How to create an algorithms for fitting a network of tubes with a range of diameters into a cube, with maximum volume in the tubes? Assume:
- 30 cm minimum square side
- 1 cm minimum pipe diameter
- 1 mm pipe wall thickness.
- Cube edge length $L$
- number of different diameters $n$
- volume Pipes $= \pi r^2\times\text{length}$.
The tubes form a network in the sense of a continuous network of connected Tubes.
Tricky parts:
- Finding the ideal number of different radiuses of pipes to use.
- accounting for curves in the pipes. May be worth assuming straight pipes with U-turns at the ends? Obviously square pipes would be ideal. But we are restricted to circular pipes.
Best to break down the problem into components:
- assuming (one) 1 cm diameter pipe, how to fit most volume pipe into 30cm cube.
- assuming (multiple) diameter pipes: 1 cm and 5 cm diameter pipes, how to fit most volume pipe into 30 cm cube.
- how to determine the optimal number of diameters
i. given limited number or radii,
ii. given unlimited number of radii.