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

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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 ...
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279 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
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230 views

Points in the plane at integer distances

Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds: For all $a,b$ with ...
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99 views

How far away is that cloud?

A few weeks ago I was on an airplane and to pass the time started thinking about this problem. Using the following information, I wanted to know how far away a cloud I could see was. Under some ...
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40 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
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80 views

Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
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48 views

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point ...
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136 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
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47 views

What does the metric matrix G tell us here

Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$. $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial ...
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82 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
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43 views

The largest regular m-gon that fits inside a regular n-gon

This question just popped into my head while doing some "for fun" math. More precisely: Let $m,n\in\Bbb{Z};m,n>2$. Let $P$ be a regular $n$-gon (let's say $P$ is the convex hull of the $n$ $n$th ...
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131 views

Focus-Focus Definition for a Parabola

I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant. ...
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81 views

Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
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108 views

In how few points can a continuous curve meet all lines?

Inspired by this (currently closed) question, I'm wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for ...
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118 views

Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
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67 views

Spacing nodes by moving the shortest distance possible.

I have a list of N nodes with positions $(x, y)$ each. I want to move each node the shortest possible distance such that every node is placed on the radius $R$ from at least one other node, and is at ...
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168 views

Invariant of matrix under elementary transformations

$\DeclareMathOperator{\rank}{rank}$ Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix $$ B = \begin{bmatrix} A & b \\ b^T & c ...
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206 views

Second derivative of a metric in terms of the Riemann curvature tensor.

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial ...
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210 views

How to find the maximum diagonal length inside a dodecahedron

I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of 2.319914107*10^89 meters. I am not sure if any other information than that is needed, if it is I ...
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289 views

Reflections in regular polygons

I was thinking about regular polygons and paths beginning at a vertex such that whenever the path hits a side, it has a mirror reflection (angle of incidence equalling the angle of reflection) and ...
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219 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
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127 views

Union of two self-intersecting planes is not a surface

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose ...
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85 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1] For ...
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174 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
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73 views

Spectrum of laplacian in a parallelogram

Is the spectrum of the laplacian on an arbitrary parallelogram with dirichlet boundary conditions known?
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124 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
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165 views

Tilings of the plane

There are many possible tilings (or tesselations) of the plane: periodic ones by a - necessarily - finite number of prototiles (e.g. regular tilings) aperiodic ones by a finite number of prototiles ...
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52 views

Reference Request: Background/Extention on Bill Thurston's Lecture 'The Mystery of Three-Manifolds'

I found Thurston's lecture The Mystery of Three- Manifolds fascinating but, well, mystifying. For those with insufficient time to watch the video, he establishes correspondence (in his very intuitive ...
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273 views

“Green Globs” question

When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are ...
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927 views

Lowest surface-to-volume ratio for an uncovered vessel

It is well known that a sphere has the lowest surface to volume ratio. However, a related question is: What is the shape that gives the lowest surface to volume ratio if you do not include the top in ...
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71 views

Convex pentagons are similar if conformally equivalent.

The problem: Suppose two convex pentagons $A$ and $B$ have equal interior angles (that is, $A=A_1A_2A_3A_4A_5$ and $B=B_1B_2B_3B_4B_5$) with $\angle A_j =\angle B_j$ for each $j\in\{1,\ldots,5\}$). ...
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157 views

Convex hull of balls

The convex hull is defined as the smallest convex set containing a set of points. Now we want to generalize it to a set of balls. If these balls have the same radius, then it can be shown that a ball ...
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158 views

Can a rectangle be written as a finite almost disjoint union of squares?

Given a closed rectangle $R$ are there closed squares $(S_{i})_{1\leq i\leq n}$ such that $R=\underset{1\leq i\leq n}{\cup}S_{i}$ and $S_{i}^{\circ}\cap S_{j}^{\circ}=\varnothing$ for $i\neq j$ ?
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Curve - non singular curve and its genus

Help me please with this problem:$ X \subset \mathbb{P}^{2}$ defined as $x^{3}y+y^{3}z+z^{3}x=0$ 1.Prove X - non singular curve and find its genus. 2.Prove X - maximal curve over $F_{8}$ field, and ...
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171 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
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209 views

how do you map a sphere to a cube

I want to map a sphere to a cube in order to create a panoramic tour like the one given here But I don't know how can you obtain images like This image is one of the cube's faces. What I tried was ...
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983 views

Calculate “Volume” and “Surface Area” of a voxel-based sphere

By "voxel-based sphere" I mean a sphere made up of cubes. Sorry if that is not the correct terminology. Imagine a sphere made out of legos. Except each voxel is a cube (unlike most legos). Determining ...
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79 views

Removing highly scattered points in the plane to eliminate all high-area rectangles

The following question came up at tea today, and none of us managed to come up with an answer. I was wondering if anyone had any ideas. Does there exist a subset $X$ of $\mathbb{R}^2$ with the ...
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101 views

Can Kollros' Theorem w/ extra turns be seen via inversive geometry?

Kollros' Theorem, with extended turns allowed, says For every ring containing $p$ spheres, there exists a ring of $q$ spheres, each touching each of the $p$ spheres [...] if more than one turn is ...
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94 views

how understand if a segment is inside a lissajous curve

i am a programmer and not a math guru, but i like geometry. so if i'm not accurate in math terminology or i have folly question please sorry me. i'm drawing with a programming language the lissajous ...
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304 views

Can “anti-groups” and “anti-manifolds” be constructed? (and other “anti-objects”)

Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no ...
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44 views

Hands of the clock, Revisited.

It has already been answered (here) that it is impossible for the (continuously moving) hands of a clock to trisect the face of said clock. Even ideally the hour, minute, and second hand can never ...
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27 views

Help me visualize seven maximally-spaced points on a sphere

I'm interested in the subject of maximally separated points (e.g., the minimum of the distances between any two of the points is maximal) in various spaces, and I've been trying to think about how ...
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30 views

Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?

This question was posed to me by a friend (formulated as creating a peg to fit perfectly into holes of these shapes), and after an experiment in OpenSCAD it seems it is not possible - either one ...
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30 views

determining orientation of a sphere

I have a sphere and I am trying to find out its orientation with respect to ground frame. The sphere is as follows:- As can be seen from the image, different colors are painted on the sphere with ...
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36 views

What does the rotation group of $\mathbb{\bar{Q}}^n$ look like?

There's a structural difference between the rotation groups of $\mathbb{Q}^n$ and $\mathbb{R}^n$; in some abstract sense the former is 'small' (discrete?) while the latter is 'large'. I suspect that ...
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56 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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0answers
42 views

Area of ​​the intersection of two discs : Integral solution?

Here is the problem : We consider two cerlces that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
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72 views

Pythagorean like theorem for general spaces

There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for ...
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37 views

Ratio of the Volume of n-spherical cap to the volume of n-sphere

Assume an n-dimensional sphere with radius $R$ and volume $V^{(n)}_s$. Also assume a corresponding n-spherical cap with height $h$ and volume $V^{(n)}_c$. what is the ratio of two volumes? ...