Pythagoras Clock

Illustration of Pythagorean triple, 3, 4, 5, and units of time

The traditional clock shows the hour divided into 12 periods of 5 minutes, 6 periods of 10 minutes, 4 periods of 15 minutes ( a quarter of an hour ), 3 periods of 20 minutes, and 2 periods of 30 minutes (half an hour). This neat divisibility is a consequence of the factorizations of the number of hours in a day and minutes in an hour: one day = 12 hours = 3*4 hours – resulting in integer factors of 2, 3, 4 and 6 hours. One hour = 60 minutes = 3*4*5 minutes – resulting in integer factors of 2, 3, 4, 5, 6, 10, 15, 20, 30 minutes. If there were 61 instead of 60 minutes in an hour, then half an hour, a quarter of an hour – none of these would correspond to a whole number of minutes.

There is something else special about these numbers: 3, 4 and 5 make up the least Pythagorean triple: a right angled triangle can be drawn with sides 3, 4, and 5 units long.

A few years ago, I designed the Pythagoras Clock, based on this coincidence.

The clock is powered by an Atmel ATMega168. The clock display is made of laser cut acrylic, housing 19 colored LEDs laid out in two intersecting Pythagorean triangles.

Pythagoras Clock

The two lines in the upper right corner – excluding the diagonal – represent fractions of a day, and the three lines in the bottom left corner plus the diagonal, represent fractions of an hour.

Pythagoras clock with explanatory overlay

When the clock boots, it shows up at 9 o’clock. The following video shows the clock working at increasingly accelerated speed, starting from 9 o’clock:

The top line, of white LEDs, divides each day up into four quarters. If the leftmost of those quarters is lit, then we are in the first quarter of the day, i.e. between twelve o’clock and three o’clock; if the rightmost quarter is lit, then we are in the fourth quarter of the day, i.e. between nine o’clock and twelve o’clock. The orange line on the right edge divides each of those quarters of a day into three thirds. If the topmost of those LEDs is lit, then we are in the first third of the quarter of the day represented by the top line, and if the bottommost is lit, then we are in the last third of the quarter of a day.

In the diagram, the second white LED is lit, meaning the time is between three and six, and the first orange LED is lit, meaning the time is in the first third of that range, i.e. it is four. More mathematically, the hour is 1 x 12/4 + 0 * (12/4)/3 = 4.

Similarly, the bottom line counts time in units of a quarter of an hour, the leftmost line counts time in units of a third of a quarter of an hour (five minutes), and the diagonal counts time in fifths of thirds of quarters of an hour, i.e. minutes.

So in the diagram the hour is 4, as we have already seen, and the minutes are zero quarters (bottom line), two thirds of a quarter (left line), and four fifths of a third of a quarter (diagonal), i.e. 0 x 60/4 + 2 x (60/4)/3 + 4 x ((60/4)/3)/5 = 14.

I am releasing the source code and designs under open source licenses – the source code is available under an MIT license, and the designs are licensed under a Creative Commons Attribution-ShareAlike 4.0 International license. You can download them from Github

The circuit assigns one pin for each LED, and one for the button which is used to set the clock or place it in demo mode.

Pythagoras clock circuit

The clock spent a long time on a proto board, but I finally did get some boards made. If you’d like one let me know.

An assembled circuit for the Pythagoras Clock.

The body of the clock is a sandwich of eight pieces of acrylic, which I cut with a laser cutter at Techshop. The Corel Draw file is on github.

The acrylic cuts.

The LEDs each sit in a little pocket, with a rectangular slit set on top of and to the edge of the pocket.

During assembly of the acrylic.

This gives a uniform rectangle of light.

Light rectangle