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Re: A Point on Understanding
Posted:
Dec 26, 2012 11:38 PM
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On Wed, Dec 26, 2012 at 10:27 AM, Robert Hansen <bob@rsccore.com> wrote:
> Let's look at it this way. Consider a pizza and you are going to slice this pizza into 1, 2, 3, 4, ... slices. For each n, where n = 1, 2, 3, ... the interior angle of each slice will be 360/n, correct? And if for each n we sum up the interior angles we will get 360, correct?.
>
In the algorithm I'm talking about, we triangulate the sphere such
that each vertex has either 6 or 5 spokes coming out from each vertex,
making a wheel-like pattern.
The perimeter is either a hexagon or a pentagon. If we bashed out the
spokes, we would have like a soccer ball but with a vast number of
hexagons, yet still only 12 pentagons.
This is de-generalizing just a little, in that "all hexagons and
pentagons" (were the spokes removed) only occurs in multiples of 3,
counting the number of intervals (edges) from one pentagon center to
the next (this is called the "frequency" of the ball / sphere).
That's OK (to de-generalize a little) if it helps with the visualizations.
Here are some pictures:
Hexapents:
http://www.myspace.com/4dstudios/photos/698295
http://www.4dsolutions.net/ocn/hexapent.html
Adding back the missing spokes (so it's really all triangles):
http://donhavey.com/blog/tutorials/tutorial-3-the-icosahedron-sphere/
http://www.instructables.com/id/Construct-Geodesic-Spheres-on-Google-SketchUp/
All very apropos of New Year's by the way, as the Times Square ball is
one of these.
The higher the frequency, the more triangles, but it's still all six
slice and five slice pizzas, just more of them.
> As n -> infinity, the interior angles approach 0. The limit of 360/n as n approaches infinity is 0. Yet, the sum of the interior angles is always 360, or stated another way, n * 360/n = 360, a constant.
>
The apex of each hexagon makes the pizza slightly convex / concave
i.e. the center point is a not in plane with the points around the rim
(the rim doesn't have to be all in a plane either). Every vertex is a
local apex i.e. is surround by a little less than 360 degrees.
The number of degrees around each vertex is never quite 360, because
we're on a ball. But the triangles are so small, relatively, that
it's as flat as a bathroom floor (triangular tiles) except
mathematically (and with careful measurement) we see that it isn't.
Picture a ball the size of the Earth with triangular bathroom floor
tiles smoothly covering it -- they look almost perfectly equilateral
except around pentagons (which may be hard to find are there are only
12 of them).
Every vertex contributes some small positive finite amount to the
grand total of 720. 720 degrees apportioned to all these vertexes,
really would create perfect flatness around each of them. The skin
would no any longer have convexity and this would not be a polyhedron.
> But you are missing a step when you stated your contradiction.
>
> You are asking why does limit(n) * limit(360/n) as n->infinity not equal the limit(n * 360/n) as n->infinity.
>
360 is divided by either 6 or 5, but in such a way that | 360 - d |
where d is the total degrees around v (vertex) > 0.
I can make | 360 - d | arbitrarily close to 0 as I increase the
frequency of the triangular mesh (more pizzas, but not more slices
around each vertex -- that's 6 or 5).
Give me an epsilon, like .000000000000001 degrees, and I can give you
a frequency that is less than epsilon, i.e. | 360 - d | < e.
> This is because the limit(n) as n->infinity does not exist, and thus, you can't multiply the limits.
>
As the number of vertexes increases, | 360 - d | decreases as close to
0 as we like. But it can never be 0, because all the vertexes
together contribute a grand total of 720 degrees (unchanging
constant).
This is what's known as Descartes Deficit, i.e. if you add up the
number of degrees round each corner of a cube (90 + 90 + 90), the
local deficit (the difference from 360) is 90 and 90 times the number
of vertexes is 720. This will be true of your octahedron,
icosahedron, dodecahedron etc. etc., and true of your super high
frequency geodesic sphere.
> You did show that limit(360/n) exists and is zero.
> You did show that limit(n * 360/n) exists and is 360.
>
> But your final statement actually involves limit(n), which you never showed existed, and indeed, doesn't exist.
>
> Bob Hansen
>
n = number of vertexes on the sphere (note: if p = n - 2, then the
ratio of p:faces:edges = 1:2:3 if omni-triangulated)
720 = constant angular deficit (all local deficits added) -- per Descartes
d = number of degrees around each vertex (< 360 in all cases) but
arbitrarily close
|360 - d| = local angular deficit (a given vertex)
Sigma [local angular deficit] == 720
all v
As the limit n increases without limit, local angular deficit -> 0,
but sum total angular deficit remains fixed at 720.
Contradiction?
How could the sum of all the |360 - d| amounts be 720 (a constant) and
yet each have 0 as a limit?
I guess you could say 720 * n * (1/n) --> 720 for any n (which isn't
quite the same thing), but it's interesting nonetheless how you can
argue a limit of 0 at each vertex (perfect flatness at the limit), and
yet know there's a fraction of 720 that's always keeping us from true
0 (720/n approaches 0 too, as n -> infinity, I guess one would say).
Not a show stopper, but in the process we learn something about
subdividing a sphere, and maybe about calculus.
Kirby
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