Articles of geometry

How do you parameterize a sphere so that there are “6 faces”?

I’m trying to parameterize a sphere so it has 6 faces of equal area, like this: But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each “slice”). I can’t seem to get the $\theta$ elevation parameter correct. Help!

What is the average rotation angle needed to change the color of a sphere?

A sphere is painted in black and white. We are looking in the direction of the center of the sphere and see, in the direction of our vision, a point with a given color. When the sphere is rotated, at the end of the rotation we might see the same or a different color. The […]

Volume of irregular solid

I need to calculate volume of irregular solid which is having fix $200 \times 200$ width and breadth but all four points varies in depth. I have table which gives depth at each point. How to calculate volume of such solid? Hi, I am giving here my main problem definition. I have a grid with […]

What is the solution to the Dido isoperimetric problem when the length is longer than the half-circle?

Given $L$-the length of a curve (single-valued function) passing trough the points $x_1$ and $x_2$ on the $x$-axis. What is the curve $y(x)$ maximizing the area between this curve and the $x$-axis? The solution is, of course, well known: one formulates a variational problem with a constraint $ F[y,y’]=\int_{x_1}^{x_2}\left(y+\lambda\sqrt{1+y’^2(x)}\right)dx, $ which yields an equation of […]

Maximum area enclosure given side lengths

A peer of mine gave me the following problem : Given a sequence of $n$ lengths (i.e.,$L_1, L_2, .., L_n$ ) where each is the length of the side, find a sequence of $n$ points (where $p_k = (x_k, y_k)$) such that $dist(p_k, p_{k+1}) = L_k$ and $dist(p_1, p_n) = L_n$ where $dist(a, b)$ is […]

In this isosceles right angled triangle, prove that $\angle DAE = 45^{\circ}$

Consider the following right angled triangle with $AB =AC$. $D$ and $E$ are points such that $BD^2 + EC^2 = DE^2$. Prove that $\angle DAE = 45^{\circ}$ The obvious thing was to construct a right angled triangle with $BD, EC, DE$ as its sides. So, we draw a circle around $BD$ with $D$ as centre […]

In how few points can a continuous curve meet all lines?

Inspired by this (currently closed) question, I’m wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for $t\in\mathbb{R}$ and $x(t), y(t)$ continuous functions) which meets every line in the plane, what is the minimum number of times it must meet some line? […]

Why can't we construct a square with area equal to a given equilateral triangle?

In modern geometry, given an equilateral triangle, one can’t construct a square with the same area with the use of Hilbert tools. Why is this? The claim seems untrue to me, so there must be something wrong with my understanding. First, given an equilateral triangle of side length $s$, the area of the triangle is […]

Calippo, toothpaste and milk .. packing

This post was posing the question “The Calippo™ popsicle has a specific shape … Does this shape have an official name?”, and the accepted answer was that it is a Right Conoid which in fact is much resembling. However, a conoid is not developable and so cannot be economically made out of a cardboard sheet. […]

Confused about dimension of circle

I confused myself when thinking about the circle: It can be parameterised as $$ C(t) = (\cos t , \sin t)$$ for $t \in [0,2\pi)$. This makes it clear that the circle is one dimensional. But then the circle is also defined by $x,y$ such that $$ x^2 + y^2 = 1$$ If we try […]