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Mathematical Mysteries Behind

the Cosmic Calendar

Photo of a Kalachakra or Wheel of Time exhibited at the National Watch & Clock Museum. KEITH LEHMAN.

Would it surprise you to know that if you traveled back 2,600 years to the ancient city of Babylon and handed King Nebuchadnezzar a perpetual calendar chronograph with a moondial, he would instantly recognize it as a timekeeping device? Would it surprise you even more to know that if after converting the Arabic numbers to cuneiform, he could tell you the time? Just don’t argue with him about how our lunisolar calendar is superior to his lunar one, because he might just throw you into the fiery furnace!

An illustration from the early twentieth century depicts Nebuchadnezzar surveying the Hanging Gardens of Babylon (Corbis).

IWC Schaffhausen Da Vinci Perpetual Calendar Chronograph. IDC.

European astronomical clock with zodiac figures. King Nebuchadnezzar would have also recognized the zodiac figures of this clock as they are the very figures he saw when he consulted his star maps and astrologers. This makes the question, "What is your sign?" the oldest pick-up line in human history.

Last year in October I was honored to be one of three panelists to give a presentation on the various aspects and oddities of calendar time. My fellow panelists were Hodinkee editor Jack Forester and General Manager of FP Journe Pierre Halimi. Jack’s presentation focused more on the impossibilities of keeping perfect calendar time, and Pierre focused on the political influence on the calendar.


I’m one of the last people who should be giving a discussion on math. I was not a good student with numbers, and like most of us, I dreaded the math courses that we were forced to take in school. I took a few classes in college but only the requirements to graduate; I enrolled in a calculus class but withdrew after the first week. I decided to be an artist because what possibly could an artist need math for? In hindsight I regret this decision and realize how foolish I was. As I’ve gotten older, I began to appreciate the beauty of mathematics and apply it daily in my design and thinking.


I started to consciously apply math to my design shortly after beginning my professional career at Clipper Magazine. Designers were expected to create great looking ads in a short amount of time. Famously believed to be discovered by the Greeks and used in their art and architecture I experimented with the golden ratio and I found that it allowed me to create great looking ads in a short amount of time. Unfortunately the belief that this ratio can be found in all sorts of great art, architecture and even in the proportions of the human body simply is isn’t true, a fact that I only recently discovered. It can however be observed in in patters of plants which is illustrated below.

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. Ahecht (Original); Pbroks13 (Derivative work); Joo. (Editing).

Written in Fibonacci's book Liber Abai in 1202  AD, the Fibonacci sequence is found as math exercise using rabbits as an example. They are characterized by the fact that every number after the first adds up to the two preceding ones. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Whirling squares to illustrate decomposition of golden rectangle into smaller golden rectangles; self-made in inkscape. Author Dicklyon.




Examples of the golden ratio in observed in nature. There are many misconceptions of where the golden ratio can be found. Apocryphal examples are marked in red as false.


Despite claims that the Golden Ration was used to construct the Parthanon in Athens, Greece there is no evidence of it.

The golden ratio can be observed in how plants stagger their leaves to maximize exposure to sunlight.

Here are some examples of my artwork where I applied the golden ratio.

New Holland Auto Group commercial illustration. Keith Lehman.

Clipper Magazine ad. Keith Lehman.

Enlisting Time poster for National Watch & Clock Museum. Keith Lehman.

The Alliance fantasy illustration. Keith Lehman.

Museum Guide Book—National Watch and Clock Museum. Keith Lehman.

Odiyan Retreat Center. Cazadero County CA.


So what does this have to do with horology and calendars? After college I took the opportunity to work as an artist for the Tibetan Aid Project in Berkeley, CA, and later the Odiyan Retreat Center in Cazadero, CA. It was a life-changing experience as I discovered the rich history and the religious and artistic achievements of the Tibetan people.


During my time at Odiyan I learned about the cosmic calendar, or what the Buddhists call the Kalachakra or Wheel of Time. Buddhist teaching explains that all of creation churns within a great wheel that spins in an endless cycle of creation, maintaining, and destruction. One full turning of this wheel is called a Kalpa, and much like one earth year, a Kalpa is divided into four seasons. These seasons, which have specific qualities and events, including numbers of kings, how long the average human lives, and the quality of life, are called Yugas, which last for thousands of years to the ratio of 4:3:2:1. Religious scholars believe we are in Kali Yuga, the final Yuga of the cycle.


I always found the numbers associated with the Kalachakra interesting, but it wasn’t until I began working here at the Museum that I found the possible origins and explanation for where these figures come from. Let’s take a look at the origins mathematics, developed over 7,000 years ago and see how it possibly influenced the Kalachakra and other calendars outside the Buddhist tradition—religious and nonreligious alike, including the universal one that we use today.

Odiyan Retreat Center. Cazadero County CA.



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