More cd turbines and tessellations

 

1. Another form of the 4-blade tessellation discussed in the previous post. Here it is presented as a monohedral tiling.

2. So I had made a 4-blade tiling and was curious as to what could be done with a 3 blade version.

3. Initially, I couldn't figure out how to make an interesting tiling out of this, and I started laying it out in 2d cad with circles. But even that was confusing, and I came up with this pattern, which substitutes rectangles for circles. The different hub colours represent hubs with clockwise or anticlockwise blade placements. This bought to mind a sort of nexorade polyhedon that could be made with playing cards. I couldn't find any reference to these on the net but found a few pics on Joe Rohan's pinterest. Will report more on this later.

4. Here is the same pattern with sneaky infiltration of a hex hub.

5. Here is the same arrangement as 3 with circles and no hubs, it is a dihedral tiling.

6. With hubs, 5 looks like this, something that is makeable.

7. And I went big on this! Initially I had gathered up twelve 27" bike wheels / rims with the idea of making each rim the face of a dodecahedron. But when the overlapping circles idea came up I put this together instead.

8. There are 3/16" screws holding the rims together. The side walls of the rim are tucked into each other.  

9. A for-real version of the pattern shown in 6, and using the red and white hub parts shown in 2

10. Another view.

11. Obverse

12. Seen from this angle, 1/3 of the cds in the tiling make a series of steps. This is something that could be explored in 3d cad with different angles and shapes in the same sort of arrangements.

13. Another view

14. Here is my attempt at a qualitative generalistion of cd turbines /  tessellations. This page can be used as a reference, and deals with 2d circle packing. What happens when flat cylinders are forced together? Assume the cylinders' central axes are on the vertices of a shrinking regular polygon, and the preference is for the cylinders to remain in a flat plain. A: The cylinders are apart. B: Cylinder edges touch, and the cylinders have to twist or move to get closer. One tidy option is for all cylinders to rotate about the red lines C: Choosing the option of twisting along the red lines, these are the resulting patterns for very flat 2d cylinders which can be modeled using paper. D: Thicker cylinders can't reach the centre point and this series of sketches shows the hard limit when the cylinders are twisted through 90 degrees.

Hi

This post is continuing my at-home, in-lockdown exploration of some 2d and 3d  geometry from here and here. Will post more later, and welcome your comments and emails! Contact details are at the bottom of my home page.

 

Regards

Steve Nurse