3.0    Analytical Methodology

3.1    Mathematical Bases

The “Hailstone Theory” is based on the following equations:

n/2   if n is even

3(n)+1   if n is odd

Lothar Collatz created the “Hailstone Theory”, also known as the Collatz Problem, 3n+1 problem, and the Syracuse Problem, in 1939 [7]. When a number is run through these equations, the output is a sequence of numbers that are in random order from zero to the number of eight-counts of that specific dance routine, eventually reaching one. When the sequence of numbers arrives at one, the program stops the sequence [3].

Along with the “Hailstone Theory”, the programs use a logarithmic function for each of the four specific dances. A logarithmic function [8] is used in The Rhythm of Hailstones to simplify the mathematical calculations. The program uses a log function. The number in front of each logarithmic function is there so the numbers in the sequence do not exceed the number of eight counts for that specific dance routine. It does this by multiplying the log by a number where the highest log output multiplied by the number before it equals the number of the last eight count of the dance. For example, if there are 38 eight counts in the dance, the choreographer does not want the program to output 44 as one of the numbers.

The logarithmic functions used in the programs are as follows:

· Jazz Program (Appendix A): (6)*log(n)

· Hip Hop Program (Appendix B): (8)*log(n)

· Mix Program (Appendix C): (9.5)*log(n)

· Pom Program (Appendix D): (9)*log(n)

Where n is the integer entered by the user.

3.2    Computer Applications

All four programs were created and written by Team 082. The four Rhythm of Hailstones programs are similar in structure with different outputs. The programs all include the stdlib.h, math.h, and iosphere.h source files. All four programs allow the user to enter a starting number between 50 and 100. This range of numbers was chosen because the sequences that are produced work well with the number of eight-counts and the length of the music. The starting number is run through the formulas in a loop. Then the numbers outputted are run through the specific logarithmic function of that program. As a result, the program outputs a list of numbers that increase and decrease in a manner similar to hailstones, for example, 19, 16, 20, 18, 15, 19, 17, 14, 12, 10, 7, 5, 2, 0 (Note: log of 1 is 0). This example from the Hip Hop program used the starting number 75. The first step requires 75 to be multiplied by 3 and have 1 added to that number because the starting number of this example is odd. The final result of this equation is 226 and because 226 is even it is next divided by 2. This continues until the sequence reaches a value of 1. Then each of those numbers is run through the logarithmic function for the Hip Hop program ((8)*log(n)). That is how the smaller numbers are produced. Each number in the sequence corresponds to a video clip issued to a specific eight count. If the user entered 75 in all four programs, the output sequence would be different for each program. This happens because each program has a specific logarithmic function where the number in front of the logarithmic function is different for each dance routine. The numbers differ because each dance has a different number of eight counts.

The logic for the Rhythm of Hailstones programs is simple:

   · The user enters a beginning integer.

   · If the integer is odd, then it is multiplied by three and adds one.

   · If the integer is even, then two divides it.

   · The for() loop has this process continue until the number hits one.

   · Those numbers outputted are run through a logarithm.

   · The program outputs those integers in a list.

   · The user opens Windows Movie Maker (where the video clips are located) and put the video clips in the order of the    outputted sequence.