Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from ...

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0answers
63 views

Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...
1
vote
4answers
599 views

Characterization of Angles Trisectable with Straightedge and Compass

Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative. In the following, ...
4
votes
1answer
599 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 ...
6
votes
1answer
191 views

Law of sines for tetrahedra

Wikipedia gives a generalization of the law of sines to higher dimensions, as defined in F. Eriksson, The law of sines for tetrahedra and n-simplices. However, this generalization misses an important ...
0
votes
1answer
199 views

Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true. Since posting the question, ...
10
votes
4answers
751 views

In which geometries do triangles have an Euler line?

In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line. In which other geometries does this hold?
34
votes
14answers
8k 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 ...
1
vote
1answer
61 views

Set of General Linear Position with Nonzero Measure

I came to the following question, but I don't have quite a good idea how to approach. Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position? I believe ...
5
votes
0answers
509 views

Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
25
votes
2answers
1k views

Understanding sphere packing in higher dimensions

In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved. Admittedly it is very ...
4
votes
1answer
258 views

Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note) Consider the Bundle theorem configuration : Points $A_1, A_2, A_3, A_4$ lie on a circle, ...
7
votes
1answer
150 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
160 views

A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane). This is a generalization of the last my previous question in Three chains of six circles. ...
3
votes
0answers
167 views

A conjecture on six planes [closed]

When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...
3
votes
0answers
129 views

Conjecture generalization of Feuerbach theorem and somes another theorems

My question: I am looking for a solution of a conjecture generalization of the Feuerbach theorem in the end of the topic. But I think, I should let you know why I found this conjecture. I thank to ...
6
votes
0answers
116 views

Paradoxical spherical caps

All spherical caps (i.e. sets $C_L:=\{(x,y,z)|x^2+y^2+z^2=1, z\geq L\}$) admit a paradoxical decomposition in the sense of Banach-Tarski, meaning $C_L \tilde{} 2C_L$; here $\tilde{}$ stands for the ...
2
votes
1answer
192 views

Relation of some Euclidean geometry theorems and more conjecture generalizations

In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem and the Feuerbach-Luchterhand. Since that I posed two ...
1
vote
1answer
175 views

Conjectute: no exist an equilateral triangle such that all vertices are integer numbers [closed]

I am looking for a solution for a conjecture as follows. In Cartesian plane, no exist an equilateral triangle such that three vertices are integer numbers. I hope that you like the question ...
6
votes
0answers
154 views

Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric?

Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others under an isometry?
3
votes
6answers
1k views

Circumference of Convex Shapes

Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
2
votes
1answer
175 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 ...
-1
votes
1answer
428 views

Meeting point of the vertices of a square cloth on x-y plane

Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...
0
votes
0answers
25 views

Statistics of Length Eccess in Shortest Path Calculations

I am trying to quantify the error that arises in the following problem: let $\mathcal{T}$ be a tiling of the plane and the task is to calculate shortest paths in the network $\mathcal{N}$ of the union ...
4
votes
1answer
124 views

what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!
4
votes
2answers
162 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
209 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
307 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) ...
2
votes
0answers
108 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
827 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
149 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
100 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
388 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
65 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
61 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
337 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 ...
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 ...
13
votes
1answer
729 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
882 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
476 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
347 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
966 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
102 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 ...
3
votes
1answer
93 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
174 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 ...
19
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
176 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 ...
9
votes
1answer
160 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 ...
2
votes
0answers
47 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 ...