# Modular Art

Murals at North Hollywood High

Mathematics is the study of patterns. One of the ways in which we may use number patterns is in the creation of unique and artistically pleasing designs. In this web page, you will learn how to make designs based on modular arithmetic tables.

Modular or "clock" arithmetic is arithmetic on a circle instead of a number line. For a detailed explanation, visit Susan Addington's web page Clock Arithmetic. (Like this page, it will require a JAVA-capable browser.) A simple explanation of modular arithmetic follows.

Let us begin by examining the clock at the left. Observe that there have been two alternations. Since the number preceding 1 in ordinary arithmetic is 0 (the identity element for addition), the 12 has been replaced by a 0. Since our aim is to do clock arithmetic rather than tell time, both the hour and minute hands have been omitted from the dial.

If it is 3 o'clock and we add 5 hours, the time will be 8 o'clock. We write:  3 + 5 = 8 . But if it is 9 o'clock and we add 5 hours, the time will be 2 o'clock. So we write:  9 + 5 = 2 . Every time we go past 0 on the dial, we start counting the hours at 1 again. All multiples of 12 are equivalent to 0. To convert whole numbers to their mod 12 equivalent, we simply divide by 12 and record the remainder. It is the remainder alone that interests us.

In modulo 12 (or simply mod 12), we use only the twelve whole numbers from 0 through 11. It is a finite system. Any integer can be expressed as one of the numbers from 0 through 11. Classic sums and products can be presented visually in operations tables. The tables for addition and multiplication mod 12 are presented below.

 Addition Mod 12 Multiplication Mod 12

The addition table is rather boring. The only pattern worth mentioning is the presence of identical numbers on diagonal rows. There are many patterns in the multiplication table.

• Multiplication is commutative.
• As would be expected, the 2 row increases by 2's, the 3 row by 3's, and so forth.
• In ordinary arithmetic, the additive inverse of 4 is -4. In mod 12 arithmetic, the additive inverse of 4 is 8. In either system, the sum of a number and its additive inverse is zero.
• Rows corresponding to additive inverses are opposites of one another, save for the number 0.
• Multipliers that are relatively prime (gcd = 1) to the modulus 12 contain all whole numbers from 0 to 11 in their row.
• The other multipliers contain repeating blocks of numbers.

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The familiar 12 hour clock is very old. However there isn't anything special about the number 12. Consider the 4-hour clock at the right. The modulus 4 has been been replaced by the additive identity 0, as on the 12-hour clock. The clock has four numbers from 0 through 3. Every time we go past 0 on the dial, we start counting the hours at 1 again.

All multiples of 4 are equivalent to 0. To convert whole numbers into their mod 4 equivalent, we divide by the modulus 4 and record the remainder as before. Using the arrow buttons, set the modulus in the applet below to "4" so you can examine the mod 4 addition table. Verify the results for yourself.

Change the operation to multiplication by clicking on the "x" button, then study the resulting table.

A light switch with four positions (OFF, LOW, MEDIUM, HIGH) operates like a mod 4 system. Likewise the 7-day cycle of common weekdays operates like a mod 7 system.

• Use the applet to generate the mod 7 addition table.
• Repeat for the mod 7 multiplication table.
• As opposed to the mod 12 and mod 4 tables, every non-zero row in the mod 7 multiplication table will contain every possible result (that is, every whole number from 0 to 6 inclusive). Why is this so?
Investigate the addition and multiplication tables for the various natural number moduli permitted in the applet. The upper limit is 36.
• Look for patterns in each table, particularly those based on multiplication.
• Try generating specific tables with pencil and paper using patterns, then compare your result to the table generated by the applet.
• Extend your investigations to subtraction and division. How would you find the clock equivalent of negative integers? For what kind of moduli do the division tables exist?