Spider Mathematics

Applets show the efficiency of displaying numbers spirally on a spider web instead of a number line. The usual clock face with 12 digits is the basis of arithmetic modulo 12. An applet allows the user to change the clock base from 12 to any other number of digits. This provides a mechanism for introducing arithmetic to different moduli in a visual and engaging way.

Another applet allows the spider to jump along the spiral along the even numbers, along the multiples of 3, or generally along the multiples of any number. The user has the choice of hiding or showing the numbers at any time. When the numbers are hidden the path traced out is better revealed.

A game involves a fly that lands at some integer on the web. The spider tries to catch the fly by jumping along multiples of 2, or multiples of 3, etc., trying to minimize the number of jumps. Sometimes she catches the fly quickly; sometimes it takes a long time. A tutorial will teach that the optimal strategy is to jump along multiples of 2, 3, 5, 7, 11, etc., following the smallest primes in sequence. The colors of the multiples of 2, 3, 5, etc. will change during the process, and the viewer will see the leftover primes on the web in a different color. After only four trial paths along multiples of 2, 3, 5, 7, all the primes less than 121 will be revealed. Two more trial paths along multiples of 11 and 13 will reveal all primes less than 289. This provides an effective way of generating and discovering primes and is, of course, another way to present the Sieve of Eratosthenes.

A surprising picture is revealed when the clock base is changed to 12. All primes (except 2 and 3) appear along exactly two radial lines (through 1 and 7, and through 5 and 11). A similar alignment appears when the clock base is 6 or 18, due to the fact that the primes greater than 3 are congruent to plus or minus 1 modulo 6.