In my last post I showed how the arrangement of dots into a helical origami can be presented in a very neat way, where a circular ribbon of a pair of dots can be folded into a circular rod, where the dots are no longer in a flat ribbon, but rather as pairs of dots, where each pair is at a 90° angle to its neighboring pairs. And this origami ribbon, when simplified to equilateral triangles looked like this:
While the above scheme is surely correct, it’s rather inconvenient that it is at such an odd angle. The reason for this is that I chose the starting point with no regard to how the folding would cause the origami to grow.
As we can see that the x-y projection is tilted 30° clockwise, the first thing to straighten the origami is to tilt it 30° counterclockwise around the z-axis. Doing this, we get the below origami.
But this isn’t yet good enough. The x-z projection is still tilted 30° counterclockwise (I might have mixed the angle of the tilt, but you get my point). The second thing is to tilt the origami 30° clockwise around the y-axis. And we get this:
The curious thing about the origami is that while we again see the starting point, this arrangement of the folds makes the structure look too abstract. The origami structure is somehow hidden in plain sight.
But now that we’re in abstract mode, we still need to nudge the origami by half a vector length towards the origin along the y-axis. Now all of the points fall along a cylinder centered on the x-axis.
What this means is that from this point onwards the y coordinate squared plus the z coordinate squared is the radius of the cylinder squared.
And this means that the movement of each dot can be described as a helix, just like with light. And looking at the x-y-projection of a helix moving along the x-axis, what one observes is a wave. Just like with light!
But how to bend this already helical trajectory into a circle, I’ll have to work a bit more with my vectors. I’ll keep you updated.
Comments