Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [spheres]

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

An $n$-sphere is $$ S^n = \{\mathbf x \in \mathbb{R}^{n+1}\mid \lVert x \rVert_2 = 1\} $$ Its enclosed volume is the $(n+1)$-ball.

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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 the sphere is way bigger than you two. There are no…
Rob Audenaerde
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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! How could somebody guess something like this for…
Larry Wang
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Can you make a sphere out of a plane?

I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a…
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What does "surface area of a sphere" actually mean (in terms of elementary school mathematics)?

I know what "surface area" means for: a 2d shape a cylinder or cone but I don't know what it actually means for a sphere. For a 2d shape Suppose I'm given a 2d shape, such as a rectangle, or a triangle, or a drawing of a puddle. I can cut out a…
silph
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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 know from Calculus, but I emphasize that I have…
ccbreen
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What is the analogue of spherical coordinates in $n$-dimensions?

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by $$x_1=r \cos \theta$$ $$x_2=r \sin \theta$$ For…
becko
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Are the points moving around a sphere in this manner always equidistant?

I recently encountered this gif: Pretend that there are visible circles constructed along the paths of the smaller black and white "discs", tracing how their individual centers move as they revolve around the center of the whole design. These…
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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 picturesque way of putting it is that if one wants to…
J. M. ain't a mathematician
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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 to know is: "Is the Fibonacci lattice the very best…
Physics Ph.D.
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Probability of random sphere lying inside the unit ball

Let $n\geq2$. Let $B\subseteq\mathbb{R}^n$ be the unit ball. Randomly choose $n+1$ points of $B$ (uniformly and independently). Then (almost surely) there will be a unique hypersphere $S$ passing through all $n+1$ points. What is the probability…
Thomas Browning
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What is $\mathcal{R}$?

First of all, I am asking this question entirely out of curiosity. It basically randomly popped out of my mind. So I am asking for the value of an infinite series. Let's call it, $\mathcal{R}=\sum_{n=1}^{\infty}\mathcal{R}_n$ Now I will try to…
Rounak Sarkar
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If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to hyperspheres, so I was wondering if there's any…
James Hanson
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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 area $A$ of a $s$-dimensional sphere is $$ A(s)…
asmaier
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Picking points on a sphere at random

Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that all the points lie within any hemisphere on the…
anurag anshu
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What is the metric on the $n$-sphere in stereographic projection coordinates?

The metric on the $n$-sphere is the metric induced from the ambient Euclidean metric. Find the metric, $d\Omega^2_n$, on the $n$-sphere and the volume form, $\Omega_{S_n}$ , of $S^n$ in terms of the stereographic coordinates on $U_N =S^n − (0, . . .…
Okazaki
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