geometry

superalgebra and (synthetic ) supergeometry Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object in there. As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces. As ordinary algebraic geometry stu…

One variant of the traditional sphere packing problem in $\mathbb R^3$ asks what the density $\rho_r$ of spheres of equal radii $r$ one can fit in a unit cube without exceeding the boundary ...

I found the coordinates and angles (67 degree) via trigonometry, but I am looking for a pure, elegant synthetic geometry solution without coordinate geometry.

If we take a cube, we can surround it with the smallest sphere, and their centres will coincide. Consider points on the surface of the cube where the 8 vertices of the cube touch the sphere; these ...

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems A geometry is an (∞,1)-category equipped in a compatible way with the structure of an (∞,1)-site; the structure of an essentially algebraic (∞,1)-theory. The objects of are to be thought of as test-spaces with certain higher geo…

This is a basic question. What do we actually mean when we say "we can describe the curve parametrically" or "we parameterize the curve with a parameter $t$ or $\theta$."? What is ...

Tiny water-secreting pores appear to organize the major veins of the plant leaves into an arrangement known as a Voronoi diagram.

A similar concern as this question I asked three years ago. We can get tangent space of a point as collection of every derivative along curves running through that point, would it a similar sort of ...

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!