Let $ a_{1}, \dots a_{m} \in \mathbb{R}^{n} $ such that $ ||a_{i}-a_{j}||=1 \, \forall i \neq j $, where $ || \cdot || $ denotes the usual norm on $ \mathbb{R}^{n} $. Prove that $ m \leq n+1 $. I did it for the easy case when $ n=1 $ by explicit computation using the […]

In an exercise I was asked to find a formula of the form $F(x,y,z)=C$ for a cylinder though the axis $(t,t,t)$ and radius $R$. The formula I got seemed a bit suspicious so I wanted to ask if I have it right. Basically I used the vector formula for the distance between a line and […]

When I see answers regarding proofs such as the one mentioned here, it seems that there is a considerable diversity of ways to attempt to look at this proof. Similarly, although this sum of squares question is very different, I was wondering if there was a geometric interpretation to the following problem: Prove that given […]

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times sin(\frac{2\pi}{5})$. It then says that the vertices of an icosahedron of edge length 2 are given by: $(0, ±1, ±\phi)$ $(±1, ±\phi, […]

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. Without loss of generality, let vertices $A$ and $C$ be one side, while the vertex $B$ […]

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in the form $f(x) = mx + q$ or any other function that maps $R \rightarrow R$ resulting in a graph that looks […]

I’m trying to prove some things about the action of the Möbius group on the “circlines” in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius transformations (as $PSL(2, \mathbb{C})$), I don’t really know how I’m supposed to treat circles. Points on the sphere are […]

Given two closed shapes made up of Bézier curves (and/or straight lines), I’m looking for an efficient way of calculating the resulting shape of the following Boolean operations: union difference intersection slice (imagine each of the shapes in the Venn diagram as its own shape; this operation is optional and can be expressed as a […]

The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid. It this possible without a rhombus?

Exercise 2.5 of Izenman’s Modern Multivariate Statistical Techniques: Consider a hypercube of dimension $r$ and sides of length $2A$ and inscribe in it an $r$-dimensional sphere of radius $A$. Find the proportion of the volume of the hypercube that is inside the hypersphere, and show that the proportion tends to $0$ as the dimensionality $r$ […]

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