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-4
votes
0answers
71 views

Size of a point [closed]

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part.Now, ...
4
votes
2answers
144 views

Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that : $\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge ...
9
votes
1answer
192 views

Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...
7
votes
2answers
301 views

Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this: (ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB) ...
4
votes
1answer
502 views

Explicitly describing the region of the plane “outward of” a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$

Some Context: I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
2
votes
0answers
104 views

Which Coxeter groups can be realized as affine reflection groups?

Every affine reflection group has a Coxeter presentation (https://en.wikipedia.org/wiki/Reflection_group#Relation_with_Coxeter_groups). How do you tell which Coxeter presentations arise from affine ...
16
votes
2answers
811 views

Probability of two vectors lying in the same orthant

Let $S^{d-1} = \{x \in \mathbb R^d: \|x\| = 1\}$ denote the unit sphere in $\mathbb R^d$. Let $v$, $w$ be drawn uniformly at random from $S^{d-1}$, conditioned on their inner product being equal to ...
5
votes
0answers
137 views

Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...
17
votes
5answers
1k views

Definition of area

I am looking for an attractive, but rigorous definition of area; say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
3
votes
1answer
98 views

Existence or otherwise of a set of “sufficiently intricate” open sets

Fix $d \in \mathbb{N}$. Do there exist mutually disjoint connected open sets $V_1,\ldots,V_n \subset \mathbb{R}^d$ and $\mathbf{v} \in \mathbb{R}^d$ such that $\mathbb{R}^d \setminus ...
4
votes
1answer
369 views

Is every complex rational algebraic variety simply connected for the Euclidean topology?

Is it true that every quasi-projective rational irreducible algebraic complex variety is simply connected for the Euclidean topology? Of course, this is false if we replace "complex" with "real" or ...
0
votes
0answers
59 views

Maximum value of linear function on the intersection of a parametrized family of balls

Let $C$ be a (nonempty) closed convex subset of $\mathbb{R}^n$ and $a, b \in \mathbb{R}^n$. Using the normal cone characterization of the euclidean projection operator $\mathrm{proj}_C$ (recall that ...
3
votes
0answers
59 views

Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...
12
votes
1answer
299 views

Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle? You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
2
votes
1answer
137 views

Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses. Also it is possible to construct straight line ...
54
votes
11answers
6k 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 ...
12
votes
1answer
703 views

A problem in elementary geometry

Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle: On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise, ...
12
votes
3answers
871 views

Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. ...
12
votes
1answer
457 views

Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$. EDIT: I forgot to add that no ...
3
votes
4answers
322 views

Terminology for polygons

As you may know term "polygon" might mean few different things and its meaning has to guessed from context. By some reason I have to use few of these meaning in one place. So I converge to the ...
7
votes
1answer
964 views

Geometric meaning of trigonometric relations

According to a paper by Zhiqin Lu in the Mathematical Gazette (the British publication, not the Boston-area newsletter, if that still exists (or even if it doesn't)) in 2007(?), if $u+v+w=\pi$ and ...
0
votes
2answers
82 views

Pairwise distance distribution for point clouds (normal distribution) [closed]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$). My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
2
votes
1answer
73 views

Equality of Euclidean numbers / constructible numbers

Euclidean numbers are those real number that can be constructed from the natural numbers by a finite chain of +,-,*,/ and $\sqrt{}$. They are also called Constructible Numbers. I am now interested in ...
8
votes
0answers
173 views

Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$. The segments are open, excluding their endpoints. They are disjoint as closed segments, i.e., no pair shares an ...
18
votes
6answers
3k 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. ...
42
votes
1answer
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a ...
2
votes
1answer
167 views

What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
7
votes
1answer
126 views

Elementary proof of a triangular grid lemma

I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714). Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...
5
votes
1answer
103 views

Penrose tiling substitution is bijective

Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles. It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...
1
vote
0answers
46 views

Projecting a convex partition onto a convex set

Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of ...
13
votes
2answers
866 views

Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
5
votes
2answers
1k views

What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
6
votes
1answer
503 views

Using mirrors to make a non-convex polygon visible from a fixed interior point

Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
16
votes
1answer
384 views

What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
5
votes
2answers
653 views

Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...
2
votes
0answers
235 views

Find minimum-area ellipse enclosing a set of ellipses, all centered at the origin

Given a set of N > 2 (two-dimensional and coplanar) ellipses, all centered at the origin, how do I find the ellipse with the minimum area which encloses all of them? Background: Thanks to Will Jagy ...
5
votes
3answers
2k views

Minimum distance between two arbitrary circles in space?

What is the minimum distance between two arbitrary circles in space? I am working out the problem with Maxima, but I am surprised by how complicated this rapidly turns out to unfold for such a ...
9
votes
3answers
409 views

Mutually tangent ellipsoids in 3 space

I recently heard a claim that for any n, it is possible to arrange n ellipsoids in 3 space such that each pair of ellipsoids is kissing. Is this true, and if so, how? Edit: By kissing, I mean that I ...
28
votes
14answers
7k 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 ...
11
votes
0answers
158 views

Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...
4
votes
1answer
181 views

Segments on a family of parallel lines

Let $\{l_i:i\in I\}$ be a family of parallel lines on the plane $\mathbb{R}^2$. Suppose for each $i\in I$ there is a closed segment $s_i\subset l_i$. Moreover, for each triple $i_1,i_2,i_3$ there ...
0
votes
0answers
128 views

How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$. It seems that such a set cannot be very large and that its ...
6
votes
1answer
139 views

Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
5
votes
1answer
586 views

Elementary problem about triangles inside a convex polygon

Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
6
votes
1answer
174 views

Did Lucas discover Lucas circles?

MathWord's article on Lucas circles traces the name to a little-known 1973 publication. These interesting circles have found their way into several 21st century publications, including the online ...
4
votes
1answer
107 views

Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy ...
2
votes
0answers
118 views

Examples of Geometric Constructions in Higher Dimensions

The classical problem of geometric construction seems to be restricted to planar Euclidean Geometry with straight edge and compass as the only admissible "construction-tools". I would like to ...
10
votes
5answers
557 views

ten concurrent lines

this relates to a question asked at MSE. i was able to find an answer using complex numbers. here is the question: there are five points on a circle. take any three points, through the centroid of ...
6
votes
4answers
736 views

Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ?

I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me: 1) Are there two finite subgroups ...
4
votes
1answer
476 views

What is the correct preposition? (And is there one?)

I just stumbled upon a linguistic problem I wasn't able to resolve via web search. Suppose we're given some geometric set $A$ and subset $B\subset A$. Isn't there a compact way of saying that there ...