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Considering a circle to be an infinite sided n-gon

by acesuv
Tags: circle, infinite, ngon, sided
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acesuv
#19
May11-14, 11:37 PM
P: 53
Quote Quote by gopher_p View Post
You use phrases like "into infinity" and "go on forever" in regards to the proposed limiting process that leads to what you're calling a regular polygon with infinite sides. This leads me to believe that the object that you're proposing has countably many sides, since the countable cardinal is the limit of the finite cardinals. I reckon you'd say that each side of your infinite-sided polygon has length 0, in which case the perimeter of your object must also be 0.
well, im no mathematician, but isnt side length 0.0000000000000000000000...1? everyone is saying 0 so i guess not :(. this is a good point i think it is in similar lines to the fact it seems like a circle must have 180 degree vertices if you consider it to have infinite points... right? because 180 degrees is the limiting factor (is that the right term?!) of the measure of the vertices of an n-gon
jbunniii
#20
May12-14, 12:07 AM
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Quote Quote by acesuv View Post
well, im no mathematician, but isnt side length 0.0000000000000000000000...1?
Assuming the ... is intended to stand for infinitely many zeros, there is no such real number. It would have to be smaller than ##10^{-n}## for every positive integer ##n##, and the only nonnegative real number with this property is zero.
pwsnafu
#21
May12-14, 01:35 AM
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Quote Quote by acesuv View Post
it works if straight lines are infinitely small but nonzero :0 from what i figure like 0.00000000000000infinity1
The number you are talking about does not exist (as a real number) for the same reason that 0.99...=1.
7777777
#22
May12-14, 03:28 AM
P: 18
It is good to study a real example of approximating a circle as an infinite sided polygon.
For example, the Viete's formula:
http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula

Viete's formula represents a sequence of polygons with numbers of sides equal to [itex]2^{n}[/itex], inscribed in a circle.The Viete product is:
[itex]2/\pi = U_{1}/U_{2} \cdot U_{2}/U_{3} \cdot U_{3}/U{4} \cdot \cdot \cdot \cdot = U_{1}/U_{\infty}[/itex]

the Viete product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a [itex]2^{n}[/itex]-gon).
Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a digon [itex]U_{1}[/itex],(the diameter of the circle, counted twice) and a square [itex]U_{2}[/itex] , the ratio of perimeters of a square [itex]U_{2}[/itex] and an octagon [itex]U_{3}[/itex], etc etc up to the ratio of perimeters of [itex]U_{\infty-1}[/itex] and [itex]U_{\infty}[/itex] .

[itex]U_{\infty}[/itex] is the perimeter of [itex]2^{\infty}[/itex]-sided polygon. If the "radius"
of this [itex]2^{\infty}[/itex]-sided polygon is equal to 1, its diameter is equal to 2 (= [itex]U_{1}/2[/itex]), then its perimeter is equal to
[itex]2\pi[/itex], hence
[itex]U_{\infty}/(U_{1}/2) = 2U_{\infty}/U_{1}= U_{\infty}/2 = 2\pi/2 = \pi[/itex]
this is the same result as we obtained with the Viete's formula [itex]U_{1}/U_{\infty} = 2/\pi[/itex]

All the time a distinction is made between a circle and [itex]2^{\infty}[/itex]-sided polygon,
which is just the limiting case of [itex]2^{n}[/itex]-gon.
It might lead to an error to believe that a polygon transforms into a circle at an "infiniteth" step.
The error just seems to disappear if we are free to call a [itex]2^{\infty}[/itex]-sided polygon
a circle.

I used the Viete's formula from Jörg Arndt book Pi - Unleashed:
http://books.google.fi/books?id=Qwwc...0holes&f=false
micromass
#23
May12-14, 04:21 AM
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Quote Quote by 7777777 View Post
It might lead to an error to believe that a polygon transforms into a circle at an "infiniteth" step.
The error just seems to disappear if we are free to call a [itex]2^{\infty}[/itex]-sided polygon
a circle.
Vičte's formula does not rely on a circle being a infinite sided polygon. All we need is the circle being somehow the limit of polygons.
mrg
#24
May12-14, 04:53 PM
P: 5
I just touched on this very concept with my high school geometry class.

In order to derive the formula for the area of a circle, we assumed that a circle was an "infinity-gon." Then, using the formula A=(1/2)ap (where a is the apothem and p is the perimeter) we substituted in the radius for a (since every apothem in an infinity gon is a radius) and then the circumference formula for p. We get A=(1/2)2(pi)r^2, or, pi*r^2.

I warned the students that a circle is not, by definition, a polygon, but for the sake of calculating the area, it's useful to imagine that it is one since we can use what we already know to describe this new concept.

I'm wondering what others think about that - isn't that how they originally calculated the area of circles? They did repeated approximations which got closer and closer to a number, which they then created a formula from? Comments would be welcome.
disregardthat
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May12-14, 05:00 PM
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Quote Quote by mrg View Post
I just touched on this very concept with my high school geometry class.

In order to derive the formula for the area of a circle, we assumed that a circle was an "infinity-gon." Then, using the formula A=(1/2)ap (where a is the apothem and p is the perimeter) we substituted in the radius for a (since every apothem in an infinity gon is a radius) and then the circumference formula for p. We get A=(1/2)2(pi)r^2, or, pi*r^2.

I warned the students that a circle is not, by definition, a polygon, but for the sake of calculating the area, it's useful to imagine that it is one since we can use what we already know to describe this new concept.

I'm wondering what others think about that - isn't that how they originally calculated the area of circles? They did repeated approximations which got closer and closer to a number, which they then created a formula from? Comments would be welcome.
The only idea that needs to be grasped is that calculating the perimeter of inscribed and circumscribed polygons give upper and lower bounds of the perimeter of the circle. To go on and say that the circle is an "infinity-gon" is quite meaningless and only serves to generate confusion. You could say (arbitrarily) that a circle is an "infinity-gon", but you can't argue anything from that.

In addition to that, by increasing the number of vertices of the inscribed and circumscribed polygons, and calculating the sequences of perimeters, you see that these values converge towards a single value, which will be (or what we call) the perimeter of the circle. I don't see any logical or pedagogical reason to force the students to imagine the circle as a polygon. The polygons serve as approximations, that is the whole idea.


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