Unsolved Problems. Menagerie. Immortal Truth. How to construct (draw) the incircle of a triangle with compass and straightedge or ruler. Video. Sphere Inside out Part - II. Video. Video. Video. K-MODDL > Tutorials > Reuleaux Triangle. If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels.
Instead the object is placed on a flat platform that in turn rests on cylindrical rollers (Figure 1). As the platform is pushed forward, the rollers left behind are picked up and put down in front. An object moved this way over a flat horizontal surface does not bob up and down as it rolls along. The reason is that cylindrical rollers have a circular cross section, and a circle is closed curve "with constant width.
" What does that mean? Is a circle the only curve with constant width? How to construct a Reuleaux triangle To construct a Reuleaux triangle begin with an equilateral triangle of side s, and then replace each side by a circular arc with the other two original sides as radii (Figure 4). The corners of a Reuleaux triangle are the sharpest possible on a curve with constant width. Here is another really surprising method of constructing curves with constant width: What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree.
Clever student: I know!
Now we just plug in x=0, and we see that zero to the zero is one! Cleverer student: No, you’re wrong! You’re not allowed to divide by zero, which you did in the last step. Which is true since anything times 0 is 0. Cleverest student : That doesn’t work either, because if then is so your third step also involves dividing by zero which isn’t allowed! And see what happens as x>0 gets small. So, since = 1, that means that High School Teacher: Showing that approaches 1 as the positive value x gets arbitrarily close to zero does not prove that .
Is undefined. does not have a value. Calculus Teacher: Why-couldnt-i-have-been-shown-this-in-maths-class.gif (Image GIF, 251x231 pixels) Weierstrass functions. Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere".
Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy.
Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr. Vi Hart: Math Doodling. Remember that video about doodling dragons and fractals and stuff?
I finally finished part 2! Here is a magnet link so you can dowload it via torrent. Here it is on YouTube: You can tell I worked on it for a long time over many interruptions (travelling and other stuff), because in order to keep myself from hating what was supposed to be a quick easy part 2, I had to amuse myself with snakes. Part of working on part 2 was working on part 3 and other related material, so the next one should go faster.
Here was part 1, via Torrent or YouTube.