Questions tagged [spheres]

For geometrical problems involving spheres. Use the tag (geometry) as well. For intrinsic geometry of spheres, see (spherical-geometry).

90
votes
17answers
38k views

Why is the volume of a sphere $\frac{4}{3}\pi r^3$?

I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! ...
7
votes
2answers
286 views

Equal slicing of my spherical cake

Recently I baked a spherical cake ($3$ cm radius) and invited over a few friends, $6$ of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited ...
7
votes
3answers
9k views

Mean distance between 2 points within a sphere

I have found an answer on this site to the question of determining the mean straight-line distance between 2 randomly chosen points in a disc of radius r. (See Average distance between two points in a ...
0
votes
1answer
1k views

Volume bounded by a sphere of radius $1$ in 4 dimensions

I want to express the volume enclosed by a hypersphere of radius $1$ with a quadruple integral, in four dimensions. I know that the equation of the sphere is $x^2 + y^2 + w^2 + v^2 \le 1$. However, I ...
3
votes
1answer
420 views

what is the surface area of a cap on a hypersphere?

According to mathworld, let the sphere have radius $R$, then the surface area a spherical cap of height $h$ and base radius $a$ is given by $$S=2\pi Rh=2\pi(a^2+h^2).$$ What is this value for an n-...
104
votes
18answers
10k views

Fastest way to meet, without communication, on a sphere?

I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and ...
54
votes
5answers
11k views

False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
8
votes
2answers
2k views

Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
22
votes
0answers
6k views

Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: https://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like ...
32
votes
2answers
1k views

On nonintersecting loxodromes

The (spherical) loxodrome, or the rhumb line, is the curve of constant bearing on the sphere; that is, it is the spherical curve that cuts the meridians of the sphere at a constant angle. A more ...
5
votes
1answer
958 views

unit sphere simply connected

I want to prove that the unit sphere $S^2$ is simply connected. In order to do this I am given the following steps: 1. Let $x_1,x_2 \in S^2$ and $\gamma \in P(S^2;x_1,x_2)$ be a path. Let $p \in S^...
5
votes
1answer
774 views

Closed, simply connected manifolds which are not spheres

In 2 or 3 dimensions, every closed simply connected manifold is a sphere. In the smooth category, I suppose you could take exotic smooth structures to give examples of closed simply connected ...
8
votes
1answer
222 views

A volume form on the sphere which gives equal areas to all hemispheres is invariant under the antipodal map?

Let $\omega$ be a volume form on $\mathbb{S}^2$ with the property that the induced area (w.r.t $\omega$) of all the hemispheres is the same. Is it true that $\omega$ is invariant under the antipodal ...
4
votes
1answer
127 views

What does $S^1$ do in many branches of math?

There are many isomorphisms of $S^1$: $\hat{\Bbb Z}, \Bbb R/\Bbb Z, U(1), SO(2), SL(1,\Bbb C), \Bbb T^1, \Bbb R\cup \{\infty\},\Bbb R\Bbb P^1 $. Seeing its importance, I'd like to see a synthesis of ...
4
votes
4answers
467 views

Why do disks on planes grow more quickly with radius than disks on spheres?

In the book, Mr. Tompkins in Wonderland, there is written something like this: On a sphere the area within a given radius grows more slowly with the radius than on a plane. Could you explain this ...
4
votes
1answer
145 views

Limit of $n-1$ measure of the boundary of a sphere

The measure of a sphere of radius $R$ centered in $0_{\mathbb{R}^n}$ in $\mathbb{R}^n$ is \begin{array}{l l}\int_{B_0(R)}dx_1\ldots dx_n & =\int_0^R\rho^{n-1}d\rho \int_{-\frac{\pi}{2}}^{\frac{\pi}...
2
votes
4answers
1k views

Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$

Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$ Here is an image: Normally I can do this question, but this one is tricky because since we are ...
0
votes
1answer
112 views

Show that antipodal points remain antipodal under any isometry of $S^2$

"The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook) Show that antipodal points remain antipodal under any isometry of $S^2$. In ...
0
votes
3answers
401 views

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter

Three spheres of diameters 2,3&4 cm's respectively formed into a single sphere.Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter
13
votes
1answer
358 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
2
votes
1answer
708 views

Area enclosed by loop on $S^2$

Let $l:S^1\to S^2$ be a simple closed loop on $S^2$. How do you calculate the area enclosed by this curve (Up to exchange of which "cap"* you choose?) I wrote "cap" because the cap defined by the ...
11
votes
1answer
445 views

Infinity-to-one function

Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$? Here, $I=[0,1]$ and $S^2$ is the unit sphere. I have no idea how to do this. Note: This is ...
10
votes
2answers
957 views

intersection of hypercube and hypersphere

There is a number of similar questions already (e.g. this one), but as far as I can see, none quite cuts it for me. In $n$-dimensional euclidean space, a hypercube $H$ with side lengths $2A$ is ...
8
votes
6answers
2k views

Reflections on a sphere

There is a sphere located in a point s with radius r. The Sphere is a perfect mirror. If i'm sitting in the point c, I want to cast a ray to the sphere such that I hit the point p after bouncing in ...
17
votes
1answer
417 views

Connection between the area of a n-sphere and the Riemann zeta function?

The Riemann Xi-Function is defined as $$ \xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s) $$ and it satisfies the reflection formula $$ \xi(s) = \xi(1-s). $$ But the ...
4
votes
2answers
5k views

Area of a circle on sphere

On a (flat) Euclidean plane, the area of a circle with a radius $r$ can be described by the function $A(r) = \pi r^2.$ But how can one describe the area of the same circle on a spherical manifold? ...
4
votes
3answers
395 views

Visible Portion of the Earth's Surface

EDIT: I need help converting the right side to a function of h Let $A_h$ be the area of the zone corresponding to height h. If we set up a rectangular co-ordinate syustem with the origin at the ...
4
votes
0answers
260 views

Sphere packing question AGAIN.

This question has probably been asked before but when I searched the site I could not find the answer. Suppose we have and $n$-dimensional ball with radius $R$. How many, smaller $n$-dimensional ...
9
votes
3answers
169 views

Does a set of $n+1$ points that affinely span $\mathbb{R}^n$ lie on a unique $(n-1)$-sphere?

In $\mathbb{R}^2$ every three points that are not colinear lie on a unique circle. Does this generalize to higher dimensions in the following way: If $n+1$ element subset $S$ of $\mathbb{R}^n$ does ...
2
votes
2answers
3k views

Sphere rotating in several directions simultaneously?

By rotation I'm here only referring to an object rotating in relation to itself, not in relation to any other object. Also I should add that the axis of rotation should be through the center of the ...
2
votes
1answer
272 views

Use Quadrilateration to locate a point

I'm having lots of trouble with this. At first I thought the problem was a matter of equating four spheres to find their one common point, i.e. point of intersection. I've looked up lots of things on ...
1
vote
2answers
2k views

Understanding the formula for stereographic projection of a point.

I was wondering about the equation of line I can write which can help me finding the coordinates of Point $P'$ in relation with coordinates of points on the sphere that is $P$. Let the $P'(X,Y)$, ...
1
vote
2answers
2k views

All Intersection points of two spheres having arbitary centres?

I have read much about intersection of two spheres from spheres-intersect , circlesphere and collision-points but all are based on the assumption of spheres located at origin or $x$-axis or some ...
0
votes
3answers
321 views

Integrating unit vectors on the surface of the unit sphere

How can I calculate below integrals: $$\int \hat{r}_i\ \hat{r}_j\ \hat{r}_k\ \hat{r}_l \ d\Omega$$ in which $d\Omega =\sin(\theta) \ d \theta\ d\phi $ is the surface element of the sphere and $\hat{...
5
votes
2answers
916 views

Walk on Earth: Math Puzzle

Here's the famous math puzzle posted by Prof. Walter Lewin about a person walking on earth, quoted below for posterity: A person stands on the North Pole. She walks 10 miles South, then 10 miles ...
3
votes
1answer
249 views

How to discretize a sphere?

I would like to discretize a sphere into icosahedra whose vertices are equidistant, i.e., I want to plot $n$ equidistant points on the surface of a sphere. I am familiar with R, Python, and Matlab. ...
3
votes
3answers
243 views

What space is this Homeomorphic to?

Let $X$ be a topological space and let $A ⊂ X$. Let $\sim$ be an equivalence relation on $X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $...
2
votes
2answers
218 views

If $M$ is a compact manifold, how to prove that the unitary tangent bundle is also compact?

I have this idea: if I choose a small open set $U\subset M$ then we can consider $T^1U=U\times S^{n-1},$ where $n=\dim M.$ (This is the hint that my professor provided. I don't know why that's true). ...
2
votes
1answer
197 views

When is a loxodromic curve unique between two points?

Consider 2 arbitrary points, A and B, which are located on Earth's surface. We assume the Earth to be a perfect sphere for the purposes of my question. Each point is given by their latitude and ...
2
votes
0answers
128 views

Can you comb the hair on a 4-dimensional coconut?

It is well-known that you can't equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) "...
2
votes
1answer
118 views

Fixing this proof of the simple connectedness of the $n$-sphere

As we all know, the $n$-sphere ($n\ge 2$) is simply connected. However, the way this was proven to me seemed somewhat complicated and I tried my hand at simplifying. My professor insists that my ...
2
votes
1answer
227 views

Prove surface area of a sphere using solid of revolution surface area formula.

I have to prove the surface area of a sphere with $r=1$ using the solids of revolution through revolution abouth both the $x$ and the $y$ axis. The formulas are easy. From top to bottom, surface area ...
2
votes
1answer
128 views

Spheres as Symplectic Homogeneous Spaces

Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a ...
2
votes
1answer
102 views

Surface Area of a $p$-norm Sphere in $\mathbb{R}^n$

There are many various ways to derive the formula for the volume of a $p$-norm ball in $\mathbb{R}^n$ $$V_{n,p,R}=\frac{(2\Gamma(1+\frac{1}{p}))^n}{\Gamma(1+\frac{n}{p})}R^n$$ And I found many ...
1
vote
1answer
172 views

What is the curvature form $\Omega$ associated with the Levi-Civita connection for the $n$-sphere $S^n$ with respect to the standard metric?

What is the curvature form $\Omega$ associated with the Levi-Civita connection $\nabla^{\text{L.C.}}$ for the $n$-sphere $S^n$ with respect to the standard metric, i.e. what is $\Omega=d\theta+\frac{1}...
1
vote
1answer
355 views

Antipodal map homotopic to identity for $S^{2n-1}$

We want to show that the antipodal map on $S^n$ is homotopic to the identity for $n$ odd. My attempt was to try strong induction. For $n=1$ define $$H((x,y),t)=(\cos(\tan^{-1}(y/x)+\pi t),\sin(\tan^{...
1
vote
1answer
140 views

How to express volume of a sphere as a sum of infinitesimally thick discs?

I want to express the volume of a sphere with a radius r as an integral that adds up each infinitesimally thick disc within the volume. So I have dV = A(x) dx, where A(x) is the area of the disc that ...
1
vote
1answer
110 views

Angle between arc of two points on a unit sphere and $xy$-plane [closed]

Suppose I have two points on a unit sphere whose spherical coordinates are $A(\theta_1,\phi_1)$ and $B(\theta_2,\phi_2)$, what is the angle between $xy$-plane and arc $AB$? Maybe I can draw a ...
1
vote
1answer
126 views

Max number of points that fit in/on a sphere of radius r, with minimum inter-point distance also r

This just came up in a game, and I realized I don't know how to solve this. Given a sphere of radius r (say, 20'), what is the largest number of points that can be arranged within the sphere (the ...
1
vote
1answer
76 views

Computing a ratio of parts of a circumference on a sphere

This question is inspired by this answer I gave, where I promised I would do a computation. The computation in question is Parametrize the unit sphere by spherical coordinates $(\theta,\phi)$, ...