Euclidean Geometry| Euclidean Geometry |
Euclidean Geometry / Space En
> s.a. differential geometry / trigonometry.
* Idea: The space
\(\mathbb R^n\) with an affine structure; Choosing an origin and a positive
definite quadratic function μ: \(\mathbb R^n \to \mathbb R\)
(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) \(\times_{\rm s}
{\rm T}^n\).
@ General references: Hilbert 02 [axioms];
Posamentier 02 [advanced].
@ Related topics: Bauer & Wachter EPJC(03)mp/02 [q-deformed];
Soltan 15 [convex sets];
> s.a. decomposition [of tensor fields].
> Online resources: see Euclid's Elements in
David Joyce's site;
Johan Mebius' page [4D euclidean geometry].
Ellipsoid > s.a. multipole moments.
* Volume: In 3D, V
= \(4\over3\)πabc, and for an ellipsoid of revolution (2 equal axes),
V = \(4\over3\)π a2b;
In n dimensions, multiply the volume of the unit n-sphere by
the square root of all the semiaxes.
Polygon
> s.a. simplex [including triangle]; Surveyor's Formula.
* Constructible: The ones
with 2k, \(3 \cdot 2^k\),
\(5 \cdot 2^k\), or \(15 \cdot 2^k\) sides (k ∈ \(\mathbb N\))
are known from BC; The one with 17 sides was found by Gauss.
@ General references Agarwal et al CG(02) [Minkowski sums, algorithms];
James et al JPA(08) [almost convex].
@ Related topics:
Charles a0806 [quantization of polygon spaces];
> s.a. markov processes [polygonal Markov fields].
@ In Minkowski space: Foth JGP(08) [3D Minkowski].
> Online resources:
see Wikipedia page.
Polyhedron > s.a. Tetrahedron;
Triangulable Space.
* Idea: A subspace of E\(^n\) made
of simplices, whose intersections are faces; A special kind of cell complex.
$ Def: The union of all elements of a
(locally finite) 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; There
are three generalizations that exist in all dimensions, the hypertetrahedron,
hypercube, and hyperoctahedron, with the latter two being dual.
* Result (Minkowski): A convex
polyhedron is uniquely determined (up to translation) by the directions and areas
of its faces.
* Examples: Euclidean space
\({\mathbb R}^n\) itself is a polyhedron.
@ Platonic solids: Everitt T&A(04) [3-manifolds from identifications];
Dechant ACA-a1307 [4D analogs];
Tavakoli & Gisin a2001 [and tests of quantum mechanics].
@ Related topics: Skarke ht/00-proc,
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];
Montroll 09 [popular level, origami polyhedra];
Koca et al ARP-a1006 [quasi-regular polyhedra and their duals];
Barequet et al CG(13)
[polyhedronization of a set of points in \(\mathbb R\)3];
Sellaroli a1712
[reconstructing 3D convex polyhedra from their face normals and areas];
> s.a. Calculating Theorem;
Schläfli Formula.
@ Variations: Livine JMP(13)-a1307 [framed convex polyhedra as symplectic quotients];
Neiman CQG(13)-a1308 [convex spacelike polyhedra in Minkowski space].
Other aspects:
see discrete geometry; riemannian
geometry / quantum geometry [quantum polyhedra].
Polyhedral Complex > see cell complex; voronoi tiling.
Other Concepts and Results
> s.a. coordinates; lines;
Surfaces.
* Curious fact: On E\(^2\),
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.]
@ References: Topper & Vincent pt(17)dec [Einstein's solution of a high-school geometry problem].
> Related concepts and results: see Cassini Oval;
conical sections (ellipse, hyperbola, parabola); Pythagorean
Theorem; simplex; spheres; Spiral.
Euclidean Metric on a Manifold
> a.k.a. riemannian geometry.
$ Def: Given a vector
bundle (E, π, M), a map μ: E
→ \(\mathbb R\) making each fiber into a Euclidean vector space.
Euclidean Theories in Physics > see formulations and solution methods in general relativity; modified quantum mechanics.
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send feedback and suggestions to bombelli at olemiss.edu – modified 5 jan 2020