Articles of spheres

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 have provided equations in vector form which is far from my ability to understand. Pre-requisite: http://paulbourke.net/geometry/circlesphere/spheresphere1.gif Equation for intersection of two spheres having centres […]

Particle on a hemisphere – lagrange

A particle of mass m is on top of a frictionless hemisphere centered at the origin with radius $R$. It starts sliding down the hemisphere. Set up the lagrange equatinos of the first kind and determine the constraint force and the point at which the particle detaches from the hemisphere as well as its velocity […]

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 ball with radius $r$ can we fit in this ball. Let this number be denoted by $N$. I am aware that this is […]

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 $$ A(s) = \frac{2 \pi^{s/2}}{\Gamma\left(\tfrac{1}{2} s\right)} $$ so that we can write the Xi-Function like $$ \xi(s) = s(s-1) […]

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 way to evenly distribute N points on a sphere? So far it seems that it is the best. Does anyone know of a […]

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 not homework! The question came up when I was thinking about something else.

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 $$\xi(s) = (s-1) \frac{\frac{s}{2}\Gamma(\frac{s}{2})}{\pi^{s/2}} \zeta(s)$$ Save for the $(s-1)$, the extra factor is exactly the inverse of $V_s$. Is there any explanation for this, or is it just a funny coincidence?

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 not taken the course, so please bear with […]

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

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 to me? I think that formulas shows something totally different: The area of a sphere is $4 \pi r^2$, […]