Leap Years and Other Cyclical Oddities

When two cycles don’t exactly match, the result can be messy—and sometimes beautiful.

Illustration: Tomasz Walenta

Leap years can lead to strange situations. If you were born on Feb. 29, 1996, for instance, this year you will be celebrating your sixth birthday, even though you’re 24.

Leap years occur because the daily cycle of the earth spinning on its axis doesn’t quite match up with the yearly cycle of the earth orbiting the sun. While we count 365 days in a year, the time between vernal equinoxes is actually around 365 days and six hours. That means every four calendar years we accumulate a day of inaccuracy, which is why we insert February 29 to correct it.

However, the solar year isn’t exactly 365.25 days. It’s actually 365.24217 days, which means that every 100 years, we need to correct the calendar in the opposite direction by skipping the leap year—as happened in 1900 and will happen again in 2100.

But skipping the leap year every 100 years only gives us an average of 365.24 days a year, which means that after a few hundred years, we’ll be a day off again. That’s why every 400 years we put a leap day back in—as we did in 2000, which had 366 days. That way, the length of the year averages 365.2425, which is pretty close to the correct value.

More Everyday Math

Cycles that don’t match up are all around us. My sleep cycle seems to be slightly longer than 24 hours, so that I’m liable to go to bed later and later until I need an alarm clock to wake up. In music, a cycle of 12 intervals of a perfect fifth (the interval you hear at the beginning of “Twinkle Twinkle Little Star”) almost matches a cycle of seven octaves (an octave is the interval at the beginning of “Somewhere Over the Rainbow”)—but not quite. This is why tuning musical instruments was a conundrum for a long time, and various methods were used to try to “correct” for the discrepancy.

28365365体育投注Sometimes cycles of different lengths can’t be forced to align. If you need to take one medication every four hours and another every six hours, for instance, you will only be able to take them at the same time every 12 hours. That’s because 12 is the lowest common multiple of four and six, the lowest number that is a multiple of each one individually.

28365365体育投注The same principle enables us to work out how long other non-matching cycles will take to meet up. If you need to do something every three days, it won’t land back on the same day of the week until you’ve done it seven times. That’s because there are seven days in a week and seven is a prime number, which means the lowest common multiple of seven with any other number is seven times that number.

28365365体育投注The Spirograph drawing toy creates intricate patterns by using non-matching cycles. It consists of an inner wheel and an outer circular frame whose cycles don’t match up. Using a pen to trace the movement of the wheel around the frame produces intricate patterns that change slightly with every “orbit.”

Offset cycles can also be used to create interesting rhythmic effects in music. The idea is to pick two fairly small numbers with a relatively large lowest common multiple, such as six and 11, whose lowest common multiple is 66. If a piece of music uses both of those rhythms simultaneously, our brains find it difficult to latch onto the resulting overall cycle, and the result is a hazy and unrooted sensation, as in many of Chopin’s Nocturnes. Sometimes it’s impossible to make the neat and tidy world of math match the messy real world around us.

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