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5
votes
2answers
313 views

Generalization of plane geometric trees?

View a plane tree drawn in $\mathbb{R}^2$ as a joining of geometric (straight) segments at endpoints such that (a) they avoid intersecting one another (except where they share a vertex), and (b) they ...
2
votes
1answer
158 views

Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map

Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the ...
14
votes
2answers
899 views

Angle of a regular simplex

I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference. What is the vertex angle of a regular $n$-simplex? Background: For a vertex $v$ ...
17
votes
9answers
4k views

Open problems in Euclidean geometry ?

Which are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a ...
5
votes
2answers
521 views

Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model?

I posted this question at math.stackexchange.com but didn't get an answer. Motivation Physicists are in search for a model of discrete space(-time) for a long time. So I wondered why not start with ...
3
votes
0answers
298 views

Characterizations of Euclidean space

I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually? There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite ...
10
votes
6answers
644 views

Decomposing the plane into intervals

I posted this on Stack Exchange and got a lot of interest, but no answer. A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so ...
3
votes
1answer
448 views

Is always possible to slice a pizza in a fair way

Given a pizza, represented by the unit disk $D_1(0,0)=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\|\leqslant 1\}$, and given $N$ slices of $r$-pepperoni, represented by disks ...
5
votes
1answer
1k views

Point cloud that maximizes the minimum pairwise distance in Euclidean space

I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize $min_{i \neq j} ...
9
votes
3answers
631 views

Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis? See http://en.wikipedia.org/wiki/Constructible_number and ...
11
votes
11answers
5k views

Theorems in Euclidean geometry with attractive proofs using more advanced methods

The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...
12
votes
6answers
2k views

Euclid with Birkhoff

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms. It would be best if it would also include some topics in projective (and/or) hyperbolic geometry. ...
2
votes
1answer
2k views

Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]

How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?
8
votes
1answer
543 views

Sticks and thread

In this recent question Math puzzles for dinner we had a nice time as we were asked to provide new maths puzzles for dinners. I suggested the following: Given three equal sticks, and some ...
41
votes
6answers
4k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
2
votes
1answer
477 views

Reference: Countable Models of (Non-)Euclidean Geometry

Has there been a survey written on the model theory of first-order (non-)Euclidean geometry in the spirit of Hilbert and Tarski? I'm especially interested in two aspects of the model theory: ...
7
votes
9answers
1k views

Comprehensive reference for synthetic euclidean geometry

Euclidean geometry is a special case of the theory of Hilbert spaces; but in order to convince small children of basic facts, e.g. that the line segments from each of the vertices of a triangle to the ...
29
votes
14answers
23k views

If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...