The Pi Manifesto • Hacker News Board

Re: The Pi Manifesto

#81

Post by sesqu » Wed, Jul 06, 2011, 5:01 AM UTC

amalcon wrote: post #59

sesqu wrote: post #46

Earlier quoted context omitted.

The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot,…

I've read this a few times, and I don't think I really understand. Perhaps you could explain further? 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4…

Okay, I'll try to grasp at my thoughts again.

Algebraically, 12 is not the same as 10+2. 12 is an element of, say, ℤ, while 10+2 is one of . To make them interchangeable, we need to establish an equivalence relation. Given that relation, we then have the opportunity to express useful, non-obvious equivalences using transitivity.

Now that we have X=10+2=12, we need to choose which one represent the equivalence class of X. 12 is certainly shorter, but a seemingly magic constant. 10+2 implies that in other number systems, X=b+1+1 may be also true. If the scribe subscribes to the principle of MDL, we can speculate that this is the reason he chose the longer version, and if that is accurate, we have gained more information. If we chose 9+3, we would arguably lose information, since this expression is (hypothetically) misleading.

This is all to say that expressions are more informative than their equivalence classes, since they have been hand-picked to be representative.

To represent the equivalence class of 6.28... with 2*3.14... implies that the equivalence class of 3.14... is more important, and that the prototype likely involves two separate instances of the concept of π. This is misleading.

Re: The Pi Manifesto

#82

Post by roundsquare » Thu, Jul 07, 2011, 6:25 AM UTC

ianterrell wrote: post #77

MostAwesomeDude wrote: post #72

Earlier quoted context omitted.

Your assumptions are invalid. The radius is not the smallest amount of information which determines a circle. The radius and origin can determine a circle. So can the diameter and origin, or the circumference and origin, or the area and origin. Don't forget your geometry.

I said smallest amount of information. The diameter, circumference, and area are all functions of the radius. Sure, you can write any in terms of any of the others, but the radius is the smallest: both in terms of absolute value as well as dimensionality.

Why does smallest in terms of absolute value matter? In any event, I can define the circle via the center and r/2 or center and r/4, etc... I don't see any value in caring about the absolute value.

As for dimensionality, how are you using the word? No matter what, we need 3 numbers to define a circle in R^2. Two numbers define its position and one number defines it size. I don't see what you're getting at. If you mean that because radius is a measure of length (one dimension) and area is a measure of area (two dimensions) then I have two questions: 1) Why does that matter? Its still just a single number. 2) Even if it does matter, why is radius more fundamental than diameter.

Re: The Pi Manifesto

#83

Post by roundsquare » Thu, Jul 07, 2011, 6:31 AM UTC

amalcon wrote: post #25

natural219 wrote: post #18

I just want to ask all of the Pi/apathetic people-- how long did it take you to understand radians? For me, it was a week before I was comfortable naming any angle in radians in a reasonable amount of time (this is after a week of drilling). This is just my point of view, but calculating radians was a significant roadblock into making quick trigonometric calculations. In fact, I'd have to say it was the biggest roadb…

It only took me about a day, but I suspect that's because I was already used to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me about a week. To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or th…

"the problem is with degrees"

I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier.

So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10.

I think the point is to be able to teach geometry to kids without worrying about them getting confused by fractions and/or irrational numbers.

Re: The Pi Manifesto

#84

Post by amalcon » Fri, Jul 08, 2011, 2:27 PM UTC

roundsquare wrote: post #83

amalcon wrote: post #25

Earlier quoted context omitted.

It only took me about a day, but I suspect that's because I was already used to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me about a week. To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or th…

"the problem is with degrees" I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier. So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10. I think the point is to be able to teach geometry to kids without worrying about them getting co…

So just use cycles instead of degrees. Simple fractions or a circle. Fractions are already taught to young children as "how many pieces a circle[1] can be divided into." It would seem to kill two birds with one stone.

I think most kids who even study pre-Calc could handle applying a conversion of "2pi" from there.

[1] Where circle="cake","pie","pizza",etc.

Re: The Pi Manifesto

#85

Post by roundsquare » Fri, Jul 08, 2011, 2:55 PM UTC

amalcon wrote: post #84

roundsquare wrote: post #83

Earlier quoted context omitted.

"the problem is with degrees" I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier. So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10. I think the point is to be able to teach geometry to kids without worrying about them getting co…

So just use cycles instead of degrees. Simple fractions or a circle. Fractions are already taught to young children as "how many pieces a circle[1] can be divided into." It would seem to kill two birds with one stone. I think most kids who even study pre-Calc could handle applying a conversion of "2pi" from there. [1] Where circle="cake","pie","pizza",etc.

Good point. I forgot we learned about "slices of a pizza" that early.

By the way, the history of the degree is somewhat interesting:

http://en.wikipedia.org/wiki/Degree_(angle)