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

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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 ...
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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?
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366 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 ...
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54 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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60 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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137 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 ...
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106 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 ...
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89 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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52 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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72 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 ...
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106 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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The space of paths

Let $ (M,g) $ be a compact Riemannian manifold, and let the path space $ \Omega $ of $ M $ be the set of equivalence class of smooth maps $ \gamma : [0,1] \rightarrow M $ (equivalent under ...
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Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. ...
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197 views

Hexagonal circle packings in the plane

I have been reading the paper Spiral hexagonal circle packings in the plane (Alan F. Beardon, Tomasz Dubejko and Kenneth Stephenson, Geometriae Dedicata Volume 49, Issue 1, pp 39-70), which proves ...
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134 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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337 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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157 views

Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
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1k 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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328 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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236 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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Intersection of 2 high dimensional balls

I am working in $\mathbb{R}^n$. I have a large ball radius $2 \delta \sqrt{n}$. Centered on the boundary of this ball, I have a smaller ball radius $\delta$. What is the probability that a ...
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23 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...
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59 views

Writing circles as $|z-a| = \lambda |z-b|$ for the same $a,b$

My problem is in the context of the complex plane. I want to know if given two disjoint, not concentric circles $C_1,C_2\subset \mathbb{C}$, can you find $a,b\in \mathbb{C}$ such that $$C_1=\{z\in ...
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Generic cubic in $\mathbb{P}_{\mathbb C}^3$

I read that "two generic cubics are not projectively equivalents in $\mathbb P_{\mathbb C}^3$". I want to know if I understand well the statement. Does "generic cubics" means that all the cubics ...
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What is the name for planar curve that intersect with every line at most $2k$ time

More exactly I want to know is there research in planar (closed) curve (there maybe nodes as singularities) s.t. every line in general position would intersect it at no more than $2k$ times. $k=1$ is ...
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60 views

What's an algebraic curve's polar line for?

I can take an algebraic curve, and I can draw its first polar. By this, I mean that I can take an arbitrary point not on the algebraic curve, and then I can identify all the points on that algebraic ...
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53 views

Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...
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64 views

Centroids and Harmonic Means

A triangle $ABC$ with centroid $G$ is such that a line $l$ passing through $G$ intersects $AB$, $BC$, and $AC$ at $H, I, J$, respectively. Show that out of the 3 distances $d(G, I), d(G, H), d(G, J)$, ...
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54 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
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57 views

Circles Of Descartes

Insired by this question, where the objective is to calculate the shaded area in the above diagram, I noticed that the inscribed circle has the following relation to the circles it rests on ...
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36 views

Area covered by one disk more than by two disks

Given are three unit disks on the plane. Let $A$ be the area of the plane covered by exactly $1$ disk. Let $B$ be the area of the plane covered by exactly $2$ disks. Prove that $A\geq B$. ...
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126 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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57 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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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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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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184 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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Selecting k vectors with maximum spread out of a set of n vectors

Given a set $\mathcal{V}$ of $n$ vectors, find a subset $\mathcal{V}_k = \mathcal{V} - \mathcal{V}_{n-k}$ containing $k$ maximally spread vectors. Intuitively, these $k$ vectors should be spread as ...
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132 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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198 views

Can this be only solved by trial and error?

The following question was asked in a competitive exam Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the ...
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184 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
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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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238 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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568 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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279 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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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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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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79 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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157 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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188 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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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 ...