1+ hour, 36+ min ago  (165+ words) Tim F left a comment on my guitar pick post saying the image was a "squircle-ish analog for an isosceles triangle." That made me wonder what a more direct analog of the squircle might be for a triangle. A squircle…...

5+ day, 21+ hour ago  (221+ words) A couple days ago I wrote a post about turning a trick into a technique, finding another use for a clever way to construct simple, accurate approximations. I used as my example approximating the Bessel function J(x) with (1 + cos(x))/2. I…...

6+ day, 6+ hour ago  (362+ words) When a function is not differentiable in the classical sense there are multiple ways to compute a generalized derivative. This post will look at three generalizations of the classical derivative, each applied to the Re LU (rectified linear unit) function....

1+ week, 1+ day ago  (348+ words) 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…...

1+ week, 4+ day ago  (363+ words) The previous post looks at the nonlinear pendulum equation and what difference it makes to the solutions if you linearize the equation. If the initial displacement is small enough, you can simply replace sin " with ". If the initial displacement is…...

1+ week, 4+ day ago  (499+ words) The equation of motion for a pendulum is the differential equation where'g is the acceleration due to gravity and " is the length of the pendulum. When this is presented in an introductory physics class, the instructor will immediately say something…...

1+ week, 6+ day ago  (389+ words) 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…...

1+ week, 6+ day ago  (195+ words) The previous post gave a simple and accurate approximation for the smaller angle of a right triangle. Given a'right triangle with sides a,'b, and'c, where'a is the shortest side and'c is the hypotenuse, the angle opposite side a is…...

2+ week, 2+ day ago  (209+ words) 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…...

2+ week, 4+ day ago  (603+ words) LLM parameters have a roughly Gaussian distribution, and so evenly spaced numeric values are not ideal for parameters. Instead, you'd like numbers that are closer together near 0. The FP4 floating point numbers, described in the previous post, are spaced 0. 5 apart for…...