For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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17
votes
0answers
192 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
14
votes
0answers
205 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
10
votes
0answers
160 views

Which power means are constructible?

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean ...
10
votes
0answers
343 views

The same bit of trivial algebra in two different places?

The Villarceau circles are things whose existence is surprising. To find radii of Villarceau circles, I stupidly went through a bit of trigonometry and got a much simpler result than I expected, and ...
10
votes
0answers
282 views

Computing the volume of a region on the unit $n$-sphere

I would like to compute the surface volume of a region on the unit $n-1$-sphere: $$x_1^2 + \dots + x_i^2 + \dots + x_n^2 = 1,$$ bounded by an ellipsoid $$a_1x_1^2 + \dots + a_ix_i^2 + \dots + ...
8
votes
0answers
79 views

Lines in $n$ dimensional space

Suppose we have $n+1$ lines in $\mathbb{R}^n$ and let $\gamma$ be the smallest angle across all pairs of lines. I am wondering what the upper bound on $\gamma$ is (probably depending on $n$). When ...
8
votes
0answers
266 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
8
votes
0answers
182 views

Does this graph have a name?

Does graph shown below from the paper Dissection Graphs of Planar Point Sets by P. Erdos, L. Lovasz, A. Simmons, and E.G. Straus have a name? Does it come from a family of related graphs?
7
votes
0answers
65 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ ...
7
votes
0answers
55 views

Largest rectangle bounded under a function

Let $f$ be a positive monotonically increasing real function in $[0,1]$. Let $F$ be the area under the curve of $f$ ($F=\int_0^1{f(x)dx}$) For every $x\in[0,1]$, let $G(x)=f(x)*(1-x)$ = the area of a ...
7
votes
0answers
352 views

How prove $\sum_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum_{cyc}h_{a}}$

Question: let $\Delta ABC$,and the altitude is $h_{a},h_{b},h_{c}$,where $AB=c,BC=a,AC=b$ and for any $P$ show that $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
7
votes
0answers
116 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
7
votes
0answers
119 views

More approximately orthogonal vectors than the dimension of the space

It is impossible to find $n+1$ mutually orthogonal unit vectors in $\mathbb{R}^n$. However, a simple geometric argument shows that the central angle between any two legs of a simplex goes as $\theta ...
7
votes
0answers
150 views

Rotations of a tetrahedron

Let $P$ be a tetrahedron inside an sphere such that all of its vertices are on the surface of the sphere. Suppose that three quarters of sphere's surface is colored black. Show that there is a ...
7
votes
0answers
160 views

What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? ...
7
votes
0answers
111 views

Biggest Little Polyhedron

The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1. Let's add a dimension and find the largest volumes. What is the biggest little polyhedron ...
7
votes
0answers
193 views

What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
7
votes
0answers
129 views

The rigorization of naive geometry angles and length

There are a number of claims from elementary school that I just remembered I don't actually mathematically know. Let's start with some specific examples and perhaps the rigorization will inspire me ...
7
votes
0answers
375 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where ...
7
votes
0answers
167 views

Volume of a sphere by “adding” half-spheres of lower dimension

I'm wondering about different ways to compute the volume of an $n$-sphere. Please see the wikipedia page for one method to compute the volume via hyperspherical coordinates: ...
7
votes
0answers
792 views

How can I tell when two cubic Bézier curves intersect?

I'm working a little program that converges on vector-based approximations of raster images, inspired by Roger Alsing's genetic Mona Lisa. (I started on this after his first blog post two years ago, ...
6
votes
0answers
85 views

Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product: $$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$ Is the cosine of the angle between them. For an application I was ...
6
votes
0answers
65 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
6
votes
0answers
70 views

I would like to show that all reflections in a finite reflection group $W :=\langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}.$

I would like to show that all reflections in a finite reflection group $W := \langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}$ for some $i$ and some $w \in W$ Clearly all such elements ...
6
votes
0answers
177 views

Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $$\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad ...
6
votes
0answers
165 views

Solution set of inequalities in $\mathbb{R}^6$

Let $\theta\in (0,1)$ fixed. We define $A_1$ be the set of all $(a_1,a_2,a_3,a_4,a_5,a_6)\in (0,1)^6$ such that the following conditions hold: \begin{equation} (1) \quad a_2\le a_1, a_4\le a_3, a_6 ...
6
votes
0answers
84 views

Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. ...
6
votes
0answers
372 views

bisectors of an angle in a triangle intersect at a single point - proof verification

Let´s consider a general triangle ABC. Let´s draw two angle bisectors from vertices A and B. It is obvious that these two angle bisectors intersect at a single point X. Since X lies on the angle ...
6
votes
0answers
141 views

Family of geometric shapes closed under division

The family of rectangles has the following nice properties: Every rectangle $R$ can be divided to two disjoint parts, $R_1 \cup R_2 = R$, such that both $R_1$ and $R_2$ are rectangles (i.e. belong ...
6
votes
0answers
210 views

What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two ...
6
votes
0answers
342 views

Slices of a hypercube

Take the unit $d$-cube with vertices $\{0,1\}^d$, and restrict to the vertices that lie between (or within one of) a pair of parallel hyperplanes. These vertices form a graph whose edges are the edges ...
6
votes
0answers
126 views

Paper cylinder inside out

My question is related with paper folding: Given a cylinder of paper it is possible to turn it inside out using folding along lines. This is a Martin Gardner recreational puzzle. Secondly, ...
6
votes
0answers
131 views

Combinatorial vs. geometric symmetries of graphs and their drawings

Associated with a graph $G$ and its automorphism group $\text{Aut}(G)$ (reflecting its combinatorial symmetries) are drawings in the plane with - eventually - one or more (geometric) symmetry groups. ...
6
votes
0answers
86 views

Do balls optimize the boundary area for a fixed volume?

I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
6
votes
0answers
418 views

Good textbooks on Non-Euclidean Geometry?

I'm currently taking a class called Foundations of Geometry. We started with the stereographic projection and carried onward through fractional linear transformations, and now we are working with the ...
6
votes
0answers
395 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
6
votes
0answers
219 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
6
votes
0answers
105 views

Analytic caustics for 3D objects

Is it possible to efficiently calculate caustics for a given 3D object, like a torus, or a cube? To be more precise: let's assume that we have a 3d torus, resting on a 2d plane and a single light ...
5
votes
0answers
50 views

Three-colored plane

Let $\Pi$ is cartesian plane with the usual topology. $A, B, C$ are pairwise disjoint subsets $\Pi$, $A\cup B\cup C = \Pi$. Each of these sets is dense in $\Pi$: $\overline{A}=\Pi$, and ...
5
votes
0answers
59 views

How many unit squares can overlap a given unit square without overlapping each other?

How many unit squares can overlap a given unit square without overlapping each other? @calculus has managed to arrange 7 squares (see this GeogebraTube page). This seems like the maximum ...
5
votes
0answers
151 views

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
5
votes
0answers
62 views

How big is a tetrahedron?

Let $T$ be a tetrahedron with volume $vol(T)$ and edge lengths $a,b,c,d,e,f$ and let $sum(T) = a^3 + b^3 + ... + f^3$. We wish to compare $vol(T)$ with $sum(T)$. [ IMO (1961 #2 ) handles the case of ...
5
votes
0answers
44 views

From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
5
votes
0answers
23 views

Product to vertices in triangle maximal

Suppose we're given a triangle $ABC$. At which interior point $T$ is the product of distances $|AT|\cdot |BT|\cdot |CT|$ maximal? Is it a known point, like the centroid or incenter?
5
votes
0answers
234 views

Any weird 'modular like mathematical space' that behaves like as if it is infinite until a threshold is reached?

(Using the advice from Mathoverflow, I have rephrased and splitted up the question) (Might be a bit layman because I don't have rigorous math term to describe the concept) Generalize it to ...
5
votes
0answers
205 views

Random 3D points uniformly distributed on an ellipse shaped window of a sphere

How can I generate random points uniformly distributed on the surface of a sphere such that a line that originates at the center of the sphere, and passes through one of the points, will intersect a ...
5
votes
0answers
51 views

Regular polygons constructed inside regular polygons

Let $P$ be a regular $n$-gon, and erect on each edge toward the inside a regular $k$-gon, with edge lengths matching. See the example below for $n=12$ and $k=3,\ldots,11$.       Two ...
5
votes
0answers
124 views

How many points you should draw in the square at least?

There is a square, which side length is $2$, To ensure there exists a triangle in the square, with an area less than $0.5$, how many points should you draw in the square at least. the goal is for all ...
5
votes
0answers
104 views

A “Paradoxist Geometry”

This question is about "how badly can we 'break' the laws of geometry and still have something which is deserving of the name geometry?". It is named after something else I saw of the same name which ...
5
votes
0answers
61 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...