Infinity Permeates Everything - a futile example with the Fractal Yupana

Technically, it is not infinity. But practically, it is. 

You would waste your whole life if you tried to count all the grains of sand on the nearest beach. 

You would return to dust before you can count every individual cell in your body. Not only because you are continuously producing new ones, or because you can't see any of them. 

In that sense, you are touching infinity, everywhere around and inside; you are made of "infinity".

So... it shouldn't be that astonishing to also find this kind of "infinity" in one single row of the Fractal Yupana. 

Multiply the age of the universe by itself,
and multiply that result by the age of the universe...
and do that again with the lastest result... 10 times !
and then multiply that by 10 billions... 

The result, in years, approximately 10 billions Googols (https://en.wikipedia.org/wiki/Googol) years, is the amount of time required to count from 1 to 100, by  all the different ways there are to do that, by using a single row of a Fractal Yupana. 

(This, assuming a few things, including the speed of your manipulations on a real yupana, taking no rest at all, living 1 Googol times as long as the current age of the universe... at least the universe is set to exist long after that https://www.science.org/content/article/way-universe-ends-not-whimper-bang ) 

By comparison, there is only 1 way to count from 1 to 100 with a Soroban. So, that would not be a way to appreciate "infinity".

Illustrations for an apparently unsubstantiated claim.

Number of ways you can "write" each number using a single row of the Fractal Yupana. 

You can represent every number from 0 and 100 on one single row of a Fractal Yupana. Even though you would not do that, normally, except during multiplications, prefering to represent numbers from 0 to 9 on each row.

Click here for an interactive version

3 followed by 115 zeros... 3 quadrillons of googols !?

That is less than the total number of the different ways to count from 0 to 100 on a single row of a Fractal Yupana. 

It is the result of the multiplication of the number of representations of each of the numbers from 0 to 100. That is a multiplication of 101 terms, and it is going to be huge (even though the 3 first ones and the 3 last ones are equal to 1). For instance, you have already 20 numbers that have at least 35 different representations: those from 38 to 57. That only is already greater than 35 multiplied by itself 20 times.

(a quadrillion of googols would be the american name for 10 to the power of 115, according to Encyclopedia Britannica : https://www.britannica.com/topic/large-numbers-1765137) 

Comparison with other interpretations of the Yupana. 

Let's first show the distribution of the numbers of ways to represent each number on one single row of the yupana. Then we'll compare the total number of ways to count from 0 to the maximum value on 1 row. 

Yupana Dinamica / Inka.

You can count from 0 to 39 on a single row of the Yupana Dinamica (assuming, as for the Fractal Yupana, that you put 0 or 1 token on each spot).  For instance, 145 is the number of ways to represent 18, same thing for 21.  

click here for an interactive version

This may seem very grossly exaggerated. It is due to the fact that we count as being different two different spots in the same column, even though all spots in one column have the same value (example, each of the 5 spots of the 4th column have value 5). But we need (? not really) to do that if we are to compare with the interpretations of the Yupanas that give a different value to each spot.

If we consider as identical all the spots of one column, the number of ways to write a number falls down drastically. For instance, you'll have only 3 ways (5, 3+2, 2+2+1), instead of 12, to represent 5. And instead of 49 ways to represent 10, you'll have only 3 (5+5, 5+3+2, 5+2+2+1). 

Yupana Chirinos-Rivera.

You can count from 0 to 66 on a single row of the Chirinos-Rivera's Yupana (assuming, as always, that you put 0 or 1 token on each spot). For instance, 70 is the number of ways you can represent 33. 

click here for an interactive version clearer image

Comparison of the total number of ways to count from 0 using a single row.

Orders of Magnitude 

115: Fractal Yupana                       (~ 1 followed by 115 zeroes)

81  : Yupana Chirinos-Rivera         (~ 1 followed by 81 zeroes)

54  : Yupana Dinamica / Inka         (~ 1 followed by 54 zeroes)

(about orders of magnitude, see https://en.wikipedia.org/wiki/Order_of_magnitude)