geometry

A growing minority believes it’s a mistake to tie so many mathematical formulas to the famed 3.14... value. Another value, tau, could be better

Prove that every triangle with all its vertices lying on the graph of $y = \sin(x)$ is always an obtuse triangle. * I start my investigation with the simple case where one of its sides s is parallel ...

I've stumbled upon the following problem that made me curious for quite a while. Below is one of many possible diagrams of the problem in question. Given that $ABCD$ is a parallelogram with acute ...

Is it possible to use Graphics3D to somehow reproduce Penrose stairs or impossible staircase? It would be nice to get to the original as close as possible. Here is my start which is not right, - not ...

Suppose you have an arc a, a portion of a circle of radius r, and you know two things: the length c of the chord of the arc, and the length b of the chord of half the arc, illustrated below. Here θ is the central angle of the arc. Then the length of the arc, rθ, […] The post Circular arc approximation first appeared on John D. Cook .

Suppose we are given four coplanar points $A$, $B$, $C$, $D$, in three-dimesional Euclidean space. Suppose that we are also given four points $A'$, $B'$, $C'$, $D'$, such that $\left| AB \right| = ...

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) A topological 7-sphere equipped wi…

Here’s a thick ‘L’ shaped land where a 4 × 4 square is removed from a corner of a 10 × 10 square. The farmer who owns the land wants to divide it among her three kids such that all three pieces are ...

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) What is called the Gromoll-Meyer s…

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) Rokhlin’s theorem states that the …

Suppose you have a right triangle with sides a, b, and c, where a is the shortest side and c is the hypotenuse. Then the following approximation from [1] for the angle A opposite side a seems too simple and too accurate to be true. In degrees, A ≈ a 172° / (b + 2c). The approximation above only involves simple […] The post Simple approximation for solving a right triangle first appeared on John D…

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) The Hitchin-Thorpe inequality stat…

Nigel James Hitchin is professor of pure mathematics at Oxford. Formulation and proof of the Hitchin-Thorpe inequality: Introducing the ADHM construction for Yang-Mills instantons: On the moduli spaces of monopoles: Michael Atiyah, Nigel Hitchin, The geometry and dynamics of magnetic monopoles M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 (jstor:j.ctt7zv206) Michael Atiya…

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) The Hitchin-Thorpe inequality stat…

I came across a 10-year old question on math SE recently which concerns a net built out of congruent quadrilaterals. Here, the three longer sides on each face are of equal lengths, and the two ...

I've just rediscovered an old problem I first encountered over half a century ago: It's the problem number 191 from Julius Petersen's famous work, whose initial statement is as follows: In a given ...

A triangle, up to similarity, is completely determined by two of its internal angles. That is two degrees of freedom. A quadrilateral can be divided into two triangles, and those two triangles are ...

A few days ago I wrote a post on Newton’s diameter theorem. The theorem says to plot the curve formed by the solutions to f(x, y) = 0 where f is a polynomial in x and y of degree n. Next plot several parallel lines that cross the curve at n points and find the […] The post More on Newton’s diameter theorem first appeared on John D. Cook .

I have not posted on this site for a long time; that does not mean I have stopped discovering things. In fact, I am still discovering new theorems every day, and I may be more active here in the ...