geometry

A battle between “slimes” and “zoglins” could be the best way to calculate pi—at least for fans of this megahit game

Let $A,B,C,D$ be four distinct points in the Euclidean plane. I would like to decide, using only a compass, whether the lines $AB$ and $CD$ are parallel. I am aware of the Mohr–Mascheroni theorem, so ...

_International Journal of Professional Studies_ 21 (1):271-918. 2026This article develops a higher-dimensional research programme for Calabi–Yau geometry beyond the classical threefold setting and through the explicit CY₂₀ horizon. It integrates hypersurface and toric constructions, Hodge statistics, mirror laws, special-holonomy constraints, conditional SYZ lifting, and dimensional-saturation mo…

I'm stuck in the following problem. I don't remember where I found it, but is kind of an Olympiad problem. Let $ABC$ be a triangle where $a=\sqrt{6}, \hat{A}=\pi/6$ and $b+c=3+\sqrt{3}$. Fin the area ...

Scientists have discovered that the Chinese money plant hides a remarkable geometric system inside its leaves, revealing that nature may solve complex problems using mathematical rules similar to those found in computer science and city planning. People often see meaningful shapes and patterns in random things. Maybe you have looked at clouds and spotted a [...]

The sudden resolution of a well-known conjecture highlights the growing adoption of AI as an assistant in high-level mathematics

The famous mathematical paradox of jagged shores gets a reality check from 130,000 global islands.

I am currently solving some Putnam math exercises for fun and I wanted to visualize some geometry questions. One exercises goes like this: Let S be a spherical cap, where the distance between two ...

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) For a Riemannian manifold or pseud…

Question: Any parallelopiped (any prism indeed) obviously allows partition into $n$ mutually congruent convex pieces for any $n$. Same property is shown by any convex solid with an axis of rotational ...

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) On a finite-dimensional real vecto…

I currently write mathematics on a tablet, and have wondered whether I can use existing geometric tools on here to perform straightedge and compass constructions. The straightedge is fine - I can draw ...

In the image: ABCDE is a regular pentagon F, A, B are colinear G, E, C are colinear FA has half the length of EC BCGF is a parallelogram Which area is bigger: the regular pentagon ABCDE or the ...

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems Generalised smooth spaces are, roughly speaking, generalisations of smooth manifolds. Their raison d’etre is the following Manifolds are fantastic spaces. It’s a pity that there aren’t more of them. Many spaces that occur in mat…

Let $n$ be a natural number. Fix a system of two equations $S(n)$: $$ \begin{cases} x^2+ny^2= z^2 \\ nx^2+y^2= t^2 \end{cases} $$ where $x,y, z, t$ are positive integers. Next, $n \in M_1$ if that ...

I am trying to prove that the lateral area of an oblique cylinder is $2\pi r\ell$ (to show that its total surface area is $2\pi r^2 + 2\pi r\ell$). To do that, I am slicing the lateral area with a ...

Curvature is conceptually simple but usually difficult to calculate. For a level set curve f(x, y) = c, such as in the previous couple posts, the equation for curvature is Even when f has a fairly simple expression, the expression for κ can be complicated. If we define then the level set of f(x, y) = c is […] The post Calculating curvature first appeared on John D. Cook .

Take a right-angled triangle with hypotenuse c and the other two sides a and b. Pythagoras’ Theorem tells us that c2 = a2+b2. Let the area of the triangle be A. We know that A = ab/2 (since an a×b rectangle is cut into … Continue reading →

The previous post constructed a triangular analog of the squircle, the unit circle in the p-norm where p is typically around 4. The case p = 2 is a Euclidean circle and the limit as p → ∞ is a Euclidean square. The previous post introduced three functions Li(x, y) such the level set of each function forms a […] The post Smoothed polygons first appeared on John D. Cook .

On proof of the (categorical) geometric Langlands conjecture: Dennis Gaitsgory, Sam Raskin: Proof of the geometric Langlands conjecture I: construction of the functor [arXiv:2405.03599, pdf] Dima Arinkin, D. Beraldo, Justin Campbell, L. Chen, Dennis Gaitsgory, J. Faergeman, Kevin Lin, Sam Raskin, Nick Rozenblyum: Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE […