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That animation looks great, but it's misleading: the fundamental measurement of the circle is the radius, not the diameter.
Measured with a unit radius, we'd see 2*pi for a full "turn" of the wheel. Twice pi, or tau, is the magic number.
See http://tauday.com/ for details.
Pi is the wrong constant. That animation looks great, but it's misleading: the fundamental measurement of the circle is the radius, not the diameter. Measured with a unit radius, we'd see 2*pi for a full "turn" of the wheel. Twice pi, or tau, is the magic number. See http://tauday.com/ for details.
BTW, I don't mean to take anything away from your comment; that link was awesome.
Next question: Is it possible for a piece of string to be pi units long? How do transcendental numbers translate to the real world?
Just as difficult a question: Is it possible for piece of string to be 2 units long? In the real world, you always run into limits on how finely you can measure things before you run into quandaries about number theory.
Pi is the wrong constant. That animation looks great, but it's misleading: the fundamental measurement of the circle is the radius, not the diameter. Measured with a unit radius, we'd see 2*pi for a full "turn" of the wheel. Twice pi, or tau, is the magic number. See http://tauday.com/ for details.
Seems a bit pedantic. The post was about illustrating pi after all. I do enjoy the argument for tau though.
Pi is the wrong constant. That animation looks great, but it's misleading: the fundamental measurement of the circle is the radius, not the diameter. Measured with a unit radius, we'd see 2*pi for a full "turn" of the wheel. Twice pi, or tau, is the magic number. See http://tauday.com/ for details.
Random speculation: Maybe pi was chosen because the symbol for tau (τ) is hard to distinguish from a ‘T’ when hand-written? BTW, I don't mean to take anything away from your comment; that link was awesome.
Next question: Is it possible for a piece of string to be pi units long? How do transcendental numbers translate to the real world?
Sure. Take any circle made with a piece of string and declare it to be your reference unit circle. Blammo! Its diameter is 1 and its length is now pi ;).
Next question: Is it possible for a piece of string to be pi units long? How do transcendental numbers translate to the real world?
Given that any error bar contains some rational numbers, the case can be made that rationals are all you'll ever need for measurements in the real world. Nothing can ever be measured so precisely that it has a definitively irrational length.
However, irrational numbers do serve as a useful abstraction when dealing with the real world, since one is often working in different scales. A rational approximation of pi that works in one context -- say, construction paper projects -- might not be good enough in another -- say, orbital mechanics. Identifying the abstract, transcendental value has the practical application that you can use a rational approximation appropriate to the scale.
Earlier quoted context omitted.
Just as difficult a question: Is it possible for piece of string to be 2 units long? In the real world, you always run into limits on how finely you can measure things before you run into quandaries about number theory.
I think something could be two units long. Two is discrete, and at a microscopic level, things can be exactly discrete values (i.e. two angstroms...). I think asking if something could be exactly pi long is a different question.
Earlier quoted context omitted.
Just as difficult a question: Is it possible for piece of string to be 2 units long? In the real world, you always run into limits on how finely you can measure things before you run into quandaries about number theory.
I think something could be two units long. Two is discrete, and at a microscopic level, things can be exactly discrete values (i.e. two angstroms...). I think asking if something could be exactly pi long is a different question.
Now presumably 1 meter will be equivalent to some integer multiple of base units, though that may turn out not to be the case. Perhaps the current definition of a meter in terms of certain wavelengths of light will turn out to have a remainder of 1/7 of a base unit. ;) In that case you'd just define a slightly adjusted meter as a whole number of base units. So for simplicity let's assume that a meter is a whole number of base units.
Then it's clear that a string of any integer meter length would be possible. A string of length 1/3 meters might be possible, assuming that the number of base units in a meter was divisible by 3. But a string of length pi meters would be impossible.
As a computer programmer I tend to think of transcendental numbers in terms of processes which relentlessly converge, and that of course is the theory of limits. But I also recognize the financial constraints on running processes. So there will always be a "good enough" aspect to any physical measurement.
Next question: Is it possible for a piece of string to be pi units long? How do transcendental numbers translate to the real world?
Sure. Take any circle made with a piece of string and declare it to be your reference unit circle. Blammo! Its diameter is 1 and its length is now pi ;).
When things interact they also do it by probabilities. The lower the probability, the longer it takes to interact. (i.e. bring two deuterium nuclei near each other - will they fuse? Well, it depends on how close they are, the closer they are the less time it takes, since there is a greater chance that they are in the same spot at the same time.)
So in some sense, that's what time is.
