Articles of geometry

If any x points are elected out of a unit square, then some two of them are no farther than how many units apart?

If 5 points are randomly positioned in a unit square, no two points can be greater than square root of 2 divided by 2 apart; divide up the unit square into four squares, and, based on the pigeonhole principle, five points (pigeons) fitting into four squares (holes) means that no two points can be greater […]

Spatial angles in higher dimensions

Wikipedia gives an excellent treatise about solid angles in 1-2-3-Dimensions. But how about n-D? I read once some notes from a seminar held during WWII in Switzerland, and one result concerned spatial angles in even dimensions (I have forgotten the reference), but I would like to have a similar general definition, likely computationally nice. So […]

Geometric basis for the real numbers

I am aware of the standard method of summoning the real numbers into existence — by considering limits of convergent sequences of quotients. But I never actually think of real numbers in this way. I think of a real number as a vector in 1D Euclidean space, the good old number line. A signed distance. […]

Can a rectangle be written as a finite almost disjoint union of squares?

Given a closed rectangle $R$ are there closed squares $(S_{i})_{1\leq i\leq n}$ such that $R=\underset{1\leq i\leq n}{\cup}S_{i}$ and $S_{i}^{\circ}\cap S_{j}^{\circ}=\varnothing$ for $i\neq j$ ?

Bezier curvature extrema

For a planar cubic Bezier curve $B (x(t),y(t))$, I would like to find the values of parameter $t$ where the curvature (or curvature radius) is greatest/smallest. The formula for curvature is: $$r = \dfrac{(x’^2+y’^2)^{(3/2)}}{x’ (t) y”(t) – y'(t) x”(t)}$$ The problem is that there is that square root in it so I was wondering whether […]

Minimum radius of N congruent circles on a sphere, placed optimally, such that the sphere is covered by the circles?

What is the minimum (radius/radii/range of such) of N congruent circles that are placed (optimally) on a sphere in such a way that they cover the entire surface of the sphere? For 2 circles, a easier visualization would be having two points, growing them into circles and to keep growing them until they touch each […]

I'm looking for the name of a transform that does the following (example images included)

I’m in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I’m looking for: Suppose I have the following “snake” of 10 quadrilaterals: I now want to apply a transformation to each of these […]

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: An equilateral triangle can be divided into 3 congruent shapes. Questions: 1) a triangle can be divided into 3 congruent shapes. Is it equilateral? 2) a shape in the plane can be divided into n congruent shapes for any positive integer n. What can it […]

Form of most general transformation of the upper half plane to the unit disk.

In David Blair’s book on Inversion Theory, he write that the transformation $$ T(z)=e^{i\theta}\frac{z-z_0}{z-\bar{z}_0} $$ is the most general transformation mapping the upper half plane to the unit circle, provided $z_0$ is in the upper half plane. If $z_0$ is in the lower half plane, then the upper half plane is mapped to the exterior […]

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle’s center in its interior?