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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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Least number of circles required to cover a continuous function on a $[a,b]$

I asked this question on MSE here. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with fixed radius $r$ required to cover the graph of $f$? It is easy to ...
pie's user avatar
  • 331
5 votes
1 answer
696 views

An unpublished calculation of Gauss and the icosahedral group

According to p. 68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices ...
user2554's user avatar
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15 votes
1 answer
452 views

Dividing a polyhedron into two similar copies

The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right ...
Kepler's Triangle's user avatar
5 votes
3 answers
542 views

An inequality in an Euclidean space

For $n\geq 1$, endow $\mathbb{R}^n$ with the usual scalar product. Let $u=(1,1,\dots,1)\in\mathbb{R}^n$, $v\in {]0,+\infty[^n}$ and denote by $p_{u^\perp}$ and $p_{v^\perp}$ the orthographic ...
G. Panel's user avatar
  • 619
7 votes
4 answers
724 views

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

I asked this question on MSE here. Given the length of the sides of a quadrilateral $a,b,c,d$ ( side lengths are given in order around the quadrilateral) the area of the quadrilateral is less than or ...
pie's user avatar
  • 331
6 votes
1 answer
163 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 $...
Glen Whitney's user avatar
2 votes
1 answer
79 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 ...
Grigori's user avatar
  • 23
14 votes
2 answers
2k 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 ...
M. Sperling's user avatar
19 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 ...
Ivan Meir's user avatar
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8 votes
1 answer
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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 ...
RavenclawPrefect's user avatar
1 vote
0 answers
73 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 ...
user967210's user avatar
3 votes
1 answer
129 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 ...
Anton Kapustin's user avatar
3 votes
2 answers
174 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 ...
Anton Kapustin's user avatar
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 ...
Mozibur Ullah's user avatar
1 vote
0 answers
35 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 - \...
Titouan Vayer's user avatar
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 ...
Ethan Dlugie's user avatar
  • 1,267
1 vote
1 answer
319 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 ...
Alainty's user avatar
  • 11
2 votes
1 answer
129 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-...
Manfred Weis's user avatar
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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. ...
Anton Petrunin's user avatar
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:\ \...
Manfred Weis's user avatar
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1 vote
1 answer
231 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 ...
José Hdz. Stgo.'s user avatar
11 votes
1 answer
395 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?
trionyx's user avatar
  • 111
6 votes
2 answers
235 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 ...
Julian Newman's user avatar
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 $...
JE2912's user avatar
  • 399
4 votes
3 answers
952 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?
Humberto José Bortolossi's user avatar
3 votes
1 answer
203 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\...
Taras Banakh's user avatar
10 votes
1 answer
499 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 ...
Taras Banakh's user avatar
6 votes
1 answer
347 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, ...
Taras Banakh's user avatar
1 vote
0 answers
15 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 ...
Haoran Chen's user avatar
0 votes
0 answers
64 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-...
Ron  Tubman's user avatar
11 votes
3 answers
543 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 $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
Taras Banakh's user avatar
2 votes
1 answer
77 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 ...
Manfred Weis's user avatar
  • 12.7k
1 vote
1 answer
85 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 $q_0,q_1,\dots,q_n$...
Manfred Weis's user avatar
  • 12.7k
2 votes
0 answers
205 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 ...
Đào Thanh Oai's user avatar
3 votes
0 answers
205 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 ...
Stanley Yao Xiao's user avatar
0 votes
0 answers
24 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 ...
plm's user avatar
  • 972
6 votes
1 answer
165 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} ...
stupid_question_bot's user avatar
5 votes
1 answer
412 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 ...
Jim Conant's user avatar
  • 4,838
0 votes
0 answers
113 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 ...
Spiridon Kuruklis's user avatar
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 ...
Timothy Chow's user avatar
  • 79.3k
0 votes
0 answers
76 views

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 \...
RyanChan's user avatar
  • 550
11 votes
0 answers
435 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 ...
Arthur Queiroz Moura's user avatar
1 vote
1 answer
145 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 ...
DavidHume's user avatar
  • 741
0 votes
0 answers
27 views

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 ...
Menander I's user avatar
0 votes
1 answer
47 views

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,\,\...
Manfred Weis's user avatar
  • 12.7k
2 votes
1 answer
139 views

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 = ...
Ulissex 's user avatar
7 votes
3 answers
231 views

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 ...
Steve's user avatar
  • 81
-1 votes
2 answers
155 views

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? (...
user51232's user avatar
14 votes
1 answer
276 views

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 ...
tuna's user avatar
  • 523
16 votes
0 answers
389 views

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 ...
Noah Schweber's user avatar

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