Articles of spheres

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 $(\theta,\phi)$, where $\theta$ is the latitude (with the north pole given by $\theta=0$ and the south pole by $\theta = \pi$) and $\phi$ is the longitude. […]

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) “you can’t comb the hair on a coconut.” What about one dimensional higher? That is, embed a $3$-surface of constant radius in Euclidean […]

How to find the point on the sphere that is closest to a plane?

Consider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane? I could find the distance from the plane to the origin using the formula $D=\frac{|1\cdot 0+2\cdot 0+2\cdot 0-4|}{\sqrt{1^2+2^2+2^2}}=\frac43$, and then I can find the distance between the plane and sphere by subtracting the radius of sphere […]

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 rotating about the y-axis, and we are quadratic, when I solve for $x$ I get two answers, one positive and […]

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 of revolution about $x$ axis, and $y$ axis formulas: $$S_x=\int_a^b2\pi y\,\sqrt{1+\Big(\frac{dy}{dx}\Big)^2}\,dx$$ $$S_y=\int_a^b2\pi x\,\sqrt{1+\Big(\frac{dx}{dy}\Big)^2}\,dy$$ Where in the first formula, […]

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 proof in this easy case or in the general picture.

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 $X/A$ to be the quotient space $X/{\sim}$. (i.e. collapse $A$ to a point) Let $D^2$ denote the unit […]

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}}\cos^{n-1}{\varphi_1}d\varphi_1\ldots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos{\varphi_{n-1}}d\varphi_{n-1}\int_{0}^{2\pi} d\theta \\ & =\omega_n \int_0^R\rho^{n-1} d\rho \end{array} where $\omega_n$ is the $n-1$ measure of the boundary of the sphere. We can calculate the integral of the function $e^{-||x||^2}$ over $\mathbb R^n$ as $$\int_{\mathbb{R}^n}e^{-||x||^2}dx_1\ldots dx_n=\int_{\mathbb R^n}e^{-x_1^2-\ldots-x_1^n}dx_1\ldots […]

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 East, then 10 miles North and she is back at her starting point (the North Pole). There are more points […]

Sphere close packing with cubic sheet

In some book, it’s written that layers making the 3d structure must be hexagonal for close packing of spheres. But suppose we have a simple cubic sheet and another one on top of it, with which we try to fill depression between spheres. By calculation, I get that minimal distance between layers is $\sqrt2r$ and […]