Questions tagged [geometry]

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0answers
73 views

Minimum volume of intersection between two high-dim $\ell^1$-balls

Let $B_1$ and $B_2$ be two balls with the same radius, in $\mathbb R^n$ with the $\ell^1$ norm. The distance between the centers of $B_1$ and $B_2$ is $d(B_1, B_2)$. Is there any deterministic method ...
10
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2answers
700 views

Three half circles on the plane may not meet nicely

Let $H$ denote the union of the northen hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ ...
1
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1answer
111 views

Identity involving dot product of solid angle and gradient [closed]

How to prove following for $n\geq0$ ? $$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$ Where, at any point $\vec{r}$, the $\vec{\Omega}$...
3
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2answers
1k views

How to efficiently compute the generalized cross product?

It's possible to extend the well known cross product between two vectors in $\mathbb{R}^3$ to $n-1$ vectors in $\mathbb{R}^n$. Let $\vec{v_1}, \vec{v_2}, \dots, \vec{v}_{n-1} \in \mathbb{R}^n$ and $\...
0
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1answer
566 views

State-of-the-art geometry book? [closed]

For my best friend's birthday, I am looking for a geometry book. He's currently doing his math PhD and is really fond of geometry, especially hyperbolic or higher-dimensional ones, also interested in (...
5
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3answers
363 views

Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?

Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property: If we rotate $C$ around $p$...
26
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7answers
3k views

Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be. "Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
3
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1answer
208 views

Does there exist concept of angles and sides in Non-archimedian field geometry? [closed]

Question about Non-Euclidean Geometry, particularly about Non-archimedian Geometry: I am studying and trying to understand about Non-Euclidean Geometry for my future purpose. An example of Non-...
3
votes
1answer
213 views

“Trapping” of discs after random sequential adsorption

Imagine I perform Random Sequential Adsorption (RSA) of discs of some radius $r$ on $[0, 1]^2$, eventually covering the surface to some density $Q \leq 0.543$ with $N$ total discs (where $\approx 0....
1
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1answer
497 views

Properties of the minimum circumscribing circle for points sequentially placed within a circle of radius $r$ of previously placed points on a 2D plane

Imagine I perform the following procedure: [1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$. [2] At time point $t_2$, I center a ...
4
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1answer
602 views

In search for isotropic graphs: Straight lines and parallels

I wonder why I can find only so little attempts of concisely defining "directions" and "isotropy" of graphs. In Euclidean spaces "directions" can be identified with equivalence classes of parallel ...
21
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0answers
824 views

Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
1
vote
1answer
544 views

Decomposing a sphere (or defomed sphere) into a vertex-transitive graph with fixed-length curved edges

Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below. Motivation - I'm interested in a particular case of the problem where one wants to ...
2
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1answer
329 views

The equilibrium position for the body of a spider-like spring system after randomly perturbing the anchor positions of its legs

Take $N$ springs, $(s_1, ..., s_N) \in S$ of length $(l_1, ..., l_N)$, and for each spring, label one end "A" and one end "B". Connect the "A" ends of the $N$ springs to a point-like particle on a ...
13
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3answers
1k views

Optimal Wireframe Sphere

Suppose you have a length $L$ of metal pipe at your disposal, and you would like to build a wireframe unit-radius sphere, by bending, cutting, and welding the pipe into a connected structure $F$. Your ...
2
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1answer
217 views

What is the Sequence that Maximizes this Distance?

I have posted this question here without answer. Maybe I can get some light here. Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where $|...
3
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2answers
505 views

Threading pinholes in the wall of cylinder to pass through an internal coordinate

Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ ...
13
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4answers
1k views

When sticks fall, will they weave?

Imagine $n$ $z$-vertical sticks uniformly spaced around a unit-radius circle in the $xy$-plane. At $t{=}0$, each is randomly $\epsilon$-perturbed from the vertical, and they fall under the influence ...
25
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2answers
1k views

Why is the half-torus rigid?

The half-torus surface that results from slicing a torus like a bagel, depicted below (left), is isometrically rigid.       I know this from a remark of Alexandrov in Mathematics: Its ...
2
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0answers
113 views

Functional inequality under mean curvature flow

Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
8
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2answers
944 views

Relating curvature and torsion of a connection to those of a curve

I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion. I know that an affine connection $A$ on an $n$-dimensional manifold $M$...
22
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6answers
2k views

Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
1
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1answer
118 views

Need help maximizing distances to nearest neighbor in a cylinder

I have a cylinder and I want to maximize the number of points in the cylinder such that the distances to the nearest neighbors are maximally spaced. How do I find out how many points I can have so ...
3
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1answer
190 views

geometrically infinite ends of hyperbolic 3 manifolds

Let $M$ be a hyperbolic 3-manifold with finitely generated fundamental group. Assume $E$ is a geometrically infinite end (not of geometrically finite type, i.e. the convex core can not be separated of ...
12
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1answer
450 views

Find structure geometry of $A_1, A_2,…,A_n$ such that $\prod_{i<j} A_iA_j$ is maximum

In any triangle we have the well-known inequality: $$\sin{A}\sin{B}\sin{C} \le \frac{3\sqrt{3}}{8} (1)$$ Signification of inequality (1): Let three points $A, B, C$ lie on a circle then $AB.BC.CA$ ...
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0answers
49 views

Continuous Functions On Grassmannans under containment restrictions

Let $V$ be a vector space. Suppose that for a $x\in V$, we are given some subspace of dimension no more than d (e.g., the kernel of some operator defined on V, which varies smoothly with x), call it $\...
2
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1answer
117 views

Exist $A_1, A_2,\cdots , A_n$ be $n$ points on a sphere, satisfy if $i-j=l-m$ then $A_iA_j=A_lA_m$

Conjecture: Let $n\geq4$. Is there a set of $n$ non-planar point $A_1, A_2,\cdots , A_n$ be $n$ on a sphere (three-dimensional space) satisfying the conditions: if $i-j=l-m$ then $d(A_i, A_j) = d(A_l, ...
1
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1answer
355 views

Intersection between a line and a cube in $D$ dimensions

Say we have a space of dimension $D$. Say we have a $D$-cube of side $l$ centered at the origin and inside it we have a point $P\in \mathbb{R}^D$ and a collection of $D-1$ angles $\phi_1, \phi_2, \...
7
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1answer
344 views

From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
2
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0answers
90 views

Determine sub-polygon from line segments with known member connectivity

Test Polygon: Consider the following polygon as attached. Let the known parameter be as follows: •Member to member connectivity, i.e. it is known that A – B, X – E, F – B, etc. for all the members. •...
3
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0answers
269 views

Does a rectangle exist on any Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. Does there exist points $P,Q,R,S$ on $C$ such that quadrangle $PQRS$ is a non-degenerate rectangle?
10
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2answers
490 views

Minimum separation among $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{...
2
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1answer
127 views

A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic

I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following: Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...
6
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0answers
196 views

An inequality in cyclic polygon and tangential polygon

I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following: Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let P be ...
4
votes
1answer
371 views

A generalization of Erdős–Mordell inequality [closed]

I proposed my conjecture generalization of Erdős–Mordell inequality as following: Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ ...
21
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4answers
2k views

Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that $$\mathop{\rm area} \Sigma\ge ...
9
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3answers
2k views

About MF Atiyah and R Bott's 1983 paper

I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
91
votes
6answers
4k views

Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I ...
12
votes
0answers
1k views

Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
7
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3answers
852 views

How can we count lines in an n-x-n rectangular array?

Is there a formula for the number of lines that contain exactly two points through an n x n rectangular array of points?
2
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0answers
115 views

An elementary question about metrics on the real plane [closed]

Given the metric $d_p$ on the real plane, i.e. $$ d_p(x,y) = d_p((x_1, y_1), (x_2, y_2)) = [|x_1 - x_2|^p+ |y_1 - y_2|^p]^{1/p} $$ for which values of $p$ ($\geq 1$) is it true that the following ...
9
votes
4answers
1k views

What is the “right” definition of the homology(cohomology) of an orbifold?

What is the "right" analog in the orbifold case of a singular homology of a topological space? We can not just take the homology of the underlying space, because it does not contain much information. ...
7
votes
3answers
921 views

Self-Intersection of closed curves

Supoose I have a closed curve $\gamma$ in the plane such that for any isometry $g$ of $\mathbb{E}^2,$ such that $g(\gamma)\neq \gamma,$ $\gamma$ intersects $g(\gamma)$ in at most two points. It ...
1
vote
2answers
153 views

Descartes' theorem and Circle Packing [closed]

There's something I am missing comparing Descartes' theorem for three isometric circles here and this wiki post on circle packing of 3 circles here. From my calculation: $$ r_{ext} = \frac{r_{int}}{{...
77
votes
4answers
8k views

Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth ...
5
votes
1answer
351 views

Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note) Consider the Bundle theorem configuration : Points $A_1, A_2, A_3, A_4$ lie on a circle, ...
-1
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1answer
170 views

When is the orbit space of a manifold still a manifold of the same dimension?

$\mathbf{Question}$. Let us assume that $M^n$ is a topological manifold of dimension n, with a group action $\Gamma$, which acts discontinuously and freely on $M^n$. Is the orbit space $M^n / \Gamma$ ...
2
votes
0answers
216 views

A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane). This is a generalization of the last my previous question in Three chains of six circles. (...
6
votes
2answers
515 views

English translation of Lambert's Theorie der Parallellinien?

Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's Theorie der Parallellinien? I was able to find it online in German by way of the ...
3
votes
0answers
213 views

A conjecture on six planes [closed]

When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...