Showing posts with label iteration. Show all posts
Showing posts with label iteration. Show all posts

Tuesday, July 5, 2016

What is iteration?


   Many of my Scratch projects are about fractals and mathematical chaos. Examples are given below. 
   Both topics are products of the computer age and are powered by a mathematical method known as iteration. Iteration is sometimes confused with looping but it is a much deeper and more mathematically powerful tool than looping.
   I have written a document that carefully, step-by-step, compares the process of graphing y = f(x) with the iteration of y = f(x). This information is basic for developing an understanding of the mathematics that underlie fractals and chaos.
   In the graphic shown below, the image on the left is the graph of the function given in the center of the graphic with the parameter c varying from 0 to 8 in increments of one.


      The image to the right is the iteration of the same function with c starting at -0.5 and varying to -2.0 in 0.005 increments

   For your free copy of this document send an email to www.grandadscience@gmail.com.

A Sample of My Fractal and Chaos Projects

Langton's Ant - Random Squares Experiment 
https://scratch.mit.edu/projects/41285348/

Koch Snowflake
https://scratch.mit.edu/projects/11128415/

x^2+ c Plots
https://scratch.mit.edu/projects/63919408/

Chaos Game - Midpoint Formula
https://scratch.mit.edu/projects/13649502/

Mandelbrot Set [Featured Project - 4656 views]
https://scratch.mit.edu/projects/10488782/





Wednesday, January 28, 2015

The Sumerian Square Root Algorithm


   Ancient Sumer was located in southern Mesopotamia in what is now called Iraq. Sumerian history dates to 5000 BC.  In 2340 BC, Sargon I founded the Akkadian dynasty. By 200 BC Sumer had declined to the point of eventually being absorbed by Babylonia 
and Assyria.
   The Sumerians are believed to have invented the cuneiform system of writing on clay tablets with a cut reed.
   The Sumerian algorithm for approximating square roots (also known as the Babylonian method and Heron's method) used a numerical method called iteration.
   Iteration is a special form of ‘looping’ where given an initial input (called the seed) to a function, the output becomes the next input.

   Suppose one wants to compute √(a). One picks a guess, xn-1, plugs it and a into the formula to compute xn. This output, xn becomes the new input, xn-1 and a new output is computed. Do this repeatedly and xn converges to the √(a).

   You can view and download this project by clicking on the link below.

http://scratch.mit.edu/projects/45120106/
   A free PDF document explaining how thew formula is derived can be had by sending an email request to 

grandadscience@gmail.com.