You can't simply "square both sides". That's not a valid operation (that keeps the equality). You have to multiply both sides by the same number, and you did not do that there.
The sum of the reciprocals of the squares of the natural numbers is π2/6. I'm wondering if this is just a coincidence, or if there really is an intuitive relation between this sum and the ratio between the circumference and diameter of a circle.
How about the RTS resource problem?
You have a turn based game where you can either wait each turn or queue up a new worker unit.
You start with F number of resources.
Each worker unit costs C resources and takes D turns to become active and produces E resources per turn once active.
The goal is to reach N resources in the shortest number of turns.
The sum of the reciprocals of the squares of the natural numbers is π2/6. I'm wondering if this is just a coincidence, or if there really is an intuitive relation between this sum and the ratio between the circumference and diameter of a circle.
I was thinking about quickly replying with something to the tune of, "Probably not, because I can't think of any formulas in geometry that involve π2 rather than π alone." I decided that I should let more experienced mathematicians take a crack at it first. But I just had an epiphany in the shower and I'm a little more confident in saying there likely isn't any geometric significance.
Is there goemetric significance to every square number? (Or alternatively, every square root?)
As a very simple example, we know from physics that the kinetic energy is equal to 1/2mv2. Does that mean that lurking behind every moving object is a square with a side whose length varies proportional to the velocity? Or perhaps even more to the point, I know that when I drop an object from a height h, its speed will be v = sqrt(2gh) just before it hits the ground. Does that mean that "somewhere" there is a square I can draw with great physical significance that has a side of length sqrt(2gh)?
Probably not. Of course, I could arbitrarily introduce such a square, but I am having a hard time imagining some physical or otherwise intuitive reason for why it should have a side of that particular length. Velocity is squared in the equation for kinetic energy because, well, that's just how physics works. (You can also reason that kinetic energy should be proportional to v2 from symmetry arguments, but those don't invoke geometric squares either.)
This got me thinking-- and I may be far out of line in this conjecture-- that the insistence that equations be geometrically or otherwise significant may have been the lynchpin holding back mathematics for thousands of years. For example, I could imagine going back in time and telling a medieval geometer that kinetic energy is proportional to the square of the velocity and him replying, "Okay, but where is the square whose side is length v?" I would be unable to properly convey that the numbers are effectively confined to the chalkboard and upheld by experiment.
It seems that much of mathematical progress of the last 500 or so years has been related to the abstraction of mathematics and thinking of "numbers as numbers". A quick view of Wikipedia's page on the history of mathematics seems to confirm that math was essentially unused for anything but geometry and accounting all the way up to the time of Johannes Kepler (whose square-cube law represents perhaps the first empirical law based on the principles of geometry while lacking a geometric argument). (And as an aside, I also think it's fascinating that his first two laws are firmly rooted in classical geometry. I can imagine Kepler searching frantically for some squares and cubes that we know aren't there and just saying, "Welp, this is just the rule that it follows.")
If anyone has some knowledge of the history of mathematics and can shed some light on it, I'd be very interested in how close I am to being correct. But my official answer is that you won't find any circles or spheres that tell you why the sum of the squares of the integers is equal to π2/6.
As a very simple example, we know from physics that the kinetic energy is equal to 1/2mv2. Does that mean that lurking behind every moving object is a square with a side whose length varies proportional to the velocity? Or perhaps even more to the point, I know that when I drop an object from a height h, its speed will be v = sqrt(2gh) just before it hits the ground. Does that mean that "somewhere" there is a square I can draw with great physical significance that has a side of length sqrt(2gh)?
I think you are succumbing to a fallacy of equivocation. You are confusing the two meanings of the word "square" (one of them being "a geometric shape with certain characteristics" and the other being "raised to the power of 2". The same word may be used for both, but that doesn't mean they are synonyms.)
Granted, there's a relationship between the geometric shape and the power-of-2, namely that the area of the former is calculated using the latter, and that's the etymology of the latter. However, that doesn't make them synonyms.
There may be a connection between the area of a square and a formula like mv2/2, but not a direct one. Rather, they both have a common underlying geometric reason why they both use a power of 2.
A clearer example would be the intensity of light being proportional to the inverse of the square of the distance from the light source. Are there literal geometric squares involved? No. However, both share same underlying geometric principles (namely, that light intensity is measured by area, and areas involve powers of two.)
As a very simple example, we know from physics that the kinetic energy is equal to 1/2mv2. Does that mean that lurking behind every moving object is a square with a side whose length varies proportional to the velocity? Or perhaps even more to the point, I know that when I drop an object from a height h, its speed will be v = sqrt(2gh) just before it hits the ground. Does that mean that "somewhere" there is a square I can draw with great physical significance that has a side of length sqrt(2gh)?
I think you are succumbing to a fallacy of equivocation. You are confusing the two meanings of the word "square" (one of them being "a geometric shape with certain characteristics" and the other being "raised to the power of 2". The same word may be used for both, but that doesn't mean they are synonyms.)
Granted, there's a relationship between the geometric shape and the power-of-2, namely that the area of the former is calculated using the latter, and that's the etymology of the latter. However, that doesn't make them synonyms.
There may be a connection between the area of a square and a formula like mv2/2, but not a direct one. Rather, they both have a common underlying geometric reason why they both use a power of 2.
A clearer example would be the intensity of light being proportional to the inverse of the square of the distance from the light source. Are there literal geometric squares involved? No. However, both share same underlying geometric principles (namely, that light intensity is measured by area, and areas involve powers of two.)
The point I was trying to make was not that squaring a number and the geometric object of a square are the same thing, but rather to highlight that there are "familiar" (based in geometry or more familiar concepts) mathematical tools, functions, and constants that "coincidentally" show up in other places. For example, why does the square-root of π so frequently show up in the gamma function of half-integers? Are there circles involved here? As far as I can tell, the answer is simply "no". Your original question (if I understand it) was why the particular ratio of the circumference of a circle to its diameter shows up so conspicuously in the sum of the reciprocals of the squares of the natural numbers. Well, you're invoking geometry! It sounds like you want to know "where the circles come in". Maybe you can create some squares with sides 1, 1/2, 1/3, 1/4, etc., then cut them all up into pieces of a very special shape, rearrange them, and by a very careful argument show that they create a circle whose radius is exactly sqrt(π/6). It sounds like that's exactly the sort of argument you're looking for.
Well, I know of no such argument. Establishing that sort of connection between the conspicuous constant (π) in the theorem and the geometric object of a circle is nigh-- if not outright-- impossible. It just is because the proof says so.
Look, let me put it this way. Suppose all you know is that numbers define quantities. You are only aware that you can use them to count things: "One apple, two apples, three apples..." and so on. You know nothing of rational fractions, negative numbers, or other such things. You are, however, taught the decimal expansion of π. So you notice that the tenth digit of pi is 3 and you lay awake at night wondering, "Where did that 3 come from?" Perhaps, somewhere deep in the derivation of π, there is something that says π's tenth digit represents three of something. What is that something? Does the universe have ten apples stowed away somewhere? Maybe you're clever enough to have a sense of scale and say that π is "3 big things, plus 1 thing one-tenth as big, plus 4 things one-tenth as big as that, plus 1 thing one-tenth as big as that..." But where do these things really come in?
They don't. You can't compare apples to circles.
I have hard time imagining that π appearing in the answer is just pure coincidence.
π is such a specific number. Not only is it one specific number from the uncountably infinite amount of real numbers, not only is it irrational and transcendental, but it's very specifically a number related to mathematical circles.
If the answer had consisted of algebraic numbers, such as the square root of 2 and such, it would feel natural. After all, the sum consists of algebraic components, so one wouldn't be surprised to find out that the result is also some algebraic number. However, it is not. The result is transcendental, and from the uncountably many transcendental numbers, it's one that involves this particular number related to circles.
Why precisely this one, and not some completely unrelated number? There just has to be some kind of geometrical explanation of why the abovementioned sum is somehow connected to the geometric circle. Some underlying property. It's difficult to imagine that there is no connection whatsoever, and that the appearance of π is purely coincidental.
Not really Warp, because to a mathematician π is so much more than the ratio of a circle's circumference to its diameter. In this case, the sum is linked to Taylor series of trigonometric functions, whose solutions involve π. Indeed the simplest proof of the identity involves expressing one such series as both an infinite sum and an infinite product. It turns out that one co-efficient of this series is given by the sum of the reciprocals of the squares on one side, and π^2/6 on the other. Since the expressions are equivalent, the result follows.
Not really Warp, because to a mathematician π is so much more than the ratio of a circle's circumference to its diameter. In this case, the sum is linked to Taylor series of trigonometric functions, whose solutions involve π.
Given that sines and cosines are closely related to circles...
Not really Warp, because to a mathematician π is so much more than the ratio of a circle's circumference to its diameter. In this case, the sum is linked to Taylor series of trigonometric functions, whose solutions involve π.
Given that sines and cosines are closely related to circles...
No, not really. What's going on is more like this...
• Circles are geometric objects.
• Sine functions are geometrically related to circles.
• The Taylor series of a sine function (which lacks an intuitive geometric interpretation) follows a nice pattern.
• A piece of that pattern is followed by the sum of the reciprocals of square numbers.
Just as thatguy said, the connection is in the pattern of the series. You can see from my outline above that the connection to geometry is quite tenuous.
I mean, if you think you can view it in terms of circles, read the proof and try to come up with a geometric way of thinking of it. I just don't think it's reasonably possible.
The sum of the reciprocals of the squares of the natural numbers is π2/6. I'm wondering if this is just a coincidence, or if there really is an intuitive relation between this sum and the ratio between the circumference and diameter of a circle.
This problem has been on my mind a little lately. I haven't given it a whole lot of thought, but to the extent that I have, I'm stumped.
1) Construct a function of two real numbers that, using only addition/subtraction, multiplication/division, and absolute values that takes the value of the larger of the two numbers. (So construct the max function using only absolute values. This should be very easy.)
2) Construct a function of three numbers with the same requirements as above (using only a/s, m/d, and absolute values) that takes the value of the middle number. (This is only a bit harder.)
3) Construct a function of four numbers with the same requirements as above that takes the value of the second-largest number. (I don't know off the top of my head how one might do it.)
So for example, without loss of generality, for any a<b<c<d, we seek functions f, g, and h (corresponding to cases 1, 2, and 3 above) that only involve addition, subtraction, multiplication, division, and the absolute value, such that:
f(a,b) = f(b,a) = b
g(a,b,c) = g(b,c,a) = g(c,a,b) = g(c,b,a) = g(b,a,c) = g(a,c,b) = b
h(a,b,c,d) = {23 other permutations of the arguments} = c
Can this be done?
Edit: I found one way to do it. It's a little tricky and I'm sure there are many equivalent ways. I also suspect the process can be generalized to yet more variables.
This problem has been on my mind a little lately. I haven't given it a whole lot of thought, but to the extent that I have, I'm stumped.
1) Construct a function of two real numbers that, using only addition/subtraction, multiplication/division, and absolute values that takes the value of the larger of the two numbers. (So construct the max function using only absolute values. This should be very easy.)
2) Construct a function of three numbers with the same requirements as above (using only a/s, m/d, and absolute values) that takes the value of the middle number. (This is only a bit harder.)
3) Construct a function of four numbers with the same requirements as above that takes the value of the second-largest number. (I don't know off the top of my head how one might do it.)
...
1)
f_1(a,b) := (a + b + |a - b|) / 2 = max(a,b)
Proof:
Let a>= b:
f_1(a.,b) = (a + b + |a - b|) / 2 = (a + b + a - b) / 2 = a = max(a,b)
Let a < b:
f_1(a,b) = (a + b + |a - b|) / 2 = (a + b - a + b) / 2 = b = max(a,b)
Likewise: f_2(a,b) := ( a + b - |a - b| ) / 2 = min(a,b)
For the others: Maybe somehow construct the function from min()/max() functions?
All syllogisms have three parts, therefore this is not a syllogism.
This problem has been on my mind a little lately. I haven't given it a whole lot of thought, but to the extent that I have, I'm stumped.
1) Construct a function of two real numbers that, using only addition/subtraction, multiplication/division, and absolute values that takes the value of the larger of the two numbers. (So construct the max function using only absolute values. This should be very easy.)
2) Construct a function of three numbers with the same requirements as above (using only a/s, m/d, and absolute values) that takes the value of the middle number. (This is only a bit harder.)
3) Construct a function of four numbers with the same requirements as above that takes the value of the second-largest number. (I don't know off the top of my head how one might do it.)
...
1)
f_1(a,b) := (a + b + |a - b|) / 2 = max(a,b)
Proof:
Let a>= b:
f_1(a.,b) = (a + b + |a - b|) / 2 = (a + b + a - b) / 2 = a = max(a,b)
Let a < b:
f_1(a,b) = (a + b + |a - b|) / 2 = (a + b - a + b) / 2 = b = max(a,b)
Likewise: f_2(a,b) := ( a + b - |a - b| ) / 2 = min(a,b)
For the others: Maybe somehow construct the function from min()/max() functions?
This matches what I have for the first problem. I believe the answer is essentially unique, unlike the other problems. You are also on the right track with your idea to use the min() and max() functions to produce the second and third functions.
Joined: 12/28/2007
Posts: 235
Location: Japan, Sapporo
Bobo the King wrote:
This problem has been on my mind a little lately. I haven't given it a whole lot of thought, but to the extent that I have, I'm stumped.
1) Construct a function of two real numbers that, using only addition/subtraction, multiplication/division, and absolute values that takes the value of the larger of the two numbers. (So construct the max function using only absolute values. This should be very easy.)
2) Construct a function of three numbers with the same requirements as above (using only a/s, m/d, and absolute values) that takes the value of the middle number. (This is only a bit harder.)
3) Construct a function of four numbers with the same requirements as above that takes the value of the second-largest number. (I don't know off the top of my head how one might do it.)
So for example, without loss of generality, for any a<b<c<d, we seek functions f, g, and h (corresponding to cases 1, 2, and 3 above) that only involve addition, subtraction, multiplication, division, and the absolute value, such that:
f(a,b) = f(b,a) = b
g(a,b,c) = g(b,c,a) = g(c,a,b) = g(c,b,a) = g(b,a,c) = g(a,c,b) = b
h(a,b,c,d) = {23 other permutations of the arguments} = c
Can this be done?
Edit: I found one way to do it. It's a little tricky and I'm sure there are many equivalent ways. I also suspect the process can be generalized to yet more variables.
For f, the function f(a,b) := (a+b+|a-b|)/2 will do, and the min function can be also constructed as f'(a,b) := -f(-a,-b) = (a+b-|a-b|)/2.
For more variables, I'm just going to explain the process, not showing the functions explicitly (you can write down functions with ease, though). Now assume we have three real numbers a, b and c, and let A, B and C be the max of the pairs (a,b), (a,c) and (b,c), respectively. Then the set {A,B,C} consists exactly of the largest and the middle among {a,b,c}. So we have only to pick up the smallest of {A,B,C}, which can be done by iterating the min function of two numbers.
For the second-largest of four numbers, pick up the largest of each triple among {a,b,c,d} to form the set {A,B,C,D}, and the smallest of this set is the second-largest of {a,b,c,d}.
This is somewhat redundant but has the advantage of being generalized easily to the case of the m-th largest of n numbers.
Retired because of that deletion event.
Projects (WIP RIP): VIP3 all-exits "almost capeless yoshiless", VIP2 all-exits, TSRP2 "normal run"
This problem has been on my mind a little lately. I haven't given it a whole lot of thought, but to the extent that I have, I'm stumped.
1) Construct a function of two real numbers that, using only addition/subtraction, multiplication/division, and absolute values that takes the value of the larger of the two numbers. (So construct the max function using only absolute values. This should be very easy.)
2) Construct a function of three numbers with the same requirements as above (using only a/s, m/d, and absolute values) that takes the value of the middle number. (This is only a bit harder.)
3) Construct a function of four numbers with the same requirements as above that takes the value of the second-largest number. (I don't know off the top of my head how one might do it.)
Edit: I found one way to do it. It's a little tricky and I'm sure there are many equivalent ways. I also suspect the process can be generalized to yet more variables.
One fairly brute force way to do it is construct sorting network and then implement it using min()/max() and further using allowed primitives.
But without Common Subexpression Elimination, the complexity of resulting expression likely grows exponentially in number of inputs.
2) Middle:
mid(a,b,c) = max (min(a,b) , min(b,c) , min (b,c) ) = min (max (), max(), max())
3) Second largest is:
second(a,b,c,d) = max ( mid(a,b,c), mid(a,b,d), mid(a,c,d), mid(b,c,d))
edit: OK, mister already got it.
2) Middle:
mid(a,b,c) = max (min(a,b) , min(b,c) , min (b,c) ) = min (max (), max(), max())
3) Second largest is:
second(a,b,c,d) = max ( mid(a,b,c), mid(a,b,d), mid(a,c,d), mid(b,c,d))
edit: OK, mister already got it.
Your solution to the third problem matches my method. You actually don't need all four mid functions in there, since the two middle numbers are surely represented by three of the four functions.
Also, as a very minor technicality, min and max each take two arguments. You'd need to use them iteratively, defining something like
max(a,b,c) = max(a, max(b,c)).
For the second problem, I used
mid(a,b,c) = a+b+c - max(a,b,c) - min(a,b,c).
A cylindrical hole, 6cm deep is drilled through the middle of a solid sphere or radius r. Find the remaining volume in the sphere.
You may find this page useful. What's interesting about the solution?
A cylindrical hole, 6cm deep is drilled through the middle of a solid sphere or radius r. Find the remaining volume in the sphere.
You may find this page useful. What's interesting about the solution?
I think I remember this problem. I'll leave the answer for others to determine.
Are you sure the hole is 6 cm deep and not 6 cm in radius (or diameter) and it's bored all the way through? My recollection is that that's the "usual" problem statement.
I seem to remember the problem differently.
Was it the height of the remaining sphere (after drilling the hole top-down) that was fixed, rather than the radius/diameter of the hole?
Edit: I think that's what Flip means by "cylindrical hole, 6cm deep". A cylindrical hole is drilled top-down through the center of a sphere all the way through, such that the length of the cylindrical hole inside the remaining sphere is 6cm. In other words, the height of the remaining sphere is 6cm.
I seem to remember the problem differently.
Was it the height of the remaining sphere (after drilling the hole top-down) that was fixed, rather than the radius/diameter of the hole?
Edit: I think that's what Flip means by "cylindrical hole, 6cm deep". A cylindrical hole is drilled top-down through the center of a sphere all the way through, such that the length of the cylindrical hole inside the remaining sphere is 6cm. In other words, the height of the remaining sphere is 6cm.
I think you are correct. The width of the drill is such that the hole it bores has a length of 6 cm, but it goes all the way through the sphere. Of course, the sphere must be at least 6 cm in diameter.
There is a very similar problem in two dimensions that I'm more familiar with.
Anyway, the solution to Flip's problem is as follows:
Let r be the sphere radius. In order to have a hole of length 6, the radius of the hole must be sqrt(r2-9). After drilling the hole through the sphere, the remainder is a solid of revolution formed by taking the area below y=sqrt(r2-x2) and above y=sqrt(r2-9) from -3 to 3, and rotating around the x-axis. Disk integration gives:
volume = π*int(-3..3) {sqrt(r2-x2)2-sqrt(r2-9)2}dx = π*int(-3..3) {(r2-x2)-(r2-9)}dx = π*int(-3..3) (9-x2)dx = π*int(-3..3) sqrt(9-x2)2 dx
which is the volume of a sphere of radius 3cm (disk integration on y=sqrt(9-x2) rotated around the x-axis), which is (4/3)π*33=36π cm3.
What's interesting about the solution? It doesn't depend on r (the sphere radius). So, no matter what r is, the answer is the same as a volume of a sphere of radius 3cm (which is also the shape of the remaining sphere as r approaches 3cm).
