# Ball of string

## Main Question or Discussion Point

Can one derive the approximate ratio between space and material in a ball with diameter B, wound of incompressible string with diameter s, where B>>s?

Would the string be randomly tangled, or wound in a precise pattern?

HallsofIvy
Homework Helper
Here's the easy way: knowing the length, l, of string used, calculate its volume as $\pi s^2l$. The volume of the ball is $(4/3)\pi B^3$. The "amount" of string is $\pi s^2l$ and the "amount" of air is $(4/3)\pi B^3- \pi s^2l$.

I was looking for an (approximate) relation between B, s and l so that only the variables B and s would be needed to solve the problem. The string would be wound so as to minimize the space within.

Also, string is taut.

I would build the sphere layer by layer, in each layer the string spiraling out to form a circle with small thickness. Every layer follow the "grooves" of the spiral in the lower one.

Do any of you have any suggestions as to how I might setup parametric equations for a ball of string?

For an Archimedes spiral it is easy but when I move to 3D I get lost.

I would like to plot my ball of string in matlab. I'm happy to plot the string as a space curve (I don't need to show the string's diameter,

Actually my real goal is to fill 3D space with an expanding search path. Akin to spiral search in 2D.

I'm hanramo a t ho tmail com

Perhaps knot theory could help?