Friday, May 27, 2011

CACTUS GEOMETRY?

Disregarding crest, spines, and rootball, if one were to model a cactus as non-convex prism (say a heptagrammic (7-2) prism for a seven ribbed cactus) how could one write an equation to predict changes in surface area as additional 'ribs', each of which is composed of two faces, are added to the prism? Here is the tricky part, in this cactus prism the core circumference where the ribs differentiate and the outer circumference where the ribs terminate must remain consistent regardless of rib count. I've provided an illustration that I hope clarifies what I am describing. 

Or, for the sake of simplicity, a two dimensional model would be completely sufficient to describe a cross section of a columnar cactus. So how could one predict the changes in perimeter as additional points are added to a series of star polygons, each of which is inscribed upon a consistently sized annulus between two concentric circles? Any help writing such an equation would be greatly appreciated. Additionally, would one expect the increase in perimeter to be linear as additional points are added to the star polygon?
Thanks to Beavis for answering this question with the following equation, which has been adapted to describe the surface area of a non convex star prism sans polygon faces:

11 comments:

Beavis said...

I just wanted to be first, so this could be wrong:

r2 = radius bigger circle
r1 = radius smaller circle
n = number of spikes

perimeter(n)=(((r2-r1)^2+(tan(pi/n)*r1)^2)^1/2)*2n

Beavis said...
This comment has been removed by the author.
Beavis said...
This comment has been removed by the author.
Beavis said...
This comment has been removed by the author.
Beavis said...
This comment has been removed by the author.
Beavis said...

perimeter(n)=(((r2-(r1*cos(pi/n)))^2+(sin(pi/n)*r1)^2)^1/2)*2n

Tom - Chicago said...

I asked Vi Hart @ vihart.com if she could help with one her unique videos. Mentioned that you two might see inner eye to inner eye...

Hamilton Morris said...

Breavis, that is very cool and would imply a linear increase in perimeter, but it seems to be giving me numbers that are much too low for a four pointed star polygon (I tried manually measuring a few to get an idea of the relationship). This is probably my fault and I will have to play around with the equation more. Additionally can that equation be adapted to describe the surface area of a prism either with or without the two star faces?

Hamilton Morris said...

Nevermind it works perfectly, very cool equation. Multiplying by cactus length will give surface area of the prism sans polygon faces.

huolesoh said...

hey Hamilton, do you think this plant that's said to produce "auditory hallucinations" could contain DiPT? http://www.erowid.org/plants/sinicuichi/sinicuichi.shtml
5-meo-DMT was considered synthetic until it was found in nature...

Hamilton Morris said...

Yes the same is true for DMT and methamphetamine both were discovered in laboratories before they were discovered in plants, yet I do not think sinicuichi contains DiPT or any other tryptamines because (a) it has been chemically profiled and they were not detected and (b) the effects of sinicuichi do not resemble those of DiPT in any way.