Volume for logarithmic spiral

[nstansbury]

Hi,

 

I am looking to take a logarithmic spiral defined by [math]\varphi[/math] and calculate it's volume as a "cone". See: http://en.wikipedia.org/wiki/Golden_spiral

 

I know it's effectively a fractal so it tends to [math]\infty[/math], but within a reasonable level of accuracy I should be able to calculate its' volume, but its' flat side I think always throws my calculations from the value I'm expecting by a small value.

Edited July 3, 2008 by nstansbury

[Al Don Gate]

Hi,

I know it's effectively a fractal so it tends to [math]\infty[/math]

 

You know, it's not because an object is fractal his volume tends to [math]\infty[/math]. I think particularly to the sponge of Menger :

 

[url=http://en.wikipedia.org/wiki/Menger_sponge][/url]http://en.wikipedia.org/wiki/Menger_sponge

 

But I don't answer to your question...

[Dave]

Hi,

 

I am looking to take a logarithmic spiral defined by [math]\varphi[/math] and calculate it's volume as a "cone". See: http://en.wikipedia.org/wiki/Golden_spiral

 

I know it's effectively a fractal so it tends to [math]\infty[/math], but within a reasonable level of accuracy I should be able to calculate its' volume, but its' flat side I think always throws my calculations from the value I'm expecting by a small value.

 

Could you elaborate on precisely what you mean by "cone"?

[nstansbury]

Could you elaborate on precisely what you mean by "cone"?

 

Ok, so take this spiral as a 3 dimensional object and extend the centre point outward perpendicularly to the rest of the spiral.

 

You'll effectively end up with an infinitely decreasing "pig tail" cork screw, whose volumes should be able to be described as a "cone".

 

In a spiral defined by [math]\varphi[/math], each quarter turn of the spiral the radius reduces by a factor of 0.61803 (90o), so for every 360o the imaginary "cone's" radius reduces by a factor of 2.47212 for [math]n[/math] height.

[Deja Vu]

I'm kinda lost on you here, can you please elaborate some more on this? I don't think the logarithmic spiral would look like a cone when you add a 3rd dimension.