Thoughts on physics, math, and computers.
A famous problem in approximation theory is to approximate the function \left|x\right| as accurately as possible on the interval -1 \le x \le 1 , measured by maximum absolute error, using either polynomials or rational functions of fixed degree. Perhaps it seems silly to approximate a function as simple as this, but it serves as a useful test problem for probing how well non-smooth functions can…
The geometric series, S_{n}(x) = \sum_{k=0}^{n} x^k, satisfies a linear recurrence S_{n+1}(x) = x \, S_n(x) + 1 with constant (in n ) coefficients. This means it can also be written as a matrix power, S_{n}(x) = \begin{pmatrix} 1 & 0 \end{pmatrix} M^{n} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, with M = \begin{pmatrix} x & 1 \\ 0 & 1 \end{pmatrix}. Repeatedly squaring M can rapidly generate high…
Newton’s iteration for computing the reciprocal of a , y_{n+1} = y_n(2-ay_n), can be derived by applying Newton’s method to the function f(y)=1/y-a . Substituting a=1-x gives y_{n+1} = y_n(2-(1-x)y_n), an iteration for computing 1/(1-x) . Starting from y_0=1 , the first few iterates are \begin{aligned} y_1 &= 1 \cdot (1+x) = 1+x, \\ y_2 &= (1+x)(1+x^2) = 1+x+x^2+x^3, \\ y_3 &= (1+x+x^2+x^3)(1+x^4)…
For some reason, a six year old study claiming that dogs align with Earth’s magnetic field when they poop in “calm magnetic field conditions” is making the rounds online again. After this article was originally published in December 2013, it was reported uncritically by PBS , NPR , National Geographic , Vice , and many others. The study’s conclusions hinge on a surprising distinction: dogs don’t…
In several recent posts, I have been exploring a way of doing trigonometry using vectors and their various products while de-emphasizing angle measures and trigonometric functions. In this system, triangles are represented as sets of three vectors that add to 0 a + b + c = 0 The traditional law of cosines can be replaced with a vector equation that uses the dot product c^2 = a^2 + b^2 + 2 a\cdot b…
In previous posts, I have shown how to visualize both the dot product and the wedge product of two vectors as parallelogram areas. In this post, I will show how the dot product and the wedge product are related through a third algebraic product: the geometric product. Along the way, we will see that the geometric product provides a simple way to algebraically model all of the major geometric…
A visual way of expressing that three vectors, a , b , and c , form a triangle is and an algebraic way is a + b + c = 0 In a previous post , I showed how to generate the law of cosines from this vector equation—solve for c and square both sides—and that this simplifies to the Pythagorean theorem when two of the vectors are perpendicular. In this post, I’ll show a similarly simple algebraic route…
I recently posted a geometry puzzle about an autonomous lawn mower steered by a rope and a peg. How much rope remains unspooled from the peg when the mower collides with it? If you haven’t seen the puzzle yet, go check out last week’s post and give it a try. I received two very nice (and satisfyingly different) solutions from Suzanne von Oy and Patrick Honner ⊕ Suzanne von Oy ( blog , twitter )…
One of the joys of being an engineer at Desmos is that my collaborators occasionally bring me math problems that they need to solve to make something that they’re building work right. I love tricky little geometry problems, and I’d solve them as a hobby if they weren’t part of my job. When it helps our team get on with the work, so much the better. In today’s post, I’d like to share one of these…
I have a confession to make: I have always found symbolic algebra more intuitive than geometric pictures. I think you’re supposed to feel the opposite way, and I greatly admire people who think and communicate in pictures, but for me, it’s usually a struggle. For example, I have seen many pictorial “proofs without words” of the Pythagorean Theorem. I find some of them to be quite beautiful, but I…