## Wednesday, January 24, 2018

### How many degrees are in a Martian Year?

James Tanton gave a great talk at the JMM (Joint Math Meeting) in San Diego on

how many degrees are in a Martian Year?

but he didn't quite answer his title question. I found an excerpt of the talk on YouTube but still didn't quite answer the question, though he could have.

The talk was

How many degrees are in a Martian Year?

Here is an excerpt on You Tube: here

I feel the urge to explain and complete the talk.

Why do we earthlings have 360 degrees in a circle? Because there are 365.25 days in a year and nobody wants to work with that number so they round off. They could have chosen 360 or 365 or 370 but since 360 has the most divisors, it wins. (There is a different measure, gradians, of which there are 400 in a circle- also a nice round number, but 360 is rounder via the criteria we will soon explain.) So why pick 360? It has the most divisors.

There are 668 martian days in a martin year. So you might think they would pick 670 or 660 for there degreees. But thats Base 10 thinking!  So the answer depends on what Base the Martians use. Lets assume that, like us, it depends on the number of fingers they have on their hands. We also assume they have two hands with the same number of fingers on them (would aliens look like us or not? Here are some thoughts:  here and here). Two fingers on a hand seems like two few, and ten seems like too many so lets assume there base is either

3 fingers per hand: base 6. 668-base 10 is 3032 base 6. 3020 is answer- divided by 2,3,5

4 fingers per hand: base 8. 668-base 10 is 1234 base 8. 1240 is answer- divided by 2,3,4,8.

5 fingers per hand: base 10 668. Lets go with 660- divisible by 2,3,5

6 fingers per hand: base 12 668-base 10 is 478-base 12. 470 is answer- divided by 2,3,4,5

7 fingers per hand: base 14 668-base 10 is 35A in base 14. 140 is answer- leave it to you.

Any other good candidates or methods for this?

1. I doubt the days in a year connection.

With the 24 hour day, the circle needs a value easily divisible by 24.

Is the timing of this post related in any way to the new unit of time known as the flick? 1/705,600,000th of a second. Works great when dealing with video frame rates.

1. Where did 24 hours in a day come from? (I ask nonrhetorically). Not sure which came first.

THANKS for info on flick- did not know that. Perhaps that new unit was inspired by my post :-)

2. 24 hour days from ancient Egypt, possibly because they used base 12

2. why does this post mention Martian Year in the title and other places?

1. The YouTube video is "How many degrees are in a Martian circle?" but I think this post is merging that question with "How many days in a Martian year?" to create a meta-question.

3. 668-base 10 is 478-base 12, rather than 479-base 12.

668-base 10 is 35A-base 14, rather than 14A-base 14, which is 262-base 10. Here, a choice of 360-base 14 would be interesting. This is 672-base 10, so the factors are the same as for the 4 finger case.

In any case it would be interesting to present to prime factorization of your chosen number. For example, 470-base 12 is 660 base-10. This is 2*2*3*5*11. So there are 22 divisors: 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, and 330. 3020 base 6 is also 660 base-10, so this has the same divisors. Likewise the 5 finger choice.

For the 4 and 7 finger choices, 672-base 10 is 2*2*2*2*2*3*7. There are also 22 divisors: 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, and 336.

4. 1) I have made the corrections, thanks
2) What makes a number better- number of divisors or number of PRIME divisors? I would go with prime divisors but either is a good criteria.