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

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19
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0answers
350 views

Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
19
votes
0answers
233 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 ...
17
votes
0answers
284 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?
16
votes
0answers
229 views

What Rubik's Twist configuration has the lowest visible surface area?

The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that ...
11
votes
0answers
176 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 ...
11
votes
0answers
343 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 + ...
10
votes
0answers
94 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
10
votes
0answers
452 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 ...
9
votes
0answers
201 views

Mathematical properties of two dimensional projection of three dimensional rotated object

Please be gentle as I do not have any degree in maths. By using a compass/straighedge method to construct Metatron's cube, a regular dodecahedron can be inferred from intersecting points. I'm looking ...
8
votes
0answers
91 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
8
votes
0answers
383 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 $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
8
votes
0answers
216 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
30 views

Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
7
votes
0answers
79 views

Fast search of local positive quadruples on the sphere

Let $U = \{u_{1}, u_{2}, \ldots, u_{n}\} \subset \mathbb{R}^{3}$ be the finite set of points on the unit sphere in $\mathbb{R}^{3}$: $||u_{i}||_{2} = 1$ Definition: Quadruple of points $(u_{i}, ...
7
votes
0answers
71 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
75 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 ...
7
votes
0answers
141 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
132 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
557 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 ...
7
votes
0answers
170 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
177 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
215 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
147 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
239 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 ...
7
votes
0answers
414 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
172 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
868 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
57 views

Finding equations for projective curves, low genus, Riemann-Roch.

Let $C \subset \mathbb{CP}^n$ be a nonsingular projective curve, and let $L \subset \mathbb{CP}^n$ be a hyperplane. We have that $L \cdot C$ is a divisor $H$ on $C$ if $C \subset L$. Let $R = ...
6
votes
0answers
65 views

What would a tesseract actually look like?

Say our world is actually 4-dimensional, and I have a tesseract sitting on my table. To a being only capable of perceiving 3 dimensions, what would it actually look like? Would it just look like a ...
6
votes
0answers
74 views

Reference for Cavalieri's principle

Does someone know of a reference where I can see Cavalieri's principle (basically the principle that generalized areas can be obtained by multiplying "base times height" -- for constant ...
6
votes
0answers
78 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
6
votes
0answers
97 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
77 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
29 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?
6
votes
0answers
358 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 ...
6
votes
0answers
66 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 ...
6
votes
0answers
184 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
171 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
92 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
163 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
224 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
358 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
134 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
138 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
89 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
579 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
401 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
645 views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots ...
6
votes
0answers
107 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
69 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?