Euclidean Geometry |

**Euclidean Geometry / Space E ^{n}**
> s.a. differential geometry / trigonometry.

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**Ellipsoid** > s.a. multipole moments.

* __Volume__: In 3D, *V*
= \(4\over3\)π*abc*, and for an ellipsoid of revolution (2 equal axes),
*V* = \(4\over3\)π *a*^{2}*b*;
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 2^{k}, \(3 \cdot 2^k\),
\(5 \cdot 2^k\), 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].

@ __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 21 oct 2019