Let's assume 60 RPM, might be a bit fast but whatever, that's one rev per second that's Tau radians per second.
Why would you want any other multiple of the answer?
Tau and Pi. Yin and Yang. Go half way around your journey, that's Pi, go all the way round your journey and you are where you started from, that is Tau.:)
Two PI, or not two PI, that is the question:
Whether 'tis Nobler in the mind to suffer
The Slings and Arrows of multiplication by two,
Or to take Arms against a Sea of troubles,
And by opposing them: to use TAU, to multiply
No more; and by using TAU, to say we end
The Heart-ache, and the thousand Natural multiplications
That math is heir to? 'Tis a consummation
Devoutly to be wished.
Clearly a circle is the simplest most fundamental out of the bunch. You can make any of them into a circle by putting their foci at a coincident point. Therefore, whatever is most fundamental for the most fundamental shape is what we are looking for.
Sir Hein
Not fond of FB, but if there was a 'Like' button I would surely click it for your poem - or is it a sonette?
Heater.,
Thanks for the circle dissertation. One could argue that a bitmap of an 'analogue' geometry cannot be done with a bitmap, but that's not correct is it - since down at the bottom, reality is granular. The circle is in there anyway e.g. as a string of ASCII character codes that translates into the mathematical definition of Pi. Obviously I love infinite non-repeating numbers. All the aliens in 'Contact' needed to do was to run a powerful computer (forget Neumann) to crunch out the Pi until the magic sequence was found.
Diverging still further from the thread; a long time ago the number system and the measurement systems was defined different - to best serve the purpose of engineering and constructing challenging constructions at that time - like pyramids and cathedrals (Cubits etc.). Now we have moved on, and Dec, Hex, and Bin serves us better. Is there any similar change that supports moving over to Tau - or more interestingly: what system would serve us best as we move on into the future?
Well, we don't know that for certain yet.. it looks like it should be granular, what with the minimum-length Planck length.. and it has been theorized that at the bottom there's just "quantum foam". But that's a testable prediction, you can look at light coming from very (very!) far off, and high-energy photons should be affected by the supposed granularity. IIRC all tests to detect this, save one, have failed so far. So we need more evidence before we can assert that reality really is granular.
We might want to be careful when thinking about that.
Physicists have pulled a length value out of the air which they call the "Plank Length" which they state is the shortest measurable
length. It becomes impossible to determine the difference between two locations less than one Plank length apart.
That is not to say that space is granular in the way that pixels are arranged on video screens or in that bit map we are contemplating. Hence Physicists start talking about a "foamy" universe at that scale.
So let's say in this bit map we are pulling out of the binary expression of Pi that the distance between the bits represents less than one Plank length in our final image. That means that the jaggy lines made out of bits that we can find in search of a circle are
actually indistinguishable from the points of an actual circle in some deep physical sense.
That means that on a small enough scale the "circle" I presented above, made out of 25 bits, is actually indistinguishable from a true circle.
Bingo! We have found the circle bit map in Pi! Not only that we see that there are loads of them in the first page of digits we print out. They are everywhere. We have achieved the impossible already. The aliens in "Contact" were looking in the wrong place.
On the other hand. We might be looking for a circle on a more human scale, say one meter across. We have to scale the thing so that the "jaggies" in the points on the circumference are less than the Plank length. That means we need a bit map that is of the order of one over the Plank length in resolution. That is 10^35 by 10^34 or 10^70 bits.
Given a chunk of 10^70 bits of Pi it has 2^10^70 possible states so there is a 1 in 2^10^70 chance it might have a clean one meter circle in it.
Given that the number of atoms in the observable universe is only a paltry 10^80 or so it's unlikely we could build a computer to do this:)
Martin, I cannot die happy until I do that with a Scribbler, even at significantly reduced RPM. That is, in a word, bitchin'.
I would love to try that whirling dervish line follower as well. But buying a 3Pi is frankly too easy for my tastes. Most of the challenge is figuring out how to build a robot that can spin and not fly apart at the seams. You need a low mass robot with a low rotational inertia. So keeping the diameter small, the batteries towards the center, and using small Pololu style motors and wheels seems like a must.
Good point. In order to accomodate a transition to tau, we'd have to recall all the scientific calculators out there to retrofit a tau button. Must be hundreds of them, at least!
This kind of thing happens all the time with curve representations. The analytic approach requires we get to the fundementals on things. Circles really aren't ovals because they are defined as a set of points that lie at a distance equal to a radius. A circle is a sub-set of an ellipse. The latter can represent a circle as a special case.
We see this in CAD systems too, where basically all the entities are boiled down to "curves" Lines are curves, circles are curves, arcs are curves, splines are curves, the conics are curves, etc.... A line in the analytic sense is represented by either two end points, or one and a slope. In the software / machine / task sense, a line is simply a curve without enough definition to present any curvature. Two node points define a line, where three can represent everything up through the conics....
After reading this discussion and the papers Heater linked, I see this as more task oriented. I'm highly likely to use Tau for some things and maybe use Pi for some others depending on which representation makes the most sense and is the most lean. My math skills vary depending on interest and use. This discussion got me thinking about a lot of basic things and it's fun and worth while to do that. +1 for all involved.
The geometry purist in me won't accept a circle as being defined by it's diameter. The radius is the core thing that brings us a circle. And I'm inclined to treat the arc just like I do a line segment. Lines are infinite, and we use points to define a segment of interest. A circle is a closed, pure shape, and the arc is simply defined as a segment like lines are, leaving diameter entirely out of the thing, but for the few real world case where diameter makes a lot of sense. The majority of them involve measuring, calculating and thinking in terms of radii and that better aligns with Tau for a lot of the things I think about regularly. --which are more basic things, as I rarely need to go to advanced math cases, which involve a review, lots of work to grok, etc.... Unless I need to go there, I don't and Tau presents as easy for easy stuff. I like it!
In Greece, thousands of years ago, some clear-minded guys deducted by pure logic that down at the bottom reality is granular. They called the bits atoms, nowadays we call them Planks or whatever. They have to be one-dimensional - just points, only then can they be the final stop downwards. On the other hand, if something is truly solid, I can think of at least one problem: push at the one end of it and the other end of it must move at the same time - immediately. So, that information must travel through the stuff at an infinitely high speed. Sorry, Einstein.
So, to the poem competition:
Even if an issue is probably serious,
it still can be sometimes hilarious.
Even if humor often is funny,
and could make you look like a dummy,
it does' not imply you're injurious.
Well, the definition of a point is that is has no dimensions, so a point can't be one-dimensional. It's zero-dimensional, and it has no area, and thus can't be the source of granularity either. A line with zero width would be one-dimensional. (Planck length though is far from zero-dimensional, it's on the scale of 10^−33 cm).
And on the other hand the closer you look at anything the less solid it becomes, on the surface a table looks like a solid, hard thing, but what stops your hand when you bang it on the table is for the most part electrons refusing to push their charge against the charge of other electrons. Everything we think of as solids is practically empty in reality.
- Yes, sorry, I agree it has to be 0-dimensional - has to - otherwise it cannot be the final stop. The alternative is an infinite sequence of things-made-by-things. Neither the 'end stop' nor the 'eternal road' concepts make (human) sense, but I'd vote for 'end stop'. So, where along the road were the 3 other dimensions lost?
Come to think of it theres the uncertainty principle, which means it is not a point, only a probability distribution of a point.
Sorry, other Forum'ers, for off-topicing to the extreme.
Quote by Erlend
...On the other hand, if something is truly solid,...
M = E/C2 is false outside of the equation itself. The nucleus, in relative terms, is as far from its electrons as the sun is from the planets. Far from being solid I think we are more like 'cosmic Legos' that would resemble uncountable galaxies if our theoretical microscope could zoom in enough. An even weirder thought is that if you plot the ratio of time to the speed of light on a graph time would equal zero at the speed of light at least on the graph. Tau or Pi is rational in comparison.
Heater.
It is encouraging to know that she actually took the time to make that video.
lardom,
You all should agree with me that at the bottom there is bits (0-dimensional points), i.e life and everything is all software.
Are we still in beta?
Erlend, I'm not clear on what is meant by 'granularity'. I still think "Zoom in a bit more". I personally have to be careful not to wander too far off onto a metaphysical side-street. For that reason I force myself to 'preview and delete'.
I watched the video too and thought it would be interesting if she could join this discussion.
Points and edges do not exist in the real world. Well, let's just say the 3D one we live in. Where edges are concerned, it's always a radius. The discussion is how big of one. Points are constructs we use to identify specific places on topology always subject to measurement error.
Planes, axis, etc... are similar things.
Real world shapes are typically represented as bounded surfaces whose edges are shared by two and only two surfaces. Real world representations are actually quite expensive! We take a lot of short cuts using points, edges, planes, axis to render idealized things. The reality is much more complex. Always found that fascinating. Nothing is round, nothing is flat, etc... It's all about round enough, or flat enough, or we are stacking molecules.
I liked the video. There were some new ideas in there for me. Love that kind of stuff. You know, somewhere down deep, I see atoms as those quantized edges, and perhaps when we get much better at working with them, some of our core representations of things may well change. The woman might be a bit ahead of her time. In any case, presentations like that are so easy to understand and a lot of fun. Thanks for linking.
I don't get the impression that vihart is ahead of her time. She is a musician by training with an interest in maths.
But she has has done a wonderful job of introducing otherwise dry mathematical ideas in fantastically captivating way.
Which I guess is ahead of it's time in it's own way. Which one of us hasn't thought there must be a better way of teaching maths than the stereotypical boring classes we have all been victim of.
It sure was. I think Matt knocked it out of the park with his statement that a circle's diameter is the only direct measure possible. I use my calipers to measure diameters, but fiddling around with a center finder to find a radius is nuts.
Comments
Let's assume 60 RPM, might be a bit fast but whatever, that's one rev per second that's Tau radians per second.
Why would you want any other multiple of the answer?
Tau and Pi. Yin and Yang. Go half way around your journey, that's Pi, go all the way round your journey and you are where you started from, that is Tau.:)
Proof that tau gets you nowhere, and pi takes you as far as you can go!
You say taumato, I say pitato...
Whether 'tis Nobler in the mind to suffer
The Slings and Arrows of multiplication by two,
Or to take Arms against a Sea of troubles,
And by opposing them: to use TAU, to multiply
No more; and by using TAU, to say we end
The Heart-ache, and the thousand Natural multiplications
That math is heir to? 'Tis a consummation
Devoutly to be wished.
Circle is defined as all the points on a plane that lie at the same distance from some other point, the center.
Ellipse is the set of points on a plane for which the sum of distances to two other points, the foci, is a constant.
One can generalize this to the k-Ellipse which is the set of points on a plane for which the sum of distances to k other points is a constant.
A k-Elipse can be a very wonky looking curve. See here for example:
http://math.stackexchange.com/questions/124333/what-are-curves-generalized-ellipses-with-more-than-two-focal-points-called-an
Clearly a circle is the simplest most fundamental out of the bunch. You can make any of them into a circle by putting their foci at a coincident point. Therefore, whatever is most fundamental for the most fundamental shape is what we are looking for.
Thank you, I knew something brilliant would come out of all this meditation.:)
Not fond of FB, but if there was a 'Like' button I would surely click it for your poem - or is it a sonette?
Heater.,
Thanks for the circle dissertation. One could argue that a bitmap of an 'analogue' geometry cannot be done with a bitmap, but that's not correct is it - since down at the bottom, reality is granular. The circle is in there anyway e.g. as a string of ASCII character codes that translates into the mathematical definition of Pi. Obviously I love infinite non-repeating numbers. All the aliens in 'Contact' needed to do was to run a powerful computer (forget Neumann) to crunch out the Pi until the magic sequence was found.
Diverging still further from the thread; a long time ago the number system and the measurement systems was defined different - to best serve the purpose of engineering and constructing challenging constructions at that time - like pyramids and cathedrals (Cubits etc.). Now we have moved on, and Dec, Hex, and Bin serves us better. Is there any similar change that supports moving over to Tau - or more interestingly: what system would serve us best as we move on into the future?
Philosophical Erlend
-Tor
Physicists have pulled a length value out of the air which they call the "Plank Length" which they state is the shortest measurable
length. It becomes impossible to determine the difference between two locations less than one Plank length apart.
That is not to say that space is granular in the way that pixels are arranged on video screens or in that bit map we are contemplating. Hence Physicists start talking about a "foamy" universe at that scale.
So let's say in this bit map we are pulling out of the binary expression of Pi that the distance between the bits represents less than one Plank length in our final image. That means that the jaggy lines made out of bits that we can find in search of a circle are
actually indistinguishable from the points of an actual circle in some deep physical sense.
That means that on a small enough scale the "circle" I presented above, made out of 25 bits, is actually indistinguishable from a true circle.
Bingo! We have found the circle bit map in Pi! Not only that we see that there are loads of them in the first page of digits we print out. They are everywhere. We have achieved the impossible already. The aliens in "Contact" were looking in the wrong place.
On the other hand. We might be looking for a circle on a more human scale, say one meter across. We have to scale the thing so that the "jaggies" in the points on the circumference are less than the Plank length. That means we need a bit map that is of the order of one over the Plank length in resolution. That is 10^35 by 10^34 or 10^70 bits.
Given a chunk of 10^70 bits of Pi it has 2^10^70 possible states so there is a 1 in 2^10^70 chance it might have a clean one meter circle in it.
Given that the number of atoms in the observable universe is only a paltry 10^80 or so it's unlikely we could build a computer to do this:)
I would love to try that whirling dervish line follower as well. But buying a 3Pi is frankly too easy for my tastes. Most of the challenge is figuring out how to build a robot that can spin and not fly apart at the seams. You need a low mass robot with a low rotational inertia. So keeping the diameter small, the batteries towards the center, and using small Pololu style motors and wheels seems like a must.
Good point. In order to accomodate a transition to tau, we'd have to recall all the scientific calculators out there to retrofit a tau button. Must be hundreds of them, at least!
This kind of thing happens all the time with curve representations. The analytic approach requires we get to the fundementals on things. Circles really aren't ovals because they are defined as a set of points that lie at a distance equal to a radius. A circle is a sub-set of an ellipse. The latter can represent a circle as a special case.
We see this in CAD systems too, where basically all the entities are boiled down to "curves" Lines are curves, circles are curves, arcs are curves, splines are curves, the conics are curves, etc.... A line in the analytic sense is represented by either two end points, or one and a slope. In the software / machine / task sense, a line is simply a curve without enough definition to present any curvature. Two node points define a line, where three can represent everything up through the conics....
After reading this discussion and the papers Heater linked, I see this as more task oriented. I'm highly likely to use Tau for some things and maybe use Pi for some others depending on which representation makes the most sense and is the most lean. My math skills vary depending on interest and use. This discussion got me thinking about a lot of basic things and it's fun and worth while to do that. +1 for all involved.
The geometry purist in me won't accept a circle as being defined by it's diameter. The radius is the core thing that brings us a circle. And I'm inclined to treat the arc just like I do a line segment. Lines are infinite, and we use points to define a segment of interest. A circle is a closed, pure shape, and the arc is simply defined as a segment like lines are, leaving diameter entirely out of the thing, but for the few real world case where diameter makes a lot of sense. The majority of them involve measuring, calculating and thinking in terms of radii and that better aligns with Tau for a lot of the things I think about regularly. --which are more basic things, as I rarely need to go to advanced math cases, which involve a review, lots of work to grok, etc.... Unless I need to go there, I don't and Tau presents as easy for easy stuff. I like it!
We are defined by our ellipses:
Hmm. It's much less magical at low RPM, but it does work! (Slow upload in progress)
then
Tau B or not Tau B ... that is the question.
Tau for now, I say: Tau Tau for now.
So, to the poem competition:
Even if an issue is probably serious,
it still can be sometimes hilarious.
Even if humor often is funny,
and could make you look like a dummy,
it does' not imply you're injurious.
And on the other hand the closer you look at anything the less solid it becomes, on the surface a table looks like a solid, hard thing, but what stops your hand when you bang it on the table is for the most part electrons refusing to push their charge against the charge of other electrons. Everything we think of as solids is practically empty in reality.
-Tor
Come to think of it theres the uncertainty principle, which means it is not a point, only a probability distribution of a point.
Sorry, other Forum'ers, for off-topicing to the extreme.
http://www.youtube.com/watch?v=D2xYjiL8yyE
It is encouraging to know that she actually took the time to make that video.
lardom,
You all should agree with me that at the bottom there is bits (0-dimensional points), i.e life and everything is all software.
Are we still in beta?
I watched the video too and thought it would be interesting if she could join this discussion.
Planes, axis, etc... are similar things.
Real world shapes are typically represented as bounded surfaces whose edges are shared by two and only two surfaces. Real world representations are actually quite expensive! We take a lot of short cuts using points, edges, planes, axis to render idealized things. The reality is much more complex. Always found that fascinating. Nothing is round, nothing is flat, etc... It's all about round enough, or flat enough, or we are stacking molecules.
I liked the video. There were some new ideas in there for me. Love that kind of stuff. You know, somewhere down deep, I see atoms as those quantized edges, and perhaps when we get much better at working with them, some of our core representations of things may well change. The woman might be a bit ahead of her time. In any case, presentations like that are so easy to understand and a lot of fun. Thanks for linking.
But she has has done a wonderful job of introducing otherwise dry mathematical ideas in fantastically captivating way.
Which I guess is ahead of it's time in it's own way. Which one of us hasn't thought there must be a better way of teaching maths than the stereotypical boring classes we have all been victim of.
Tau vs Pi Smackdown
It sure sounded familiar after reading this thread.