Rader's NUMBERNUT.COM 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. 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. Other symmetrical curves with constant width result if you start with a regular pentagon (or any regular polygon with an odd number of sides) and follow similar procedures. Figure 1: Platform resting on cylindrical roller
Mathematical Atlas: A gateway to Mathematics Welcome! This is a collection of short articles designed to provide an introduction to the areas of modern mathematics and pointers to further information, as well as answers to some common (or not!) questions. The material is arranged in a hierarchy of disciplines, each with its own index page ("blue pages"). To reach the best page for your interests, use whichever of these navigation tools ("purple pages") you prefer: For resources useful in all areas of mathematics try 00: General Mathematics. There is a backlog of articles awaiting editing before they are referenced in the blue pages, but you are welcome to snoop around VIRUS WARNING: The Mathematical Atlas receives but does not send mail using the math-atlas.org domain name. Please bookmark any pages at this site with the URL This URL forces frames; for a frame-free version use
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! 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: For all , we have Hence, That is, as x gets arbitrarily close to (but remains positive), stays at On the other hand, for real numbers y such that , we have that That is, as y gets arbitrarily close to Therefore, we see that the function has a discontinuity at the point . but when we approach (0,0) along the line segment with y=0 and x>0 we get Therefore, the value of is going to depend on the direction that we take the limit. that will make the function ! . as is whatever
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. Here was part 1, via Torrent or YouTube. 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. 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.