¡A Plausible Way For Ponies To Develop Using Balanced Ternary As A Base! · 4:06am Feb 17th, 2015
A BlogPost by Pineta about pony-bases and interactions with MSPiper Inspired this BlogPost:
> “Balanced ternary suffers from the same issues as base three, plus the problem of being completely contrary to the seeming ’default’ modes of thought. While it makes sense enough once you’ve gotten used to it, I’ve never heard of any culture that uses a similar method as their normal counting system;”
——
MSPiper
I have a plausible path to Balanced Ternary for ponies:
Ponies start counting using their forehooves (Base2). The Ponies discover the concept of 0. They discover the concept of negative numbers. A pony-mathematicians discovers that if negative -1 has its own symbol and ponies switch to Base3, the mathematics gets much simpler because multiplication is still easy and carries occur only 2/9ths of the time:
* As you know, the carry-rate for addition in Binary is ¼ (it is 2/5ths for decimal). The carry-rate is only 2/9th for Balanced Ternary.
* As you know, in Binary, , when one multiples 2 numbers, one can replace every 1 in one number with the whole number of the other number, shift the radix-point appropriately, and add the partial sums to get the final answer:
11010001*00100100=
0001101000100
+
1101000100000
=
1110101100100
In Balanced Ternary let:
▲:
-1
◯:
0
⬛︎:
+1
The rules for multiplication are the same except one switches the sign of digits when the digit is negative:
▲⬛︎▲*◯⬛︎⬛︎=
0▲⬛︎▲
+
▲⬛︎▲◯
=
▲◯◯▲
We get negative -28.
¡The ponies see the advantages of Balanced Ternary and switch from Binary to Balanced Ternary!
I believe that this is a plausible path to Balanced Ternary.
In Balanced Ternary, truncation is the same as rounding:
Let us suppose that one calculates something involving circles. The 1 true CircleConstant is τ (Tau):
c / r = τ ≈ (Base10: 6.283185307179586) ≈ (BaseBalanced3: ⬛︎▲◯.⬛︎◯▲,▲◯▲,⬛︎⬛︎◯;◯▲⬛︎,⬛︎◯▲,▲◯▲;⬛︎▲▲,◯◯◯,◯◯⬛︎
⬛︎▲◯.⬛︎◯▲,▲◯▲,⬛︎⬛︎◯;◯▲⬛︎,⬛︎◯▲,▲◯▲;⬛︎▲▲,◯◯◯,◯◯⬛︎ is a bit much, so since truncation and rounding are the same, we can chop off extra digits like thus:
⬛︎▲◯.⬛︎◯▲,▲◯▲,⬛︎⬛︎◯
Post Scriptum:
¡I got to play with my dice for generating the digits! This reminds me about a joke:
A D&D-Player is in Geometry-Class. The Teacher asks the D&D-Player to name the 5 Platonic Solids. The D&D-Player responds thus:
> "The 5 Platonic Solids are the D04, the D06, the D08, the D12, and the D20.
——
The D&D-Player
...and then they eventually decided base 10 was pretty cool.
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If ponies would use Balanced Ternary, ¿why would they change to a base like Base10 in which the mathematics is much more complicated?:
The only advantage mathematically Base10 has is that it has more factors, so it is less likely that one has a nonterminating fractional representation.. Even there, Base10 is not very good:
One wants a number with generally many factors and specifically many prime factors. This points to highly composite numbers which are also the least common multiple of natural numbers upto some point. Here is a list of the 1st few highly composite numbers which are also the least common multples of consecutive natural numbers:
* 06
* 12
* 60
12 is almost the same size as 10, but a much better base. 60 is 6 times larger than 10, but is such a good base thatit is worth considering.
Ponies starting off with Base2, considering BaseBalanced3, and then choosing Base 10 makes do sense. Either the ponies would choose BaseBalanced3 for the mathematical advantages, or they would choose Base06, Base12, or Base60 because it requires fewer characters for writing numbers and these bases are both highly composite and the least common multiples of consecutive natural numbers. If I had to bet on their choice, assuming that they start off with binary, but decide that they need an higher base, I would bet that the ponies should choose Sexagesimal.
2831931
https://www.youtube.com/watch?v=SqhSaxKS9OQ
She is using base 10 there. So ponies use base 10. QED.
I do agree with you that it doesn't really make sense for them to use it though. I don't really know my way around the other bases, but it does make a great deal more sense to deal with base 6, 12, or 60 certainly. Much easier to be able to factor 3's out than with base 10.
What's up with the "¿" mark?
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Yes, Twilight Sparkle uses Base10, but that is not the point of this BlogPost:
¿Can I find a way for equine society to develop in such a way that the ponies can plausibly use Balanced Ternary? Assuming that they start with Base2, the answer is yes.
As for what we see on the show, if we assume that it is translated for us, then we cannot trust what we see and hear. This means that we should not take certain statements literally:
Princess Celestia banished NightMareMoon to the Moon for 1,000 years. Perhaps, it was Base10 2,187 which is 3^7. It also might explain how Ponyville can do WinterWrapUps for hundreds of years and have Granny Smith being 1 of the founders of Ponyville:
In Balanced Ternary, every number between Decimal 41 and Decimal 121 has 5 digits. Maybe, the literal meaning is that Ponyville requires 5 digits to represent its age and Granny Smith is 1 of its founders.
If we run with Decimal not making sense and assume that ponies really use another base, we can make some interesting FanFiction:
Imagine a mystery with numerical clues written in pony numerals (we can use UnicodeCharacters like I did) in a different base. Clever readers can solve the mystery before the big reveal by figuring out the numerals and the base and doing simple mathematics of the kind one can do on a keep 4-Function (Addition, Subtraction, Multiplication, and Division) Calculator.
Mathematics is easiest in Balanced Ternary, but if wants an higher base so that one can write big numbers using fewer characters, some bases are better than others. Let us start with representations of 1/n in various bases with n being a natural (counting) number and the base being a natural number too:
The representation will only terminate if n is a composite of the prime numbers which are factors of the base. As an example, in Decimal, 1/5 is .2. 1/2 is .5. 1/20 (20 is 5*2*2) is .05/ 1/3 is .33333333333333333333333 and an infinite number of more 3s. In Duodecimal (Base12) 1/3 is.4 because 3 is a prime factor in the base. For making terminating fractions, of needs the prime factors in the base.
1 way to get a base with lots of terminating factors is to use a primorial (a number which is the product of primes with each prime used only once) number consisting of consecutive primes such as 30 (2*3*5). Now, we must look at the total number of factors:
Let us suppose that we have as many things as the value of the base. ¿How many ways can we evenly divide the number? The more ways the better. We should choose an highly composite number:
An highly composite number is a number with more factors than numbers smaller than itself. As an example, 12 is highly composite because it is the 1st number with 6 factors (1, 2, 3, 4, 6, 12).
We want a base with consecutive prime numbers as factors in its base which is also highly composite, so that its has many total factors. 1 trick we can use is use a number which is both the least common multiple of consecutive natural numbers and highly composite:
The least common multiple is how one finds howmany packages of buns and hotdogs to buy:
Hotdogs com in 10-packs. Buns com in 8-packs. We factor the numbers:
10:
2*5
8:
2^3
5*2^3=40
If we buy 5 8packs of buns and 4 10-packs of hotdogs, we shall have 40 buns and 40 hotdogs.
Let us write the least common multiples of consecutive natural numbers ans see which are also highly composite:
1:
1
2:
2
3:
6
4:
12
5:
60
6:
60
7:
420
6, 12, and 60 are all highly composite. 420 has 24 factors, but the first number with 24 factors is 360, so 360 is highly composite, while 420 is not highly composite. 60 is particularly interesting:
The prime factors of 60 are 2, 3, 5, to in the case of 1/n if n is a composite of 2, 3, and 5, the representation will terminate in Sexagesimal (Base60). On can divide a group of 60 12 even ways (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60).
Assuming that the ponies start off with Binary (Base2) and deliberately move to a larger base so that they can write larger numbers with fewer digits, Sexagesimal (Base60) is by far the most appealing base.
Indeed, 6, 12, and 60 are better that way. If one uses Sexagesimal, of can factor 5 too as well as 2 and 3.
It is an “¿Inverted Question-Mark?”. Inverted “⸘Interrobangs‽” and “¡Exclamation-Points!” exist too.
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Lots and lots of info there. It is rather hard for me to actually understand the implications, but I think I see the general trend.
This is related to an idea I had of "lenses of reality" Where the actual events and things that happened are presented through the filter of the show. This is especially evident in something like Magical Mystery Cure where it feels like they cut out a whole bunch of information to make the True True Friend song work. I'd like to think if not for that song we'd have had another episode of content.
You've made it sound sensible enough. I'd probably be able to muster more enthusiasm if I actually used non-base 10 math at any point. Mathematics is rather dense reading though.
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Basically, the more factors, especially prime factors, the better. The rest is just a way of maximizing the factors generally and prime factors particularly.
Indeed. This “the show translated for us” opens so many possibilities for stories.
One just has to play with the bases. In fiction, the funnies stories come from deliberately making the way bases interrelate difficult:
In Harry Potter, the coinage is all in the ratio of large prime numbers, just for making the mathematics more difficult for humorous effect.
In Babylon 5, the Minbari use Undecimal (Base11), which means that 1/n only if n is a power of 11 does the fraction terminate. This is a joke because 10 is an okay base and 12 is a good base, but 11, which fall between 10 and 12, is a terrible base.