Additional Math Pages & Resources

Monday, June 6, 2011

Degrees, Gears and Ratios, Part I

I had the chance to photograph a complicated Swiss watch recently. It's called an IWC GST Perpetual Calendar Chronograph. Here is a photo of its dial [click on it for a larger version]:

As you can see, this watch has lots of hands. The large white hands show 8 hours and 9 minutes. The small white hand at 9 o'clock (partly hidden) shows about 25 seconds.

The white hands (at 12, 6 and center) indicate elapsed time on the stopwatch. Here they are showing 10 minutes (12 o'clock hand) and nearly 23 seconds (center hand).

The yellow hands indicate the date is Monday (9 o'clock hand) June (6 o'clock hand) 6th (3 o'clock hand), and the year window shows 2011.

We teach kids geometry in our Excel Math curriculum. They learn a circle is measured in units called degrees. A complete circle is 360 degrees. Now take a close look at the 5 dials of this watch - how many degrees make up each segment of each circle? [click for larger]

A dial designer has to do the math to make sure the lines are clearly and evenly sub-dividing the various indications on the dial.

I did this for you on the diagram above - the stopwatch [white] shows quarter seconds and seconds on a 1-minute scale (outside of the dial), minutes on a 30-minute scale (at 12), and hours on a 12 hour scale (at 6).

The calendar [yellow] must indicate days of the week (360/7), days of the month (360/31), months of the year (360/12), and days of the moon cycle. On this dial the moon is shown rising and setting in a half-circle, so the formula is 180/29.5.

After doing the math I had to draw all the lines too, which was the harder part for me.

All this fancy stuff has to be accurately depicted on the dial seen by the wearer. It is also shared with the watchmakers, who are designing the unseen mechanisms behind the dial to move the hands appropriately, and to keep time correctly!

Since most of the hands on this watch jump from place to place it could be very distressing to have them missing the markers on the dial. Both the mechanisms and the placement of the hands on the arbors have to be done very precisely. A few of the indicators appear to move smoothly (the second hands; the moon) so they don't have to "hit the mark".

Here's a behind-the-dial view of this particular watch mechanism. The red spots show you where the shafts that the hands fit onto (the arbors) are located. You can see a few of the gears that drive the hands. The rest of the mechanism is further underneath this layer.

Original art from "The Grand Complication" by Manfred Fritz

We will do more some watch math with degrees, gearing and ratios in the coming days (but I cannot guarantee you will earn a math degree reading this blog).