An Abstract Kind of Clock: The Chinese Remainder Clock

Hackaday likes clocks, a lot. Speaking personally, from my desk I can count at least eight clocks, of which seven are working. There’s normal quartz movement analog clocks, fun automatic wristwatches, run-of-the-mill digital clocks, a calculator watch, and a very special and very broken Darth Vader digital clock/radio combo that will get fixed one day — most likely. Every clock is great, and one of life’s great struggles is to see how many you can amass before you die. The more unique the clock is, the better, and nothing (so far) tops [Antonella Perucca]’s Chinese Remainder Clock.

What separates [Antonella Perucca]’s clock from the rest lies in the Chinese Remainder Theorem, an eighteen hundred year old idea from Chinese mathematician Sunzi. Basically, when applied to the positive integers, the Chinese Remainder Theorem states that the set of integers modulo some positive integer N is equivalent (isomorphic as rings) to the product of the sets of integers modulo factors of N such that these factors are all relatively prime to each other and their product is N.

It sounds a mouthful, but it actually makes a lot of intuitive sense. Take for example N = 6. The set of integers modulo 6 is S = {0, 1, 2, 3, 4, 5}.  2 and 3 are relatively prime to each other and their product is 6. The set of integers modulo 2 and 3 are {0, 1} and {0, 1, 2} respectively. If you form the product of these two sets, you get the set T = {(0,0), (1,0), (0,1), (1,1), (0,2), (1,2)}. If you take any element x from S and send it to the element (x mod 2, x mod 3) in T, you quickly see how the two sets can be considered equivalent. Each element in S is sent to one and only one element in T, and both sets have the same number of elements. Taking N to be 6 may seem like an easy number to work with, but the beauty of the Chinese Remainder Theorem shows this same concept holds for any positive integer.

Now how does this apply to clocks? Well, two relatively prime factors of 12 are 3 and 4. Three relatively prime factors of 60 are 3, 4, and 5. We’re doing clock arithmetic modulo 12 and 60 anyway. Thus we can directly apply our theorem. Each hour can be represented uniquely by its remainder modulo 3 and 4. Each minute and second show up as their remainders modulo 3, 4, and 5. In the diagram above, imagine each circle starts at 0 and goes up clockwise by 1 for each bubble. The hours are the inside circles, the minutes are the outside circles, and the seconds are the tiny bubbles following the minutes. A clock starts to emerge then. Of course, you don’t have to stick with the somewhat familiar circular clock-face. You can use the same concept digitally, as in the image the very top of the article. Or you can go even further off the beaten track and rely completely on shaded polygons for that true abstract feel.

[Antonella Perucca] introduced the idea in “The College Mathematics Journal” less than a year ago, so it hasn’t seen too much use yet. However, anything that’s been done with regular clocks can be done with this one, and by sharing its existence, hopefully we get some really intriguing projects. It can even be in binary, for old time’s sake.