I've never understood peoples' fear of -1 and imaginary numbers. They aren't fake, they just aren't part of the Real Number Set.
I think it's unfortunate that they are called "imaginary". After all, all numbers are imaginary. Then putting them together with regular numbers and call them "complex" numbers sends shivers down the spine. Nobody likes complex things.
Personally I think the Real numbers are misnamed as well. That includes root 2 and PI and all that. Which you can't even write down. You can't even count them all! You can't even count how many you have between 0 and 1. How imaginary can a thing be?
What do you mean "you can't even count how many you have between 0 and 1" ?
I have 1 watermelon. Now I can count from 0 watermelons to 1 watermelon. Next I cut the watermelon into 10 equal sized pieces. Now I can count from 0 to 1 in tenths !!!
Thing is we have lots of different types of numbers. They have technical definitions and names:
"Counting" or "Natural" numbers: 1, 2, 3, 4, ...
"Whole" numbers: 0, 1, 2, 3, 4, ...
Note the inclusion of that troublesome zero.
"Integer" numbers: ...,–4, –3, –2, –1, 0, 1, 2, 3, 4, ...
Now we are getting weird, how can I have a negative number of fingers?
"Rational" or "Fractional" numbers: Can be written as p/q where both p and q are integers.
"Irrational" numbers: Cannot be written as integers p/q. Typical examples are PI and root 2.
"Real" numbers: All of the above. As we imagine as a continuum of points along the number line.
Given all that, when you cut your 1 water melon into into 10 equal parts and count them you are counting how many of some rational number (p/q = 1/10) you have between 0 and 1. You have missed out all the Real numbers between those points.
You can hit more points on the continuum between 0 and 1 buy cutting your tenths into tenths. Then you hit a 100 points between 0 and 1. Cut your 100ths into tenths, then you hit a thousand points between 0 and 1. And so on recursively, forever. You end up hitting an infinite number of points between 0 and 1.
But wait, you now have an infinite number of points between 0 and 1. But I wager you have missed all the points (Real numbers) that your would have hit if you started dividing by 3 instead of 10. 1/3, 1/9, etc.
By diving by 10 you have ended up with an infinite number of points but you have an infinite number of gaps between them!
OK, you might say, I'll hit all the points by dividing by 2, then those I get by dividing by 3, then 4, then 5 .... then I will have counted them all and have no gaps.
Well no. Even that is not good enough. If you do that you will miss all the points/Real numbers you could have hit by dividing by PI or root 2 or e or any other Irrational number.
Conclusion, there are an infinite number of points/Real numbers on the number line between 0 and 1. Or indeed between any other two points. Uncountable. Mostly imaginary.
It's disturbing enough that there are an infinite number of Irrational numbers between any two Rational numbers. No matter how close together those Rationals are!
All of which is why floating point is so troublesome on computers.
I've never understood peoples' fear of -1 and imaginary numbers. They aren't fake, they just aren't part of the Real Number Set.
I think it's unfortunate that they are called "imaginary". After all, all numbers are imaginary. Then putting them together with regular numbers and call them "complex" numbers sends shivers down the spine. Nobody likes complex things.
Personally I think the Real numbers are misnamed as well. That includes root 2 and PI and all that. Which you can't even write down. You can't even count them all! You can't even count how many you have between 0 and 1. How imaginary can a thing be?
I wonder what would happen if we removed irrationals from the Reals. Alternately pi could be expressed as 2^3x392699x10^-6 and be 99.9999% accurate. I guess it's all a matter of required precision.
Along the same point how real is it that an infinite series of 9s beyond a decimal is exactly equal to 1?
I wonder what would happen if we removed irrationals from the Reals.
Nooo... if you remove all the irrational points from the real number line the universe collapses. You can no longer have squares because you can't put opposite vertices a distance of root 2 apart. You can't have circles because the circumference cannot wrap around once without a gap between the ends. You just disconnected all the points of the fabric of space from each other with an infinite number of gaps and all the remaining remaining points fall apart!
As it happens I do have an answer to your question. It's explained by Prof Norman Wildberger here: "Inconvenient truths about sqrt(2)"
Which explains why root 2 upset the Greeks, who loved rational numbers, and includes Euclid's classic proof of why root 2 is not a Rational.
For those curious about numbers, types of numbers and where they come from and why the concept of Real numbers is broken Norman has a foundation series:
Alternately pi could be expressed as 2^3x392699x10^-6 and be 99.9999% accurate.
Why would one do that? That is the same as writing:
8 * 0.392699
Which is PI with an error of 0.00002%
Might as well remember 355/113 which is PI with an error of 0.000008% and only requires remembering 6 digits and a "/"
Along the same point how real is it that an infinite series of 9s beyond a decimal is exactly equal to 1
Hmm...let's see:
Let x = 0.99999....
Then multiply both sides by 10:
10 * x = 10 * 0.99999...
So:
10 * x = 9.99999...
Which we can write as:
10 * x = 9 + 0.99999...
Notice that part on the right of the + is the same as the x we started with above. Substituting we have:
10 * x = 9 + x
Solve for x:
10 * x - x = 9
9 * x = 9
x = 9 / 9
x = 1
We started with x = 0.99999... above so if follows that:
0.99999 = 1
QED.
Seems pretty real to me.
If we don't accept that result then we must be claiming there is a flaw in our rules of arithmetic an algebra. But where?
Show me a circle, and I will point out that under proper magnification, there are straight lines connecting all of the points.
I will admit that although circles do not occur in nature, circular objects certainly do.
Certainly, an object which tends toward a circle will have a Pi that tends toward the computed value.
Mathematics should reflect what is known of the natural world. I have no problem with "Pi"... as long as we include it within the proper punctuation.
Show me a circle, and I will point out that under proper magnification, there are straight lines connecting all of the points.
This does not totally convince me of anything.
If we were looking through a microscope at the straight lines of such a regular polygon, then I could suggest that there is a circle that is just big enough to contain all the vertices of that polygon. That we can't draw it or see it because your microscope is not good enough.
You get could get a better microscope and do the same again. I could then argue the same way again....
At some point the sides of that polygon will get down to such a small scale that your microscope will need so much energy to resolve them that your microscope and you will form a black hole! Thanks quantum mechanics.
I will admit that although circles do not occur in nature, circular objects certainly do.
I have been thinking about this all day and I'm still not sure what you mean by it.
Mathematics should reflect what is known of the natural world.
On the contrary. Or at least maths should not be limited by what is known of the natural world.
Historically mathematicians have explored all kinds of maths that has no analogy in known natural world. Should be glad they did because it happens that many, many years later that maths turns out to be very useful in describing new things we find out about the world. Think non-euclidian geometry and imaginary numbers as examples.
I have no problem with "Pi"... as long as we include it within the proper punctuation.
...Until someone fixes it, it is nonsense.
As far as I know it is fixed. That is why mathematicians have the idea of limits.
Anyway, why worry about Pi? If your view is that circles are made of little straight edges then we can simplify the problem. With that view even a squares are not possible. A square of side 1 should have a diagonal of root 2. But that is an irrational number. So if you try tho make a square you find, under your microscope, that the diagonal can never be the right length to fit where it should!
I wrote the solution to Phil's problem as,
r/R = sin(pi/2n) / (1 - sin(pi/2n)
That is just to emphasize that the the ratio of the radii is the important thing.
The case of n=6 is quite remarkable in that the ratio r/R is an integer. I wonder if it can be proven that is a unique, or even that it is the only solution in rational numbers? Indeed, it would seem that the ratios are more likely to be transcendental numbers. Cases like sqr(2) are algebraic numbers, that is, numbers that are the solution of some polynomial of any degree. Take away all the algebraic numbers from the real number line (or from the complex plane), and the vast majority of numbers are still there, numbers like pi and e that are not solutions of any polynomial. (The algebraic numbers are said to have Lebesgue measure of zero.) Most numbers are transcendental.
Ok... forget the logic. If we just ban circles, we get absolute values for Pi, based upon irregular polygons, THEN we don't have this ridiculous pseudo number to shove down the throats of innocent children.
Pi is worse than mathematical malpractice... it is child abuse.
... I wouldn't take it as a grain of salt... we are talking about children... some of whom may be here illegally... but what does that have to do with mathematics?
Once we get rid of the circle, we can get rid of "=",
I was sitting there in first grade as the teach tried to explain "="
Not a all convincing: " 1+3 is exactly the same as 4."
On the left you have an operation... on the right you have a result. Not at all the same.
I just stopped listening, because I knew she was wrong... why listen?
IF you can't explain it properly to a first grader... you shouldn't be trying.
Or better yet... why formulate mathematics that a first grader disagrees with?
Absolutely true but sounds like a complete lie... don't remember exactly how I phrased it to myself,
but something along the lines "no wonder this place is such a mess."
Be careful of your first graders... they are judging you!
It would have to be a perfect rock in a perfect pond... but neither actually exists.
It all has something to do with the processes of creation... for every ying there isn't actually a yang. It just
looks that way. It turns out that in order to have time... certain asymmetries must be enforced. That yang is missing
something.
But why should there be a creation at all?...Apparently, that has something to do with a conversation (or heated argument)... all bending upon
the meaning of a single word... go figure.
The way I see it is that "=" or at least the concept of equal is the one thing that all kids know instictively. It seems to be hard wired into our minds. Even if we don't have a concept of numbers and such.
This is easily demonstrated. Get a bunch of kids to each do some chore, promising them some candy as a reward. When they have done their chores give half the kids 4 candies and the other half only 2. Be sure they all know who gets what. Soon you will have a riot on your hands as the kids that only got 2 start complaining that it's not fair.
Surely you have heard young kids crying and whining "But it's not fair!...wah, wah,..."
They soon understand the meaning of the "=" sign and the rest follows, number symbols, operators...
Maths is based on fairness and at that level very much linked to our concepts of justice.
Recently I was reading a paper about how chimp's understand this concept of fairnes as well. In the case of chimps they were rewarded with nice bannans or crappy carrots. When they find out other chimps are getting a better reward for the same thing they start throwing a hissy fit and thowing the carrots back at you!
What does an infinite set of infinitesimal quantities amount to?
Anything you want it to. Depends how you approach it (pun intended). No, really.
If one can argue that there is a 1:1 ratio of even to whole numbers in the set of reals (excepting 0). Then one could argue that the spaces between rational numbers only matter if you don't have accurate enough rationals and that as you increase the accuracy of your rationals you can rationalize more decimal values, at least one more each time. If that's the case then as precision approaches infinity irrationals must approach zero.
The erroneous supposition is that the set of rationals is countable.
If one can argue that there is a 1:1 ratio of even to whole numbers in the set of reals (excepting 0). Then one could argue that the spaces between rational numbers only matter if you don't have accurate enough rationals and that as you increase the accuracy of your rationals you can rationalize more decimal values, at least one more each time. If that's the case then as precision approaches infinity irrationals must approach zero.
I'm not sure I follow you.
Our decimal number system can only represent rational numbers. No matter how many digits you have. The are of the form:
a/10^n + b/10^(n-1) + c/10^(n-2)...
for a, b, c, ... as integer 0 to 9, and integer n.
Euclid shows how there are numbers that cannot be expressed as p/q for integer p and q. See Norman Wildberger's video I linked above for that simple proof. For example root 2.
Ergo, our decimal system cannot represent irrationals even if you have an infinite number of digits.
The erroneous supposition is that the set of rationals is countable
Hmm...Cantor's diagonal method convinced me that the set of rationals is countable when I was fifteen. That is to say that the rationals can be placed into one to one correspondence with the integers. It also shows that the set of irrationals is not countable, they cannot be put into one to one correspondence with the integers/rationals.
Forty years later I have yet to find fault with Euclid or Cantor
I wrote the solution to Phil's problem as,
r/R = sin(pi/2n) / (1 - sin(pi/2n)
I think my maths teachers would have docked you a couple of points for not simplifying that into an actual expression for R, as requested.
Mind you, they might have docked me a point for not factoring out that r into:
R = r(1/sin(PI/n) - 1)
The case of n=6 is quite remarkable in that the ratio r/R is an integer. I wonder if it can be proven that is a unique, or even that it is the only solution in rational numbers?
There is the trivial solution when n = 2. We have:
R = r / sin(PI/n) - r
R = r / sin(PI/2) - r
R = r / 1 - r
R = r - r
R = 0
But that gives me headache.
We now have two circles of radius r touching each other. By definition of the problem.
But they are separated by that circle of radius R=0 between them!
So are they separated or not? Is that zero radius circle in the middle a thing or not?
Of course you will want to discount that result as your ratio r/R becomes r/0 which an undefined form.
Except that two marbles touching each other with a zero sized thing between them is something we can imagine quite easily. It's just two marbles touching each other.
@Heater,
The extra factor of 2 must have been due to the fortified eggnog!
r/R = sin(pi/n) / (1 - sin(pi/n))
In the limit of large n that becomes
r/R ~= pi/n / (1 - pi/n)
For example, at n=100, the approximation overestimates: 0.032434899 versus 0.032429391.
and at very very large n
r/R ~= pi/n
That is the same as lining up small circles of radius r along a straight line of length L = 2*pi*R
2*r*n = 2*pi*R
It worked out great, thanks, despite a few false starts building it and the marbles occasionally jamming. The non-uniform marble sizes were a bit of an issue. Only real problem is that it was too easy: my friend's daughter solved it in a few hours. I thought it would take days.
The idea is that you have five Olympic rings, but the colors are all jumbled up. You have to roll the marbles around the tracks until the rings are the correct colors. Here's a photo of the puzzle almost solved:
I say "almost" because the black, green, and red rings aren't yet linked properly. My puzzle is not original, BTW. Smaller plastic versions are available commercially.
This puzzle is an expanded version of the "Hungarian Rings" puzzle. I needed to make sure that all permutations were accessible before I loaded the marbles at random and glued the final layers of wood into place. For that I relied upon this 322-page (!) "lecture notes" document:
Hmm...if I may be so bold, do I see a gap between you black balls?
Yes, because one has partially slipped into the yellow ring. And I had to leave a little slack in the design so the balls would roll freely. Except at the intersections, there are detents in the bottom track. If you shake it while horizontal, the balls will settle into the detents. This was a help in preventing jamming, but not the entire solution.
So you got me wondering: How would this work out with Borromean rings?
I think I now have a project for next year -- if I can remember it that long!
Yes, I realized the reason for the gap when I looked at it a bit longer. When the balls are rotated half a "notch" in one ring the other ring has a bit more wiggle room.
Actually, now I'm wondering if it's even possible to build it with Borromean rings. We then have three rings and six intersections. Is it even possible to get all those intersections to line up around the n parts of the rings circumference? Does it fit at all? Does it fit some special values of n?
Comments
I have 1 watermelon. Now I can count from 0 watermelons to 1 watermelon. Next I cut the watermelon into 10 equal sized pieces. Now I can count from 0 to 1 in tenths !!!
Thing is we have lots of different types of numbers. They have technical definitions and names:
"Counting" or "Natural" numbers: 1, 2, 3, 4, ...
"Whole" numbers: 0, 1, 2, 3, 4, ...
Note the inclusion of that troublesome zero.
"Integer" numbers: ...,–4, –3, –2, –1, 0, 1, 2, 3, 4, ...
Now we are getting weird, how can I have a negative number of fingers?
"Rational" or "Fractional" numbers: Can be written as p/q where both p and q are integers.
"Irrational" numbers: Cannot be written as integers p/q. Typical examples are PI and root 2.
"Real" numbers: All of the above. As we imagine as a continuum of points along the number line.
Given all that, when you cut your 1 water melon into into 10 equal parts and count them you are counting how many of some rational number (p/q = 1/10) you have between 0 and 1. You have missed out all the Real numbers between those points.
You can hit more points on the continuum between 0 and 1 buy cutting your tenths into tenths. Then you hit a 100 points between 0 and 1. Cut your 100ths into tenths, then you hit a thousand points between 0 and 1. And so on recursively, forever. You end up hitting an infinite number of points between 0 and 1.
But wait, you now have an infinite number of points between 0 and 1. But I wager you have missed all the points (Real numbers) that your would have hit if you started dividing by 3 instead of 10. 1/3, 1/9, etc.
By diving by 10 you have ended up with an infinite number of points but you have an infinite number of gaps between them!
OK, you might say, I'll hit all the points by dividing by 2, then those I get by dividing by 3, then 4, then 5 .... then I will have counted them all and have no gaps.
Well no. Even that is not good enough. If you do that you will miss all the points/Real numbers you could have hit by dividing by PI or root 2 or e or any other Irrational number.
Conclusion, there are an infinite number of points/Real numbers on the number line between 0 and 1. Or indeed between any other two points. Uncountable. Mostly imaginary.
It's disturbing enough that there are an infinite number of Irrational numbers between any two Rational numbers. No matter how close together those Rationals are!
All of which is why floating point is so troublesome on computers.
I wonder what would happen if we removed irrationals from the Reals. Alternately pi could be expressed as 2^3x392699x10^-6 and be 99.9999% accurate. I guess it's all a matter of required precision.
Along the same point how real is it that an infinite series of 9s beyond a decimal is exactly equal to 1?
As it happens I do have an answer to your question. It's explained by Prof Norman Wildberger here: "Inconvenient truths about sqrt(2)"
For those curious about numbers, types of numbers and where they come from and why the concept of Real numbers is broken Norman has a foundation series:
Why would one do that? That is the same as writing:
8 * 0.392699
Which is PI with an error of 0.00002%
Might as well remember 355/113 which is PI with an error of 0.000008% and only requires remembering 6 digits and a "/"
Hmm...let's see:
Let x = 0.99999....
Then multiply both sides by 10:
10 * x = 10 * 0.99999...
So:
10 * x = 9.99999...
Which we can write as:
10 * x = 9 + 0.99999...
Notice that part on the right of the + is the same as the x we started with above. Substituting we have:
10 * x = 9 + x
Solve for x:
10 * x - x = 9
9 * x = 9
x = 9 / 9
x = 1
We started with x = 0.99999... above so if follows that:
0.99999 = 1
QED.
Seems pretty real to me.
If we don't accept that result then we must be claiming there is a flaw in our rules of arithmetic an algebra. But where?
0.9999.... can be written as:
0.9 + 0.09 + 0.009 + 0.0009 + .....
This is the infinite geometric series
9/10 + 9/100 + 9/1000 + 9/10000 +
Or
9/10^1 + 9/10^2 + 9/10^3 + 9/10^4 + ...
Or the sum of
9 / 10^n
For n = 1, 2, 3, ... to infinity.
The sum of such an infinite series is:
S = a / 1 - r
Where a is the first coefficient, and r is the ratio between terms.
In our case:
S = (9/10) / (1 - 1/10) = 1
Ergo:
0.9999.... = 1
See here for proof of the sum of an infinite geometric series:
https://www.khanacademy.org/math/calculus-home/series-calc/geo-series-calc/v/infinite-geometric-series
Depends on if your in an airplane trying to take off from a conveyor belt or not.
I will admit that although circles do not occur in nature, circular objects certainly do.
Certainly, an object which tends toward a circle will have a Pi that tends toward the computed value.
Mathematics should reflect what is known of the natural world. I have no problem with "Pi"... as long as we include it within the proper punctuation.
Until someone fixes it, it is nonsense.
Mike
If we were looking through a microscope at the straight lines of such a regular polygon, then I could suggest that there is a circle that is just big enough to contain all the vertices of that polygon. That we can't draw it or see it because your microscope is not good enough.
You get could get a better microscope and do the same again. I could then argue the same way again....
At some point the sides of that polygon will get down to such a small scale that your microscope will need so much energy to resolve them that your microscope and you will form a black hole! Thanks quantum mechanics. I have been thinking about this all day and I'm still not sure what you mean by it. On the contrary. Or at least maths should not be limited by what is known of the natural world.
Historically mathematicians have explored all kinds of maths that has no analogy in known natural world. Should be glad they did because it happens that many, many years later that maths turns out to be very useful in describing new things we find out about the world. Think non-euclidian geometry and imaginary numbers as examples. As far as I know it is fixed. That is why mathematicians have the idea of limits.
Anyway, why worry about Pi? If your view is that circles are made of little straight edges then we can simplify the problem. With that view even a squares are not possible. A square of side 1 should have a diagonal of root 2. But that is an irrational number. So if you try tho make a square you find, under your microscope, that the diagonal can never be the right length to fit where it should!
r/R = sin(pi/2n) / (1 - sin(pi/2n)
That is just to emphasize that the the ratio of the radii is the important thing.
The case of n=6 is quite remarkable in that the ratio r/R is an integer. I wonder if it can be proven that is a unique, or even that it is the only solution in rational numbers? Indeed, it would seem that the ratios are more likely to be transcendental numbers. Cases like sqr(2) are algebraic numbers, that is, numbers that are the solution of some polynomial of any degree. Take away all the algebraic numbers from the real number line (or from the complex plane), and the vast majority of numbers are still there, numbers like pi and e that are not solutions of any polynomial. (The algebraic numbers are said to have Lebesgue measure of zero.) Most numbers are transcendental.
Ok... forget the logic. If we just ban circles, we get absolute values for Pi, based upon irregular polygons, THEN we don't have this ridiculous pseudo number to shove down the throats of innocent children.
Pi is worse than mathematical malpractice... it is child abuse.
... I wouldn't take it as a grain of salt... we are talking about children... some of whom may be here illegally... but what does that have to do with mathematics?
Ban the CIrcle... save the children.
all at once now:)
We had got as far as calculating the areas of squares, rectangles and even advanced stuff like the area of triangles and polygons.
I stuck my hand up and asked the obvious question: How can we find the area of a circle?
Teacher brushed me off with something about how we would learn that in high school.
That was child abuse. Or at least a shame. Someone could have taken me aside and satisfied my young curious mind. To some level anyway.
Those pesky transcendental numbers are really disturbing.
There is infinitely many more transcendental numbers than any other class of number yet we pretty much can't find any of them !
That kind of leaves the nice continuum of our real number line in tatters, falling apart with gaps.
Once we get rid of the circle, we can get rid of "=",
I was sitting there in first grade as the teach tried to explain "="
Not a all convincing: " 1+3 is exactly the same as 4."
On the left you have an operation... on the right you have a result. Not at all the same.
I just stopped listening, because I knew she was wrong... why listen?
IF you can't explain it properly to a first grader... you shouldn't be trying.
Or better yet... why formulate mathematics that a first grader disagrees with?
but something along the lines "no wonder this place is such a mess."
Be careful of your first graders... they are judging you!
What does an infinite set of infinitesimal quantities amount to?
It would have to be a perfect rock in a perfect pond... but neither actually exists.
It all has something to do with the processes of creation... for every ying there isn't actually a yang. It just
looks that way. It turns out that in order to have time... certain asymmetries must be enforced. That yang is missing
something.
But why should there be a creation at all?...Apparently, that has something to do with a conversation (or heated argument)... all bending upon
the meaning of a single word... go figure.
The way I see it is that "=" or at least the concept of equal is the one thing that all kids know instictively. It seems to be hard wired into our minds. Even if we don't have a concept of numbers and such.
This is easily demonstrated. Get a bunch of kids to each do some chore, promising them some candy as a reward. When they have done their chores give half the kids 4 candies and the other half only 2. Be sure they all know who gets what. Soon you will have a riot on your hands as the kids that only got 2 start complaining that it's not fair.
Surely you have heard young kids crying and whining "But it's not fair!...wah, wah,..."
They soon understand the meaning of the "=" sign and the rest follows, number symbols, operators...
Maths is based on fairness and at that level very much linked to our concepts of justice.
Recently I was reading a paper about how chimp's understand this concept of fairnes as well. In the case of chimps they were rewarded with nice bannans or crappy carrots. When they find out other chimps are getting a better reward for the same thing they start throwing a hissy fit and thowing the carrots back at you!
If one can argue that there is a 1:1 ratio of even to whole numbers in the set of reals (excepting 0). Then one could argue that the spaces between rational numbers only matter if you don't have accurate enough rationals and that as you increase the accuracy of your rationals you can rationalize more decimal values, at least one more each time. If that's the case then as precision approaches infinity irrationals must approach zero.
The erroneous supposition is that the set of rationals is countable.
Our decimal number system can only represent rational numbers. No matter how many digits you have. The are of the form:
a/10^n + b/10^(n-1) + c/10^(n-2)...
for a, b, c, ... as integer 0 to 9, and integer n.
Euclid shows how there are numbers that cannot be expressed as p/q for integer p and q. See Norman Wildberger's video I linked above for that simple proof. For example root 2.
Ergo, our decimal system cannot represent irrationals even if you have an infinite number of digits. Hmm...Cantor's diagonal method convinced me that the set of rationals is countable when I was fifteen. That is to say that the rationals can be placed into one to one correspondence with the integers. It also shows that the set of irrationals is not countable, they cannot be put into one to one correspondence with the integers/rationals.
Forty years later I have yet to find fault with Euclid or Cantor
Mind you, they might have docked me a point for not factoring out that r into:
R = r(1/sin(PI/n) - 1)
There is the trivial solution when n = 2. We have:
R = r / sin(PI/n) - r
R = r / sin(PI/2) - r
R = r / 1 - r
R = r - r
R = 0
But that gives me headache.
We now have two circles of radius r touching each other. By definition of the problem.
But they are separated by that circle of radius R=0 between them!
So are they separated or not? Is that zero radius circle in the middle a thing or not?
Of course you will want to discount that result as your ratio r/R becomes r/0 which an undefined form.
Except that two marbles touching each other with a zero sized thing between them is something we can imagine quite easily. It's just two marbles touching each other.
The extra factor of 2 must have been due to the fortified eggnog!
r/R = sin(pi/n) / (1 - sin(pi/n))
In the limit of large n that becomes
r/R ~= pi/n / (1 - pi/n)
For example, at n=100, the approximation overestimates: 0.032434899 versus 0.032429391.
and at very very large n
r/R ~= pi/n
That is the same as lining up small circles of radius r along a straight line of length L = 2*pi*R
2*r*n = 2*pi*R
I wonder how Phil's gift project worked out.
The idea is that you have five Olympic rings, but the colors are all jumbled up. You have to roll the marbles around the tracks until the rings are the correct colors. Here's a photo of the puzzle almost solved:
I say "almost" because the black, green, and red rings aren't yet linked properly. My puzzle is not original, BTW. Smaller plastic versions are available commercially.
This puzzle is an expanded version of the "Hungarian Rings" puzzle. I needed to make sure that all permutations were accessible before I loaded the marbles at random and glued the final layers of wood into place. For that I relied upon this 322-page (!) "lecture notes" document:
http://www.sfu.ca/~jtmulhol/math302/notes/302notes.pdf
It's everything you ever wanted to know -- or didn't even know you wanted to know -- about permutation puzzles.
-Phil
Hmm...if I may be so bold, do I see a gap between your black balls?
So you got me wondering: How would this work out with Borromean rings? Three rings that interlock but no two of them are interlinked.
That would be a compact version. Would it be easier or harder to solve? Any time you rotate one ring you stir up two others, not just one.
https://en.wikipedia.org/wiki/Borromean_rings
Yes, because one has partially slipped into the yellow ring. And I had to leave a little slack in the design so the balls would roll freely. Except at the intersections, there are detents in the bottom track. If you shake it while horizontal, the balls will settle into the detents. This was a help in preventing jamming, but not the entire solution.
I think I now have a project for next year -- if I can remember it that long!
-Phil
Actually, now I'm wondering if it's even possible to build it with Borromean rings. We then have three rings and six intersections. Is it even possible to get all those intersections to line up around the n parts of the rings circumference? Does it fit at all? Does it fit some special values of n?