Con-mathematical systems

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Radius Solis
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Re: Con-mathematical systems

Post by Radius Solis »

Salmoneus wrote:The area of a circle might be a basic unit, at least for mathematicians
This, THIS! I just got a result that rather shocked me. I wanted to understand the relationship between areas of circles of various bases... which of course is equivalent to what we call circumference, so I set about investigating that with a system of equations, the old familiar circle formulas C = 2pi*r and A = pi*r2. The first of those can be restated as r = C/2pi, which then plugs into the other: A = pi(C/2pi)(C/2pi), and the pi on top cancels out one of the lower ones, leaving A = C2 / 4pi.

Then, defining a circle of base=1 to be the basic unit of area - let's name this unit to help us keep track, we can call it the "smurf" (sorry, running out of smart ideas here) - I want to know how many smurfs there are in a circle of base=2. I got slightly lost at this point and pharazon rescued me, but here it is: the relationship of Area2 to Area1 is the ratio A2 / A1, which means (C22 / 4pi) / (C12 / 4pi), the 4pi terms cancel and we get C22 / C12.

Still with me? A2 / A1 = C22 / C12. Now C1 and A1 were set equal to one as the circle we started from, and 1 squared equals 1, so the denominators go away too and we can rewrite the entire thing as A = C2. That means that, even within this alien circular cyclical modular math, area is still the "square" of distance. So if we have a circle where the circumference is 10, its area is 100 smurfs. And we arrived at that relationship without ever having to think about any four-sided figures, and it doesn't even have to involve pi! :o That Area = Circumference2 is just as true in circles as Area = Length2 is in rectangles is surely something plenty of mathematicians are already aware of, but I wasn't.

Aside from surprising me, why is it important? First - because it could be this area relationship rather than that of rectangles that is first hit on by our hypothetical con-people, so it shows that they can arrive at an understanding of the squaring relationship by a totally different path, even if I don't yet know what the details should look like. (Whats above is proof of its possibility, not the path itself.) And second - because it suggests that quite a lot else might be gotten to from the cycle system. And third, though I have no interest in putting in the work to verify it, it also suggests that sphere volume varies simply by C3 and so on up the dimensions, so our conpeople can probably have a fully fleshed system for dimensions and exponentiation based on this too. And lastly, it implies that pi only enters the picture as a conversion factor between line-based and circle-based math (which is apparently true according to pharazon), so our conpeople don't actually need it at all just because they're counting around circles instead of in lines, they'll only need it for converting, just like we do.

So thank you, thank you, for providing the big hint I needed for how to approach area in this cyclic math.

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Re: Con-mathematical systems

Post by alice »

pharazon wrote:Nancy Blackett: Here's a list of all the English words appearing in the first 100000 base-36 digits of pi in base 36,

(snip)

(why yes, this WAS a colossal waste of time)


Well, 100000 is pretty insignificant compared to infinity, so it's not surprising you didn't find very much. You need a bit more patience :-)
Zompist's Markov generator wrote:it was labelled" orange marmalade," but that is unutterably hideous.

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Re: Con-mathematical systems

Post by linguofreak »

Radius Solis wrote:That Area = Circumference2 is just as true in circles as Area = Length2 is in rectangles is surely something plenty of mathematicians are already aware of, but I wasn't.


Yup. It works for *any* two dimensional figure, however crazy its shape, as long as the shape remains the same as you scale it up or down. If you just scale along one dimension (turning your square into a rectangle or your circle into an oval, for example), this relationship won't hold, but otherwise, it holds for *any* two dimensional shape.

And for n dimensions, your area/volume/whatever is equal to length/circumference/whatevern, however crazy the shape, as long as all dimensions are scaled by the same amount.

Everything else in the formula for the area/volume/whatever of a shape is just a scaling factor determined by what units you choose for length and area/volume/whatever.

There are even other ways that you could use a square as a base shape, that would need a conversion factor to be converted to our system: For instance, if you define one unit of area to be the area of a square with a *diagonal* of 1 unit of length, your unit of area ends up being half the size it is in our system. (Exercise for the reader: How did I arrive at this answer?). Converting between the two systems, you'd have a conversion constant of one half (going between the "diagonal system" and our own) or two (going between our own system and the "diagonal system").

In fact, given a two dimensional shape, however complex, as long as you scale it evenly in both dimensions, you can use the area of the shape as your unit of area, and *any* line (or part of a line) in that shape, curved or straight (length of a side, perimeter, radius, circumference, etc.), as your unit of length, and you will always get the result A = L2, and all you will change is what conversion constant you need for different shapes.

This property also generalizes to any number of dimensions. (The volume of a 3 dimensional shape, if you scale it evenly in all dimensions, can be calculated from the cube of the length of any line in that shape).

If every human were shaped *exactly* the same way, and differed only in size, then you could define your unit of volume to be "the volume of a person with a 1-inch-long pinkie finger", and the equation V = L3 would hold.

Finding the volume of a cube would require a rather obnoxious conversion constant, though...

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Re: Con-mathematical systems

Post by Salmoneus »

Radius Solis wrote:
Salmoneus wrote:The area of a circle might be a basic unit, at least for mathematicians
This, THIS! I just got a result that rather shocked me. I wanted to understand the relationship between areas of circles of various bases... which of course is equivalent to what we call circumference, so I set about investigating that with a system of equations, the old familiar circle formulas C = 2pi*r and A = pi*r2. The first of those can be restated as r = C/2pi, which then plugs into the other: A = pi(C/2pi)(C/2pi), and the pi on top cancels out one of the lower ones, leaving A = C2 / 4pi.

Then, defining a circle of base=1 to be the basic unit of area - let's name this unit to help us keep track, we can call it the "smurf" (sorry, running out of smart ideas here) - I want to know how many smurfs there are in a circle of base=2. I got slightly lost at this point and pharazon rescued me, but here it is: the relationship of Area2 to Area1 is the ratio A2 / A1, which means (C22 / 4pi) / (C12 / 4pi), the 4pi terms cancel and we get C22 / C12.

Still with me? A2 / A1 = C22 / C12. Now C1 and A1 were set equal to one as the circle we started from, and 1 squared equals 1, so the denominators go away too and we can rewrite the entire thing as A = C2. That means that, even within this alien circular cyclical modular math, area is still the "square" of distance. So if we have a circle where the circumference is 10, its area is 100 smurfs. And we arrived at that relationship without ever having to think about any four-sided figures, and it doesn't even have to involve pi! :o That Area = Circumference2 is just as true in circles as Area = Length2 is in rectangles is surely something plenty of mathematicians are already aware of, but I wasn't.

Aside from surprising me, why is it important? First - because it could be this area relationship rather than that of rectangles that is first hit on by our hypothetical con-people, so it shows that they can arrive at an understanding of the squaring relationship by a totally different path, even if I don't yet know what the details should look like. (Whats above is proof of its possibility, not the path itself.) And second - because it suggests that quite a lot else might be gotten to from the cycle system. And third, though I have no interest in putting in the work to verify it, it also suggests that sphere volume varies simply by C3 and so on up the dimensions, so our conpeople can probably have a fully fleshed system for dimensions and exponentiation based on this too. And lastly, it implies that pi only enters the picture as a conversion factor between line-based and circle-based math (which is apparently true according to pharazon), so our conpeople don't actually need it at all just because they're counting around circles instead of in lines, they'll only need it for converting, just like we do.

So thank you, thank you, for providing the big hint I needed for how to approach area in this cyclic math.


Just to illustrate the reason why I suggested this in the first place, here's how to describe a square with sides of length L.

If the sides are L, the diagonal is root-2 times L. So the area of the circumcircle is pi*(L(2^-2))^2 = 2pi(L^2). Compared to the count-circle's area of UB^2, where U=the length of one number, and B=the number of numbers in a circle. One way to get these to match is to let U=L, in which case B^2=2pi. Or, we could let B=L, in which case U^2=2pi.

So, three ways of description spring to mind. Let's say that L=4. Then a square could be:
- a polygon with vertices at 1,2,3,4 in a base-4 circle at a 'scale' of 1/(2pi^-2)
- a polygon with vertices at (1/4)X, (1/2)X, (3/4)X and X, where X = 1/(2pi^-2), in a base-1/(2pi^-2) circle
- a polygon with vertices at 1,2,3,4 and a height of 1/(2pi^-2) in a base-4 helix.

Don't know if that helps, though. And it could well be wrong.
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Re: Con-mathematical systems

Post by finlay »

pharazon wrote:
finlay wrote:
finlay wrote:This is the maximum expansion that wolfram alpha is capable of:
3.53i5ab8p5fsa[...]swg6buyw[...]rs1xclck34ji4[...]m197rgfyg09uqi3v1pa..._36

Zompist, you haven't been smuggling advertising into the fundamentals of mathematics, have you?? :o


No, it is his nemesis, although this only becomes apparent after looking at more digits:

3.53i5ab8p5fsa...swg6buyw...rs1xclck34ji4...qi16not

What are you using to find the extra digits, by the way?

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Re: Con-mathematical systems

Post by finlay »

Radius Solis wrote:stuff

Kinda like if you re-write it with proportions, I think, right?

so you have like A C² rather than necessarily A = C², and then it's all to do with units...

When I was making a spreadsheet that would calculate stuff about my con-planets I used cheats like this a lot, because you can define something as a relationship between the size or area of the other planet and the size of Earth, and then multiply by a conversion factor to get it in a certain unit.

This made a lot of the calculations easier because to calculate area or volume, instead of working out how to do it exactly from a value in metres, you take the value in metres, divide by Earth's radius so Earth is 1 and everything else is proportional to it, and then just square for area and cube for volume, because A ∝ r² and V ∝ r³. This is equivalent to A ∝ C² because C ∝ r. Obviously, you have to know what Earth's area and volume are so that you can then calculate the value of those in km² or whatever, and you could probably calculate those with pi. You're basically doing the same calculation but with more steps and less pi.

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Re: Con-mathematical systems

Post by pharazon »

finlay wrote:
pharazon wrote:
finlay wrote:
finlay wrote:This is the maximum expansion that wolfram alpha is capable of:
3.53i5ab8p5fsa[...]swg6buyw[...]rs1xclck34ji4[...]m197rgfyg09uqi3v1pa..._36

Zompist, you haven't been smuggling advertising into the fundamentals of mathematics, have you?? :o


No, it is his nemesis, although this only becomes apparent after looking at more digits:

3.53i5ab8p5fsa...swg6buyw...rs1xclck34ji4...qi16not

What are you using to find the extra digits, by the way?


Maple. Annoyingly, it will only convert integers between bases, so to get 100000 base-36 digits I had to tell it to compute 36^100000 * Pi, round that off, and then convert.

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Math in Conworlds

Post by SHiNKiROU »

How do the people in your conworld perform mathematical operations (+, -, *, /, square root) on paper? Do they use a different layout for long divisions, or do they even completely new methods.
What estimation methods are there? What is the value of pi? (some use 6.282... )

see also: http://en.wikipedia.org/wiki/Long_divis ... _countries
http://en.wikipedia.org/wiki/Fourier_division

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Re: Math in Conworlds

Post by con quesa »

Kingdom of Magic-Era Saimi use polish notation for basic arithmatic (instead of, say, 4+5 you'd write + 4 5, essentially). This comes naturally from the Saimiar language, where you'd say i trêsec xoike soi xil, "IMP add four and five" or i karøpec xil ŋês, IMP-empty-CAUS five one-PREP, "lessen five by one". The symbols for numbers are logographs borrowed from the Elésu writing system, which is also the source of the (alphabetic) Saimi writing system. The operators are the stylized Saimi letters t, k, e, and m, for trêsec, "add", karøpec, "lessen", ebesfi, "increase" (= multiply) and mybesfi, "unincrease" (= divide), respectively.

Polish notation (or for the Saimi, the natural way you'd go about writing math on a page) has the nice advantage that you don't need grouping operators, and it's very natural to think of numbers as arguments to mathematical functions, which is something the Saimi do have a sense of by the Kingdom of Magic era. The most common word for a mathematical function is perencir, literally "relationship", but the symbol for one on a page borrows the Elésu logograph for the Elésu word for "relationship" (borrowing Elésu logographs for mathematical or scientific use is a common theme in KoM Saimi scholarship). Arguments are separated by a stylied letter s, for soi, "and", just like you'd use a comma in standard western notation, but there is no notion of parentheses in the Saimi notation.

The guy who formalized the calculus (a mage by the name of Kêxuźi Śilu Dhenølidhat) made up symbols out of whole cloth for differentiation and integration, that looked sort of like an Elésu logographs but were not actually on any official list of them.
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