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        riddles >> medium >> Value of pi
        
(Message started by: Christine on Jan 28^th, 2013, 10:50am)

Title: Value of pi
Post by Christine on Jan 28^th, 2013, 10:50am

We know that the ratio pi = C/D = C/2r
is constant for circles on <U>flat</U> surfaces.

What would be the value of pi on non-flat surfaces?
Say, on the surface of a sphere, or on the surface of a cube with centre one of the corners of the cube.

Title: Re: Value of pi
Post by towr on Jan 28^th, 2013, 1:07pm

If you measure the radius of a circle over a sphere, then it's not constant. A circle on the equator has "pi" = 2, whereas an infinitesimally small circle has "pi" = pi.

Title: Re: Value of pi
Post by peoplepower on Jan 28^th, 2013, 2:34pm

As for a circle centered on the corner of a cube, we might have to be careful about what metric is being used. To make my point, consider a vertex and a point on some face not visible to the vertex, which direction do we take in measuring the distance between the two points? This matters because the circle is usually defined as being the set of points a fixed distance from a given point. If you have some other way to define a circle, go ahead.

There is one case we can cover without worrying about those considerations. If the radius is less than or equal to the edge length of the cube, the circle will be split evenly between three faces. On each face it is just a quarter "planar" circle. Hence the circumference on a cube is 3/4 what it is on the plane, the value of pi_cube is accordingly 3/4 of pi_plane. Remember, though that this might differ if the radius of the circle is greater than the edge length.

Title: Re: Value of pi
Post by Christine on Jan 28^th, 2013, 4:08pm

on 01/28/13 at 13:07:15, towr wrote:

A circle on the equator has "pi" = 2, whereas an infinitesimally small circle has "pi" = pi.

How does the value of pi change?

peoplepower,

Let's take the unit cube.
How does the value of pi change as r increases?

3 cases to consider:
-> 0 < r < 1
-> 1 <= r < sqrt(2)
-> r => sqrt(2)

Title: Re: Value of pi
Post by peoplepower on Jan 28^th, 2013, 6:49pm

My concern is it is not clear how to define distance. For instance, is distance the length of the line in 3d-space connecting those two points? Or does it have to be a "path" along the surface of the cube to count? Or do we use something entirely different? I would take the second to be most natural, so here is how I would formalize this problem so that we can show it is equivalent to a very easy problem.

Thus, I will define distance between two points on a cube to be the length of the shortest path along the surface of the cube connecting the two points.

We unwrap the cube into a net (not exactly, it needs an extra cut) in a very specific way. First, label the vertex at the center of our circle v, and call the faces touching this vertex E, F, G. Let O be the face opposite E. Consider the point x of intersection FGO and the vertex y opposite it on O; cut the line xy. Cut all edges except those touching the face E and the edges OF, OG. If I described this properly, it should be possible to open this up into something resembling a net of the cube.

It has the property that for any point on the cube, the distance between that point and the vertex v is the length of the corresponding line on the net.

That means a circle on the cube centered at this vertex is simply the intersection of a circle on the plane with the net. So you measure the circumference of the circle and subtract the portion which does not lie in the net.

Title: Re: Value of pi
Post by towr on Jan 28^th, 2013, 10:37pm

on 01/28/13 at 16:08:29, Christine wrote:

How does the value of pi change?

I think it's pi sin(r)/r, with 0 <= r <= pi on the unit sphere. Although it's hard to say whether you have a circle with radius > pi/2 or < pi/2

Title: Re: Value of pi
Post by Christine on Jan 29^th, 2013, 1:32am

on 01/28/13 at 22:37:59, towr wrote:

I think it's pi sin(r)/r, with 0 <= r <= pi on the unit sphere. Although it's hard to say whether you have a circle with radius > pi/2 or < pi/2

I've just discovered this site.
Circumference = R sin(r/R)
the circumference of a circle is less than the Euclidean value of 2*pi*r

http://www.astro.virginia.edu/class/whittle/astr553/Topic16/t16_circumference.html

Title: Re: Value of pi
Post by Christine on Jan 29^th, 2013, 11:45am

The circumference of a circle of radius r is:
3/4 * 2*pi*r

http://nrich.maths.org/5654

I posted earlier 3 cases:
-> 0 < r < 1
-> 1 <= r < sqrt(2)
-> r => sqrt(2)

So these cases don't matter!?

Title: Re: Value of pi
Post by peoplepower on Jan 29^th, 2013, 4:18pm

We actually need four cases (and a fifth if you count too large radii).
r <= 1
1<r<=sqrt 2
sqrt 2 < r<= 2
2<r <= sqrt 5

The expressions for the latter cases are somewhat messy, so I will probably tex it up along with a diagram of the net I was referring to.

P.S. The topological notion of curvature, which is what is explained in your link, might help with finding circumferences of small circles, but it does not evaluate the circumference of an arbitrarily large circle for you.

When points of nonzero curvature are inside the circle, you have to account for the curvature at these points even when you are just calculating circumference, and the interrelations between the effects of the curvature are more complicated the more points of nonzero curvature you have.

Title: Re: Value of pi
Post by rmsgrey on Jan 30^th, 2013, 6:03am

on 01/29/13 at 16:18:45, peoplepower wrote:

We actually need four cases (and a fifth if you count too large radii).
r <= 1
1<r<=sqrt 2
sqrt 2 < r<= 2
2<r <= sqrt 5

The vertex at distance 2 by one route is at distance sqrt(2) by another.

To get a suitable net, imagine a + sign, fill one quadrant with a square and the neighbouring quadrants with right-isosceles triangles so you end up with a non-convex hexagon (equivalently, take the familiar "house" pentagon - a square with a right-isosceles triangle on top - and remove the triangular bottom quadrant of the square). That's a third of a pseudo-net for a cube, with the center of the circle at the convex right-angle - three copies meeting at that right-angle will fold to form a cube with added diagonal edges from the antipode of the circle's center.

With this net, points that meet when you fold it are at the same distance from either side of the net.

Title: Re: Value of pi
Post by peoplepower on Jan 30^th, 2013, 4:37pm

In my net the lack of symmetry with respect to the faces which share the distinguished vertex should have been a red flag for me.

Perhaps a robust proof that the pseudo-net you described is accurate is to show in general that one only needs to check the points along the boundary of the diagram; this seems to have something to with it being star-shaped. Then, we just have to check by cases the three types of boundary segments, taking advantage of the symmetry present. Oops, this seems to be an elaboration of your last line, so you would probably agree anyway.