Questions tagged [euclidean-geometry]

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 these.

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1 vote
1 answer
317 views

Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
6 votes
1 answer
136 views

Inscribing one regular polygon in another

Say that one polygon $P$ is inscribed in another one $Q$, if $P$ is contained entirely in (the interior and boundary of) $Q$ and every vertex of $P$ lies on an edge of $Q$. It's clear that a regular $...
14 votes
2 answers
1k views

How can I (semi-formally) convince myself that Euclidean geometry comports with visual intuition?

I originally posted this question on Math.SE and received some interesting comments but no answers. Now that some time has passed I thought that it might be appropriate to post here as well; perhaps ...
2 votes
1 answer
68 views

Difference of probabilities of two random vectors lying in the same set

Suppose I have to random vectors: $$\mathbf{z} = (z_1, \dots, z_n)^T, \quad \mathbf{v} = (v_1, \dots, v_n)^T$$ and set $A \subset \mathbb{R}^n$. I want to find an upper bound $B$ for the following ...
3 votes
1 answer
201 views

Another implication of the Affine Desargues Axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
11 votes
3 answers
541 views

Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
18 votes
1 answer
1k views

Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

Does a function from $\mathbb{R}^2$ to $\mathbb{R}$ which sums to 0 on the corners of any unit square have to vanish everywhere? I think the answer is yes but I am not sure how to prove it. If we ...
25 votes
1 answer
2k views

The abc-conjecture as an inequality for inner-products?

The abc-conjecture is: For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have: $$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
7 votes
1 answer
120 views

Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?

(Originally on MSE.) Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial ...
1 vote
0 answers
66 views

The intersection of $ n $ cylinders in $ 3D$ space

I posted the question on here, but received no answer I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
3 votes
1 answer
126 views

Bounding distance to an intersection of polyhedra

Let $P$ and $Q$ be polyhedra in ${\mathbb R}^m$ with a non-empty intersection. I believe there should exist a constant $C_{PQ}>0$ such that for any point $x\in {\mathbb R}^m$ the following ...
3 votes
2 answers
164 views

Bounding distance to a polyhedron

I need to estimate the Euclidean distance from a point $x\in {\mathbb R}^m$ to a polyhedron $P\subset {\mathbb R}^m$ in terms of distances from $x$ to the tangent hyperplanes which define $P$. By a ...
3 votes
1 answer
880 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
17 votes
3 answers
2k views

Is symmetric power of a manifold a manifold?

A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^n(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_m$, where product is done $...
16 votes
4 answers
2k 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 http://en.wikipedia....
54 votes
2 answers
5k views

Automatically solving olympiad geometry problems

Warning: I am only an amateur in the foundations of mathematics. My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about ...
1 vote
1 answer
226 views

"On models of elementary elliptic geometry"

While perusing p. 237 of the 3rd ed. of Marvin Greenberg's book on Euclidean and non-Euclidean geometries, I learned that it can actually be proven that "all possible models of hyperbolic ...
10 votes
3 answers
2k views

Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem ...
1 vote
0 answers
29 views

Characterization of Gaussian Gram matrices

From Euclidean geometry we know that a matrix $C$ is a matrix of squared Euclidean distances between some points if and only if $-\frac{1}{2} H D H \succeq 0$ (positive semi-definite) with $H = (I - \...
12 votes
2 answers
1k views

Group generated by two irrational plane rotations

What groups can arise as being generated by two rotations in $\mathbb R^2$ by angles $\not \in \mathbb Q\pi$? If the centers of the rotations coincide, then the rotations commute and generate some ...
2 votes
1 answer
119 views

An alternative to Cayley Menger determinant for calculating simplex volume

I recently came across the determinant of a symmetric $3\times 3$ matrix $\begin{pmatrix} 2a^2& a^2+b^2-c^2& a^2+d^2-e^2\\ a^2+b^2-c^2& 2b^2& b^2+d^2-f^2\\ a^2+d^2-...
35 votes
4 answers
3k views

Psychological test for Euclidean geometry [closed]

There is the so-called FCI test. It contains a list of questions such that anyone who can speak will have an opinion. Based on the answers one can determine if the answerer knows elementary mechanics. ...
13 votes
9 answers
4k 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 ...
0 votes
4 answers
994 views

Characterization of angles trisectable with straightedge and compass

Lindemann's proof 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, ...
0 votes
0 answers
32 views

Enumeration of flat integral $K_4$

Question: What is known about the enumeration of all $(a,b,c,d,e,f)\in\mathbb{N}^6_+: \\ \quad\operatorname{GCD}(a,b,c,d,e,f)=1\ \\ \land\ \exists \lbrace x_1,x_2,x_3,x_4\rbrace\subset\mathbb{E}^2:\ \...
6 votes
2 answers
234 views

Does "perpendicular phase incoherence" satisfy the triangle inequality?

I asked this question at https://math.stackexchange.com/q/4783968/222867, but even after a 200-point bounty, no solution was provided, only some thoughts regarding possible directions. So I'm now ...
11 votes
1 answer
388 views

Smallest sphere containing three tetrahedra?

What is the smallest possible radius of a sphere which contains 3 identical plastic tetrahedra with side length 1?
16 votes
1 answer
481 views

A textbook on foundations of geometry in spirit of Tarski

I am interested in a textbook for studying (and teaching) foundations of geometry in the spirit of Tarski. I know that there is a rather old German book [W. Schwabhäuser, W. Szmielew, A. Tarski, ...
8 votes
2 answers
1k views

Quadrature of the Lune

What is a good reference for the following result which I believe is proved by Tchebotarev. There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were ...
4 votes
3 answers
922 views

Is there a pyramid with all four faces being right triangles? [closed]

If such a pyramid exists, could someone provide the coordinates of its vertices?
15 votes
1 answer
3k views

3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2 -piece dissection of a square to an equilateral triangle. Can anyone confirm that this is ...
11 votes
1 answer
509 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 ...
10 votes
1 answer
489 views

A projective plane in the Euclidean plane

Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is ...
1 vote
1 answer
143 views

$1$-Lipschitz map from hyperbolic to Euclidean plane

I'm trying to find a reference to the following statement. Define a function $f$ from the hyperbolic plane (in the Poincaré unit disc model using polar coordinates) to the Euclidean plane (using polar ...
6 votes
1 answer
341 views

Desargues ten point configuration $D_{10}$ in LaTeX

I want to draw the Desargues configuration $10_3$ in LaTeX using the standard picture environment, which allows only lines with the slopes $n:m$ where $\max\{|n|,|m|\}\le 6$. Is it possible? If not, ...
1 vote
0 answers
13 views

Estimate on the minimum distance from integer points on some fixed hyperplanes to a moving hyperplane

Suppose in $\mathbf{R}^n$ there are $m$ given hyperplanes $\Pi_j:\sum_{i=1}^n c_{i,j}e_i=0$ all of which go through the origin, and all the coefficients $c_{i,j}$ are rational (you can make them all ...
0 votes
0 answers
58 views

Finding a point that minimizes sum of distances to a given set of lines

Given a set $L$ of size $n$ of lines in $\mathbb{R}^d$, find a point $x \in \mathbb{R}^d$ that minimizes: $$\sum\limits_{l\in L}\min\limits_{y\in l} {\lvert \lvert x-y \rvert\rvert}^2$$ I wrote a 1.5-...
25 votes
6 answers
2k views

Are there infinitely many "generalized triangle vertices"?

Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
2 votes
1 answer
73 views

Calculating a relaxed Delaunay Triangulation

The triangles of a planar Delaunay Triangulations are essentially characterized by the property that no triangle's corner is inside another triangle's circumcircle; Delaunay Triangulations can be ...
1 vote
1 answer
83 views

Is every triangulation the projection of a convex hull

Question: given the triangulation $T$ of a set $P$ of $n$ points $p_1,\dots,p_n$ in the euclidean plane whose convex hull is a triangle, can we always find a set $Q$ of $n+1$ points $q_0,q_1,\dots,q_n$...
2 votes
0 answers
202 views

A generalization of the Archimedean circle

I proposed a generalization of the Archimedean circle : In this figure $M$ is the midpoint of $AB$, $DE$; $(G)$, $(H)$, $(M)$ are the semicircles. Then two yellow circles are congruent. Question: Is ...
3 votes
0 answers
201 views

Which manhole covers fall through their holes?

Apparently one of the reasons why all manhole covers are shaped like discs is because for any other shape, the manhole cover would fall through its own hole. As stated this is not necessarily a ...
0 votes
0 answers
23 views

Piecewise affine-isometric maps of polytopal graphs into the plane

There are well-known "relatively faithful" representations of the polytopal metric subgraphs $C^n\subseteq\mathbb R^n$ (with the euclidean distance, for all $n\geq 0$) of hypercubes into the ...
17 votes
0 answers
725 views

Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
22 votes
1 answer
1k views

Aperiodic monotile without reflections?

The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
6 votes
1 answer
164 views

$\mathbb{Q}$-rank of the space of angles of pythagorean triples

A pythagorean triple is a triple of integers $(a,b,c)$ with $a^2 + b^2 = c^2$. Given a triple, $(a/c, b/c)$ is a point on the unit circle, so we may associate to it the normalized angle $$\theta_{a,b} ...
90 votes
5 answers
4k views

Does this property characterize straight lines in the plane?

Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
11 votes
0 answers
434 views

What sequence maximizes the final distance?

This problem was created by professor Ronaldo Garcia from Universidade Federal de Goiás (UFG) and he showed it to me at an event in my university. This problem has a lot of history and he told me he ...
5 votes
1 answer
403 views

On the aperiodic monotile

One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
0 votes
0 answers
111 views

Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle

The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...

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