geometry

superalgebra and (synthetic ) supergeometry A supermanifold is a space locally modeled on Cartesian spaces and superpoints. There are different approaches to the definition and theory of supermanifolds in the literature. The definition is popular. The definition has been argued to have advantages, see also the references at super ∞-groupoid. See at geometry of physics – supergeometry the section …

Let $\mathcal{P}$ be a polygon, and we add CCW direction to its boundary $\partial\mathcal{P}$. Let us measure the angle $\theta_i$ along each edge of the polygon for $i=1,...,N$ where N is the number ...

In the figure below of a semi-circle, show that ##h=\sqrt{ab}##. In the special case where ##a=1##, then ##h=\sqrt{b}##. A complete discussion can be found in chapter 2 of the book An Imaginary Tale by Paul J. Nahin.

Mathematicians have considered how to watch every corner of a space—but soccer adds moving players, blocked views and constant action

Let $Q$ be a convex polygon all of whose sides are tangent to a circle of radius $1$. The inscribed square conjecture says that every Jordan curve has an inscribed square. It is already known that $Q$ ...

if i take a circle in 3d , and view with some angle $\alpha$ ( along the y axis with context to the image ) , it looks like an ellipse i want an proof that the shown image is exactly a ellipse #btw ...

I have started to read about Geometric Langlands from Frenkel's book https://arxiv.org/abs/hep-th/0512172. As said in Theorem $3$ of the book, the Langlands correspondence is stated as follows ...

On the surface of the sphere $S_2$, every point has exactly one antipode (a point at the maximum distance from it). I believe this is also true in a circle $S_1$, and a flat torus $S_1 \times S_1$. Is ...

Such a shape looks a bit like a hypocycloid, but is not curved the same. There are names for the lens, lune, Reuleaux triangle, arbelos, and salinon Surely this simple figure has one too. It is a bit ...

World territories have been redrawn so that a region is determined by the closest capital city. This is calculated using a spherical Voronoi diagram, which takes into account the curvature of the Earth when computing distances. See also United States of Voronoi.

Scientists have cracked a mathematical puzzle, completing a theoretical framework for understanding how humans perceive colour. The Los Alamos National Laboratory, headed by Roxana Bujack, employed geometric principles to construct a rigorous mathematical definition of colour perception. The research was built around the three fundamental qualities of hue, saturation and lightness. Their findings…

Here is a little puzzle from Pierre Berloquin’s book 100 Geometric Games. The goal is to cut the figure once to make two identical parts. Have fun!

In non-orthogonal coordinate systems, can we say that changing a coordinate could result in changing another coordinate? That is, the coordinates are dependent on each other. As I understood, non-orthogonal systems will have unit vectors (which are defined to point in the direction of... Read more

In a cyclic quadrilateral ABCD, a line L intersects AB, BC, CD, and DA at P, Q, R, and S, respectively. If four points A', B', C', and D' on the circle satisfy the following conditions: A'B' passes ...

I am considering a connected Riemannian manifold $M$ of dimension $n \geq 2$. Let $\gamma: [a, b) \to M$ (with $-\infty < a < b < \infty$) be a $C^\infty$ curve that is inextendible to the ...

Suppose we have a point $Q(x_0,y_0,z_0)$ and a plane $Ax+By+Cz+D=0$, we have the formula of the distance from the point to the plane $$D_{dist}=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$$ it can ...

Given the knowledge of the geodesics of the Beltrami-Klein model for hyperbolic geometry (a disk with straight geodesics) over a euclidean ordered field, is it possible to recover the distance metric ...

I am studying an article written by S. Brendle called The isoperimetric inequality for a minimal submanifold in Euclidean space (https://doi.org/10.1090/jams/969) and I am struggling to ensure the ...

Why does this $$h_z = \begin{cases} -z(L+x) \; , & x\in \left[-L,-z\right]\\ (L-z)x \; , & x\in \left[-z,z\right]\\ z(L-x) \; , & x\in \left[z,L\right] \end{cases}$$ define a $2L$-periodic ...

Two Ways to Draw Infinite Jest's Sierpinski Gasket Table of Contents 1. About infiniteJest dfw fractals sierpinski In a 1996 Bookworm interview with Michael Silverblatt, David Foster Wallace confirmed something Silverblatt had spotted about Infinite Jest: Silverblatt, talking about Infinite Jest's structure: I said to myself, this must be fractals. DFW replied: I've heard you're an acute reader….…