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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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 - \...
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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 ...
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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 ...
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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-...
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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. ...
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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:\ \...
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"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 ...
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Smallest sphere containing three tetrahedra?
What is the smallest possible radius of a sphere which contains 3 identical plastic tetrahedra with side length 1?
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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 ...
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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 $...
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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?
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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\...
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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 ...
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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, ...
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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 ...
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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-...
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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 $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
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1
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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 ...
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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 $q_0,q_1,\dots,q_n$...
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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 ...
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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 ...
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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 ...
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$\mathbb{Q}$-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
$$\theta_{a,b} ...
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On the aperiodic monotile
One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
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Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle
The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...
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Aperiodic monotile without reflections?
The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
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Geometry of inner products between the unit vector and several given vectors
Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
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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 ...
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$1$-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 ...
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0
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Alternative equivalence results for the constructibility of real numbers
Everyone is aware of the standard result from undergraduate field theory that a real number $\alpha$ is constructible by straightedge and compass if and only if there exists a finite sequence of field ...
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Constructing a polygon from another with collinearity constraints
Let $P$ be a closed polygon defined by the sequence
$p_0,\,\dots,\,p_{n-1},p_0$ of points.
Question:
how can one construct, with straightedge and compass alone, another sequence of points $q_0,\,\...
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Finding angle with geometric approach [closed]
I would like to solve the problem in this picture:
with just an elementary geometric approach. I already solved with trigonometry, e.g. using the Bretschneider formula, finding that the angle $ x = ...
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3
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Minimum diameter of set inscribed in a unit sphere
For a study of the stability of certain maps taking values in a sphere I have the following question.
Let $A$ be a subset of $\mathbb{R}^n$. Suppose $A$ lies in a unit ball, but in no ball of smaller ...
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Condition to be concyclic [closed]
What condition would you impose upon $n$ points on a plane of which no three points are collinear so that they are
concyclic if the distances of each point from the all remaining points are known? (...
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1
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How many distances are required to calculate all distances among $n$ points in the Euclidean plane?
I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
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Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
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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, ...
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1
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Orthogonal projection of a point centrally-symmetric closed convex subset of $\mathbb R^n$ never expands the coordinates of the point
Let $C$ be a closed convex subset of $\mathbb R^n$ which is symmetric about the standard coordinate axes. For example, think of $C$ as the unit-ball for an $\ell_p$-norm, for some $p \in [1,\infty]$. ...
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A 'natural' enumerable metric space with integral distances which is essentially the Euclidean space
It is easy to construct a metric space $E_d$ such that all points
of $E_d$ are at mutually integral distance and such that there is a map $\varphi$ from $E_d$ into the $d$-dimensional Euclidean space ...
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How many equilaterals have vertices intersections of angle trisectors of a triangle?
The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral.
In the paper Trisectors like Bisectors with Equilaterals ...
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0
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Fashioning higher precision tool from a lower precision tool
I'm not sure where to ask this question.
Suppose I have one or more rulers with which I can measure distances with up to 1 mm error. Is there a way I could make another tool of higher precision (e.g. ...