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: Discrete Math Lecture Notes. Stephen Wolfram: Computing a theory of everything. Make way for mathematical matter - physics-math - 05 January 2011. Editorial: "The deep value of mathematics" WE ALREADY have solid, liquid, gas, plasma and Bose-Einstein condensate. Now it seems we may be on the verge of discovering a whole host of new forms of matter - all based on mathematics. Nils Baas, a mathematician at the Norwegian University of Science and Technology in Trondheim, has unearthed a plethora of possibilities for the way the components of matter can link together.
He made the discoveries while researching the field of topology - the study of the properties that objects share because of their shape. It is particularly concerned with the various shapes that can be formed while squashing and bending an object. A ring doughnut and a teacup share the same topology, for example: it is possible to squish the doughnut into a teacup shape without doing away with the hole, as it becomes the hole in the handle. Baas was studying "Brunnian rings" - collections of rings that are linked together but can all be separated if only one ring is cut. Lattice Multiplication.
6174 (number) 6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property: Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.Subtract the smaller number from the bigger number.Go back to step 2. 9990 – 0999 = 8991 (rather than 999 – 999 = 0) 9831 reaches 6174 after 7 iterations: 8820 – 0288 = 8532 (rather than 882 – 288 = 594) 8774, 8477, 8747, 7748, 7487, 7847, 7784, 4877, 4787, and 4778 reach 6174 after 4 iterations: Note that in each iteration of Kaprekar's routine, the two numbers being subtracted one from the other have the same digit sum and hence the same remainder modulo 9.
Sequence of Kaprekar transformations ending in 6174 Sequence of three digit Kaprekar transformations ending in 495 Kaprekar number Bowley, Rover. "6174 is Kaprekar's Constant". Jain's True Value of Pi. I will be releasing a new body of work that gives the True Value of Pi, based on the Harmonics of Phi (1.618033...), a value close to 3.144... The ancient Mathematics masters have always known that the two most important transcendental numbers Pi and Phi are intimately related. As shown on this website, The Book of Phi, volumes 1 and 2 are available, but the upcoming, unpublished volumes of 3 and 4 will reveal this Pi Phi Connection and how 3.144... is derived from the Square Root of Phi (1.272...). These two books, THE BOOK OF PHI volumes 3 and 4 are based on the 24 Repeating Pattern of the Digitally Compressed Fibonacci Sequence that encodes the frequency of 108, that anointed Vedic Number sonically encrypted in the prayers for enlightenment known as the GAYATRI MANTRA.
Visualize a Hexagon in the Circle which has 6 straight lined chords. It is encoded in the mathematics of the Cheops Pyramid. Until then, Part 2 Original Title on Manuscripts: JAIN of AUSTRALIA'S VALUE of PI. Deep meaning in Ramanujan's 'simple' pattern - physics-math - 27 January 2011. The first simple formula has been found for calculating how many ways a number can be created by adding together other numbers, solving a puzzle that captivated the legendary mathematician Srinivasa Ramanujan. The feat has also led to a greater understanding of a cryptic phrase Ramanujan used to describe sequences of so-called partition numbers. A partition of a number is any combination of integers that adds up to that number. For example, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1, so the partition number of 4 is 5. It sounds simple, yet the partition number of 10 is 42, while 100 has more than 190 million partitions.
So a formula for calculating partition numbers was needed. Previous attempts have only provided approximations or relied on "crazy infinite sums", says Ken Ono at Emory University in Atlanta, Georgia. Pattern in partition Without offering a proof, he wrote that these numbers had "simple properties" possessed by no others. Mathematical metaphor More From New Scientist More from the web. Mathematicians Solve 140-Year-Old Boltzmann Equation.
PHILADELPHIA –- Two University of Pennsylvania mathematicians have found solutions to a 140-year-old, 7-dimensional equation that were not known to exist for more than a century despite its widespread use in modeling the behavior of gases. The study, part historical journey but mostly mathematical proof, was conducted by Philip T. Gressman and Robert M. Strain of Penn’s Department of Mathematics. The solution of the Boltzmann equation problem was published in the Proceedings of the National Academy of Sciences. During the late 1860s and 1870s, physicists James Clerk Maxwell and Ludwig Boltzmann developed this equation to predict how gaseous material distributes itself in space and how it responds to changes in things like temperature, pressure or velocity. The equation maintains a significant place in history because it modeled gaseous behavior well, and the predictions it led to were backed up by experimentation.
The study was funded by the National Science Foundation. 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. This is how to do it: 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 ! . As . . ? 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. Hypotrochoid_R_equals_7,_r_equals_2,_d=3.gif (GIF Image, 400x400 pixels) Animated Trigonometry | Jason Davies | Web Design and Development.