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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Solving special multivariable limits by Euclidean geometry
General Problem: Inspect function $L(a)$ for $a\in \mathbb{R}^{n}$, given that:
$$L(a)=\lim_{x\to 0_{+}}\frac{x^{a}}{F(x)}$$
Notation legends:
$x=(x_1,\ldots,x_n),\ a=(a_1,...,a_n),\ x^a=x_{1}^{a_1}\...
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Does this result above six points follow have a name?
Does this result above six points follow have a name?
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ ...
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Pólya's orchard problem among Gaussian primes
Quoting myself from an earlier post:
Pólya's orchard problem asks for which radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of ...
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Sum of Simplex Volumes with Corners from Points in Convex Configuration
Question:
given $k,\,k>n$ points in convex configuration and general position in $n$ dimensional Euclidean space, i.e. no $n+1$ points of which are co-hyperplanar,
what can be said about how the ...
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How can one test whether a given analytic curve in the plane is algebraic or not?
Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
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Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
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Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
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Is every connected subgroup of a Euclidean space closed?
The question listed above (in the context of the complex numbers, but it is a reasonable question to ask in any dimension) was asked by a student in my complex analysis class, and I did not have an ...
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Upper bounds for minimum angle
What are the latest and best results on the asymptotic upper bound for the minimum angle between any pair of rays among $n$ rays in $\mathbb{R}^3$?
Any helpful answer would be appreciated. Thank you!
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A necessary and sufficient condition for three diagonals of a hexagon to be concurrent
When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric ...
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Is this a new result about hexagon?
Let a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent:
Three lines $AA', BB', CC'$ are concurrent (let the point of ...
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Another Butterfly theorem — Conway like circle
Have You seen these result as follows before?
In Figure 1: $AA'=BB'=tAB$; $CC'=DD'=tCD$, where t is real number then $ABCD$ is a cyclic quadrilateral iff $A'B'C'D'$ is a cyclic quadrilateral.
In the ...
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How many Tverberg partition are in cloud of points? [closed]
Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect.
For example, $d=2$, $r=3$, 7 points:
Let $p_1, p_2,...
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Incenter-of-mass of a polygon
"Circumcenter of mass"
is a natural generalization of circumcenter to non-cyclic polygons.
CCM(P) can be defined as the weighted average of the circumcenters
of the triangles in any ...
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What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
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If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?
If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?
I’m happy to assume the polyhedron is simply connected, ...
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Triangle centers formed a rectangle associated with a convex cyclic quadrilateral
Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that:
Let $ABCD$ be a convex cyclic quadrilateral.
...
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Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
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What axiomatic system does AlphaGeometry use?
In January 2024, researchers from DeepMind announced AlphaGeometry, a software able to solve geometry problems from the International Mathematical Olympiad using a combination of AI techniques and a ...
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Shortest polygonal chain with $6$ edges visiting all the vertices of a cube
I am trying to find which is the minimum total Euclidean length of all the edges of a minimum-link polygonal chain joining the $8$ vertices of a given cube, located in the Euclidean space. In detail, ...
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Is the formula known? and can we generalized for higher dimensions?
In this configuration as follows, we have a nice formula:
$$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$
Is the formula known? and can we generalized for higher dimensions?
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$k$-subset with minimal Hausdorff distance to the whole set
Let $(\mathcal{M}, d)$ be a metric space. Let $k \in \mathbb{N}$. Let $[\mathcal{M}]^k$ be the set of $k$-subsets of $\mathcal{M}$. Consider the following problem:
$$ \operatorname*{argmin}_{\mathcal{...
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Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
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In how many ways is it possible to order the sides and diagonals according to their length for all n-gons?
If we'd do it for example for 4-gons, for quadrilaterals, we could start with all the possible quadrilaterals.
We could say that the four vertices are a,b,c and d.
And then we'd have 6 lines, I mean,
...
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Least number of circles required to cover a continuous function on $[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 closed circles with fixed radius $r$ required to cover the graph of $f$?
It is ...
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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 ...
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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 ...
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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 ...
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What are the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?
I asked this question on MSE here.
Given a quadrilateral with side lengths $a,b,c$ and $d$ (listed in order around the perimeter), t's known that the area, is always less than or equal to $\frac{(a+...
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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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