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.
   A free PDF document explaining how thew formula is derived can be had by sending an email request to

Thursday, January 22, 2015

The Lorenz Attractor and Sensitivity to Initial Conditions

    I have previously posted on the Lorenz Attractor in Scratch (see:
   In that post I described the equations that define the attractor and that the attractor models a simple convection cell powered by a source of heat. 
    The importance of the Lorenz Attractor is that it is a nonlinear system that exhibits a sensitivity to initial conditions. In a linear system, small changes in inputs produce small changes in outputs.

    In this project, two points that start very close together—small changes in inputs—quickly diverge—large changes in outputs.
   At first, the two “particles” (each representing a slightly different starting state of the convection cell at time t) remain close together. But, after approximately 1400 iterations, the two points start to separate (see small circles in above graphic). 
   This behavior demonstrates how a very slight difference in the initial positions (x, y, z coordinates) of the two particles quickly lead to two different states of the cell.
   In the display, the two lobes can be thought of as representing a rotation of the convection dell in one direction and the other lobe as the rotation of the cell in the opposite direction.
   You can view and download this project by clicking on the following link.
   I have written a more detailed description of this project (including a sample of Dr. Lorenz's original data) that can be had in PDF form by emailing me at

Wednesday, January 14, 2015

Bertrand's Random Chord Paradox Methods 2 and 3

   In my April 21, 2014 post, Bertrand’s Random Chord Paradox 1  ( I presented the first of three methods for randomly selecting a chord in a circle with an inscribed equilateral triangle and then using the Monte Carlo method (coded in Scratch) to experimentally determine the probability that a randomly selected chord is longer than the side of the inscribed triangle. For the random points on the circumference method discussed in the post, the probability is one-third (1/3).
   The paradox arises from the fact that two other equally valid methods for picking the chord each give a probability different from one-third. 
   In my Scratch project Bertrand’s Random Chord Paradox 2 (
the Monte Carlo method correctly approximates the probability that a randomly selected chord, using the random radius method, is one-half.
   In my Scratch project Bertrand’s Random Chord Paradox 3 
using the random midpoint method, the probability is found to be one-fourth!
   The theoretical derivations for the three methods are presented in a three free, PDF files that can be obtained by sending an email request to

Saturday, January 3, 2015

Langton's Ant Random Black Squares Experiment

   I have one more Langton's Ant in Scratch project to share.
   With the exception of experiments involving probability, one doesn’t usually associate the word experiment with mathematics. Although the behavior of Langton's Ant is clearly determined by its rules, its behavior is unpredictable! For that reason, the ant is a mathematical system that lends itself to experimentation. This project lets you explore for yourself what happens if random square are distributed on the coordinate plane.
   To start the project, click on the green flag. The ant shrinks to the size of one of the squares that make up the grid, ten additional black squares are randomly placed within a 100-step square centered on the origin (0, 0). Langton's Ant, starting at the origin and facing to the right, starts moving as shown in the following diagram.
   The diagram below shows the normal behavior of Langton's Ant when no black squares are randomly added to the plane. This is the control pattern for experiments.
   This diagram shows the path of the ant when ten black squares have been randomly added to the plane. Note its differences from the control pattern.
   This project can be viewed and downloaded by clicking on this link.
   I have written a detailed description of the Scratch code and ideas for experiments you can do with the ant. To request your free PDF, send an email to
   For a deeper description of Langton's Ant and why it's an important mathematical system, download and read the following article by Ian Stewart.