8 sided Drive Train footprint

[Oblarg]

Originally Posted by cadandcookies

Geometry is such a lovely thing, isn't it?

It can be, though it's easy to overlook its subtleties and a often pain to iron out the details.

(If you want to be convinced of this, try actually proving the seemingly-obvious thing I posted above).

[DampRobot]

Originally Posted by Oblarg

It can be, though it's easy to overlook its subtleties and a often pain to iron out the details.

(If you want to be convinced of this, try actually proving the seemingly-obvious thing I posted above).

Challenge accepted!

In a regular n-gon, the central angle for each side is 180/n degrees, and if the ngon has an inradius r, the side length of each side is S=2r*tan(180/n). I proceeded to construct a isosceles triangle where the center of the n gon is connected with equal length legs to one side of the n gon. The inradius of the n gon is the altitude of that triangle. Thus, r=S/(2tan(180/n)). This means the area of that triangle is A=rS/2=S^2/(4tan(180/n)). Therefore, the area of an N gon with side length S and side number n is A=nS^2/(4tan(180/n)).

The perimeter of a n gon is p=nS. Therefore, p/n=S. Thus, A=p^2/(4n*tan(180/n)).

With p held constant, the limit of A as n approaches infinity converges upwards to the Area of a circle with radius r.

Therefore, a circle is the optimum shape for an n gon of a given perimeter, and the more sides a regular n gon has, the more closely the it approaches the greatest possible area it can have for a given perimeter.

[Oblarg]

Originally Posted by DampRobot

Challenge accepted!

In a regular n-gon, the central angle for each side is 180/n degrees, and if the ngon has an inradius r, the side length of each side is S=2r*tan(180/n). I proceeded to construct a isosceles triangle where the center of the n gon is connected with equal length legs to one side of the n gon. The inradius of the n gon is the altitude of that triangle. Thus, r=S/(2tan(180/n)). This means the area of that triangle is A=rS/2=S^2/(4tan(180/n)). Therefore, the area of an N gon with side length S and side number n is A=nS^2/(4tan(180/n)).

The perimeter of a n gon is p=nS. Therefore, p/n=S. Thus, A=p^2/(4n*tan(180/n)).

With p held constant, the limit of A as n approaches infinity converges upwards to the Area of a circle with radius r.

Therefore, a circle is the optimum shape for an n gon of a given perimeter, and the more sides a regular n gon has, the more closely the it approaches the greatest possible area it can have for a given perimeter.

This is one portion of the correct proof, but a crucial (and rather more difficult) piece is missing:

This proves that a circle has a greater area than any regular n-gon, and that the area of a regular n-gon approaches that of a circle as n increases. It does not show that a circle has greater area than any n-gon, regular or irregular. So, to complete the proof, you must show that a regular n-gon has the greatest area of any n-gon with a fixed perimeter.

Actually, if we're going to be really precise, there's an additional step in showing that if a circle has greater area than any polygon with fixed perimeter, then there can be no shape with fixed perimeter with greater area than a circle, but this follows rather trivially from the fact that we can easily approximate any (convex, piecewise-smooth) shape to arbitrary precision with an inscribed polygon.

[BBray_T1296]

I would much rather stick to an octagon than bending 3/4" plywood around a 17.825353626292277606114981497722" inside radius




(since we are getting so precise that we say a circle is more useful than any reasonable N-gon, I figure you want to be spot on 112" perimeter)

[Oblarg]

Originally Posted by BBray_T1296

I would much rather stick to an octagon than bending 3/4" plywood around a 17.825353626292277606114981497722" inside radius




(since we are getting so precise that we say a circle is more useful than any reasonable N-gon, I figure you want to be spot on 112" perimeter)

If you find a supplier who'll sell you plywood with 32-sigfig tolerance on the sizing, let me know.

[BBray_T1296]

Originally Posted by Oblarg

If you find a supplier who'll sell you plywood with 32-sigfig tolerance on the sizing, let me know.

I know. I still wouldn't want to bend plywood to a 18" radius