Remember the blue and green hyperbolic plane made of triangles? In every corner, seven triangles meet. Because a flat plane can be made with six (regular) triangles meeting in every corner, and seven is more than six, there’s too much fabric everywhere to keep flat and the resulting ‘plane’ surface (thanks, Wilfred!) has a saddle-shape.
I’m still working on this thing, and after I ran out of green yarn, I continued with the orange-yellow stripes I also used for the other hyperbolic thing.
As if all these colours aren’t wild enough, I went a bit crazy with the shape of it and let the plane surface meet itself on a few places, turning the original flared tube into a ridiculous complicated messy more interesting structure.
Doing that probably ruined any real hyperbolic-ness that was there (is there a real mathematician in the house?), but at least locally, it has the same ‘negative curvature’ I started with. In every corner, seven triangles meet…
Filed under: Crochet, Design, English, Hands-on math & science
Try the word surface, and certain types of curvature do not depend upon the precise embedding into three dimensional space. Is your surface still orientable?
Ah, yes of course, a surface.
It is still orientable, with two different sides. Easy to see in real life as (with these stitches) the fabric looks different on front and back.
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This is looking so fab, I love it, I really need to look into doing some hyperbolic creations to decorate my sewing room!
Thanks, Fern!
I am just starting out. I am wondering how they get so convoluted. They are so natural looking and beautiful.