Eclipse of the Jaguar Serpent

Sometime earlier... Defectio Jaguar Serpens

The solar eclipse viewed from our roof top in San Francisco.

Eclipse of the Jaguar Serpent

Sometime later...
Voilà, Mechanical Mayan Calendar.

winkle, twinkle, little star, How I wonder what you are

I ought to explain how this works.  Each tooth represents one Earth day.  All the gears are simply coupled so that moment is propagated across the entire train.  Thus moving one gear one tooth move all the gears one tooth.  For each gear the teeth are numbered from 0 to (n-1).

Tooth counts for 365, 73 and 5 toothed gears.

The simplest example is the Earth-Sun relationship, where the 73 toothed and 5* toothed gears are compared.  An interesting Earth-Sun event occurs, say an equinox and we want to know predict when it happens again.  So, null out the dials on the 73t and 5t gear, or simply read the current setting.  I don't care, win an award.  Now you just run the clock forward until the dials get zeroed out, one day at a time.

What's going on is the each dial has it's own number base, 73 and 5, picked because 73 * 5 = 365.  The number of days in a year.  The dials will match again only after 365 teeth have advanced.

Homework: What do the dials read at half a year?

* I like to use gears with more than 12 teeth due to the physical properties of wood.  This also determines the resolution of my problem domain.  So, instead of 5 teeth I use 20 (4*5), and label the teeth 0 - 4, 4 times.

Sometime earlier...
I'm always on the look out for polynomials so recognising the diophantine equations in the Mayan Calendar was an amazing discovery to me.  Whoever designed the calendar was a genius of first order.

If you're not familiar diophantine equations, they are indeterminate polynomial equations where the variables are restricted to integers.  Perfect for making gears computers!  Look up Hilbert's Tenth Problem and you might start to see where I'm going with this.

Of immediate interest is the work of Derrick Lehmer, both of them.  This father and son team developed primitive digital computer to factor numbers and solve diophantine equations.  They where comprised of a number of loops or chains.  The driver motor is halted on a root and the result read as the position on each loop.

Check it out, a tape loop, hyperbolic space computer. 

An original Lehmer sieve.  Looks might be deceiving.

I'm adding a sextant/astrolabe attachment soon.  With a little work we'll be able to tell the day and year whenever our time machine goes.  Don't worry, I know what I'm doing and I have no intentions of destroying the universe.  In fact, I'm creating another universe to keep this one company.


In the mean time, check out this inspiring article on Continued Fractions.

I made you an Universe, but I ate it.

 Sometime earlier...

The Mayans where keen astronomers.  They had a particular fascination with the 584 day Venus cycle (584 factors into 2 * 2 * 2 * 73 ) , which will get interesting again come June 5th or 6th.  The Mayan year or Haab, has 365 days (365 factor into 5 * 73).

Oh look 73's, our time traveller is using Phillips Codes.  Hello fractal world!  From, the Mayan space telegraph operator.

Mayan Telegraphy.

The lunar cycle was a mutated (a day was dropped depending on the month) 30 day cycle (30 factors into 2 * 3 * 5) of 6 * 5 day periods.  The other major cycle was a 9 day, "Nine Lords of the Night" cycle (9 factoring into 3 * 3)

Finger hole obscura and random Mayan skull glyph.

The Mayans also tracked a 260 day Tzolk’in cycle (260 factor into 2 * 2 * 5 * 13) and 819 day that's not well understood (819 factor into 3 * 3 * 7 * 13).  There is also a concept of the Calendar Round of 52 Haabs or 18980 day (18980 factors into 2 * 2 * 5 * 13 * 73).  Which is the least common multiple of 260 and 365 or put in Mayan Maths, 73 * 260 Tzolk’in days equalling 52 * 365 Haab’ days.

 Eclipse analyser apparatus.

So, let's summarise our factors.

2 * 2 * 2 * 73 days per Venus cycle
           5 * 73 days per Solar cycle
2 * 3 * 5         days per Lunar cycle (minus one from time to time)
3 * 3              nights per Lords of the Night cycle
2 * 2 * 5 * 13 days per Tzolk’in cycle
3 * 3 * 7 * 13 days per weird unknown cycle
2 * 2 * 5 * 13 * 73 days per Calendar round
     2 * 2 * 13 Haabs per Calendar round

I'm going to guess that Mayan astronomers knew the synodic cycle lengths of all the visible planets.

2 * 2 * 3 * 5 * 13 days per Mars Synodic cycle (780 days)
3 * 7 * 19 days per Jupiter Synodic cycle (399 days)
2 * 3 * 3 * 3 * 7 days per Saturn Synodic cycle (378 days)

The common factor of 7 points to the weird cycle having something to do Jupiter and Saturn cycle.   The Mayans were already mutilating cycles with the lunar cycle so it's not difficult to imagine they did the same for Jupiter.  Theirs was a 20 base number system ( 20 is 2 * 2 * 5) and 400 (2 * 2 * 2 * 2  * 5 * 5) is very close to 399 days.  So I'm betting that the Jupiter cycle had a simple correction every 20 * 20 days.

BTW:  There is a Venus-Jupiter conjunction every 378 days (365 + 13) that I imagine the Mayan would have been interested in.

2 * 3 * 3 * 3 * 7 days between Venus-Jupiter conjunctions (378 days, oh look same as Saturn's Synodic cycle)

I'll be back later to show how we can build a mechanical Mayan calender from just a few gears. 

The cycle continues at the top...