The Rube Goldberg Model

This model was completed in Jan., 2006.  It¡¯s a sequel to the spider-math model which although interesting, seemed unlikely as something which would ever get built in a full-scale version. The model is based on two equal-angle polyhedra with chord factors provided by Joe Clinton in his ¡°Equal-angle conjecture¡± paper. I call them ¡°equal-edge¡± polyhedra which amounts to the same thing. The model uses an inflated hemisphere (ripstop nylon) as staging for the floppy stages of construction.

Click on photos to enlarge

 

1)      The hemisphere being inflated, with a partial grid laid out flat, beginning to settle into a spherical shape. 

 

2)      The inner equal edge grid, triangulated, showing hubs with empty prongs for connecting to diagonals to the outer grid.

 

3)      This shows a phase of construction involving many fudge-factors.  Not knowing how to derive the triangulation chords mathematically, I arbitrarily decided to have the radius of the outer shell be 5¡± greater than the inner shell, and made -some 5¡± posts to support the outer equal-edge polyhedron. 

 
I think of these breakdowns as being analogous to a 4-frequency triacon for the inner shell, and a 6-freqnency alternate for the outer shell. Its fascinating, but becomes intuitively obvious, that the different equal-edge grids relate in pairs which allow for triangular connection.

4)      The chord lengths of the diagonals were set empirically by seeing what seemed to fit between the inner and outer shells. There were three lengths in the inner grid (one for the ¡°equal edge¡± length, one for the pentagon spokes and one for the hexagon spokes).
 There were two lengths for the diagonals (one connecting the pents of the outer grid to the pents of the inner grid, and one connecting the hexes of the inner grid to the hexes of the outer grid). There was one length for the outer equal-edge grid. This 6-length method is not mathematically accurate, and the model required some coaxing to fit together, resulting in some visually stressed struts.

 

5)      Here the model is complete and the hemisphere is being deflated

 
 

6)      The finished model.  Startlingly strong; pulling or pushing any point, the whole structure moves as a single piece.