kaleidocycle aka rotating rings of tetrahedra

Greetings everyone-

Let me start of by saying that I know GH is not the best for solid based geometry, especially folding origami-esque configurations, but I am determined to make this work.

I am using the "kaleidocycle" as a facade component, and i need to be able to move it through its entire "rotation" in 3d space to understand where and how it is moving.

https://www.youtube.com/watch?v=4owFczeqqMQ

this is what it is doing, in general. there are 2 sets of 3 hinges, rotated 180 degrees, making up a hexagonal form.

here is a rhino model of the form. i used the trigonometric properties of the isoceles triangle to make this model very accurate (63.333, 53.333, 63.333 angles), and now i need to describe the movement.

It is TOUGH. i think i have it and it just throws me for a loop (no pun intended).

I have a ghx model set up to where it can go through part of the cycle, but the inbetween states are incorrect, and therefore it's not valid, but it shows how something like this could work. The trick is it rotates on multiple axes at different times, and its just very very tricky to figure out what it is rotating around and when.

If anyone has any ideas, or insight, please please let me know. I am working on this in my masters' studies, and I'm pretty screwed if i can't figure this out in grasshopper!

Also, please find attached a research article concerning this form. I haven't been able to apply the geometric findings of theirs, yet. But it shows it can be described mathematically.

THANK YOU!!!!

benjamin

Attachments:

tetra-rhino.3dm, 355 KB
kaleido.ghx, 185 KB

Replies to This Discussion

Permalink Reply by benjamin jenett on October 4, 2010 at 12:54pm

oops! forgot the paper!

Attachments:

retractable.pdf, 2.3 MB

Permalink Reply by Daniel Piker on October 4, 2010 at 1:45pm

I would say work with just one pair of tetrahedra to begin with.
You know the arrangement has 3-fold symmetry, so you must keep the 2 outer edges of those tetrahedra on 2 vertical planes at 120° to each other.

I would fix the centre of one of those tetrahedron edges and use its rotation on that plane, about its midpt as the controlling parameter.
Given the orientation of that one edge of one tetrahedron you need to find the 2 fold angles that will make the opposite edge of the other tetrahedron lie on the neighbouring plane. Finding those 2 angles should just be a case of intersecting some circles.
Then you array the 2 tetrahedra around the intersection of the 2 planes to get your kaleidocycle.

Permalink Reply by benjamin jenett on October 5, 2010 at 4:19am

Daniel-

Thanks for the advice! I am SO close to getting it to work. I have the fixed center point of rotation on the plane (i chose the center, top, z axis plane), and determined how the tetrahedron can rotate about this axis, as well as the hinge itself.

It goes to the right end position, and really imitates the movement in a much mroe accurate way, which is encouraging, but it still deviates from the adjacent (60 deg rotated) axis.

I think if I find the 2 fold angles I could constrain the movement of this hinge to that plane, and then, in theory, that should be it!

The problem now isn't figuring out what these angles are (it should be easy enough), but what they do. How do they constrain the movement of this already very constrained tetrahedra?

Please check out the latest ghx, it should show how far i have gotten.

Thank you so much for your help... I see now why you are "the man"!

cheers

benjamin

Attachments: