Euclidean group: The
allowed symmetry transformations (in the sense of Klein's program), the group
of rigid motions; In n dimensions, SO(n) ×s
Tn.
@ Axioms: Hilbert 02.
@ Related topics: Bauer & Wachter EPJC(03)mp/02 [q-deformed].
Ellipsoid > s.a. multipole
moments.
* Volume: In 3D, V =
(4/3)
abc, and for
an ellipsoid of revolution (2 equal axes), V = (4/3) a2b;
In n dimensions, multiply the volume of the unit n-sphere
by the square root of all the semiaxes.
Polygon > s.a. Polygon [Minkowski case]; simplex [including
triangle].
* Constructible: The
ones with 2k,
3 · 2k, 5 · 2k,
or 15 · 2k sides (k 
N)
are known from BC; The one with 17 sides was found by
Gauss.
@ References Agarwal et al CG(02)
[Minkowski sums, algorithms]; James et al JPA(08)
[almost convex].
Polyhedron > s.a. riemannian
geometry; Tetrahedron; Triangulable
Space.
* Idea: A subspace of En made of simplexes, whose intersections
are faces; A special kind of cell complex.
$ Def: The union of all elements of a simplicial complex, together
with the Euclidean subspace topology.
* Regular polyhedra:
The cube, icosahedron, Platonic solids, tetrahedron.
* Platonic solids: The
five polyhedra in 3D Euclidean space that have equal faces and equal angles
at their vertices, the tetrahedron, cube, octahetron, dodecahedron, and icosahedron.
@ Platonic solids: Everitt T&A(04) [3-manifolds from identifications].
@ Related topics: Skarke ht/00-in,
Kreuzer & Skarke RVMP(02)m.AG/00,
ATMP(02)ht/00 [reflexive];
Atiyah & Sutcliffe MJM(03)mp [in
physics, chemistry and geometry]; Grünbaum DM(07)
[polyhedra and graphs]; > s.a. Calculating
Theorem, Schläfli
Formula.
Polyhedral Complex > see cell complex; voronoi tiling.
Other Concepts and Results > s.a. conical
sections; lines; Pythagorean
Theorem; simplex;
spheres; Spiral;
Surfaces.
* Curious fact: On E2,
draw a circle and n points on it in
generic positions (vertices of a regular polygon is ok but not necessary);
Join all
pairs by a line; This divides the disk into N(n) regions;
For n =
1, 2, 3, 4, 5, N(n) = 1, 2, 4, 8, 16; What is the next one?
Answer: 31! [N(n) is given by some known polynomial.]
Euclidean Metric on a Manifold > a.k.a. riemannian
geometry.
$ Def: Given a vector
bundle (E, , M),
a map :
E
→ R making each fiber into a Euclidean vector space.
Euclidean Theories in Physics > see formulations and solution methods in general relativity; modified quantum mechanics.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jun 2008