Apparently, mathematicians had long thought that you couldn’t build (I guess: embed in our normal euclidean three dimensions) a hyperbolic surface, even though they only had to look at a sea slug or even the lettuce on their plates to know better.
As a follow-up on my earlier post with pictures of a yellow crocheted bead and cube (which at some point got quite some visitors from a knitting and crochet group on yahoo, if I remember well) I present you another cube in crochet.
This time with pictures of the making-of! Let me know if you made one yourself.
De kijkcijfers van dit blog worden de laatste tijd voornamelijk gestuurd door de populariteit van een bepaald haaksel-postje. Vandaar onderstaande foto van een al wat ouder projectje.
Oplettende lezertjes missen natuurlijk de 9, maar na de 6 is dat natuurlijk een triviale oefening….
Heel erg cool, via Marcel Tunnissen, mijn oude buurjongen die mooie, erg ingewikkelde modellen van veelvlakken maakt:
De jaarlijkse Bridges Conference (on Mathematics, Music, Art, Architecture, Culture) wordt dit jaar van 24 tot 29 juli gehouden in Leeuwarden, de geboorteplaats van MC Escher. In Nederland dus!
Ik zou er wel heen willen, maar een paper schrijven voor 1 februari is nogal… een uitdaging, zeg maar. Hm. Maar misschien willen ze mijn wiskundige haaksels wel op de tentoonstelling hebben?
Behold the face of the eeeeevil hyperbolic monster:
So how did this hyperbolic monster come to life, you wonder? And why is it hyperbolic?
Well, I made it, using the same blue yarn I used for Sylvie’s hat. This time I made a hyperbolic surface by decreasing 1 stitch in 10, starting with a circle of about 100 stitches round. This resulted in a pointy little hat. I then made two more of those hats, the three of them joined together at the edges. This results in:
Turn the whole thing upside-down and push one of the joining edges in to form a nose, et voilá! The hyperbolic monster is there.
Last weekend I went to visit my friends Sylvie and Alex who work in the Max Planck Institute in Stuttgart. And I brought them, or rather, her, a present: a 3-4-5-hat!
(Grapje voor de Nederlandstaligen: in plaats van “1, 2, 3, 4, hoedje van papier” is dit dus een “3, 4, 5, hoedje van katoen”.)
I made it from dark blue and blue-striped cotton yarn, colours that go pretty well with her sweater (which was a coincidence). This picture was taken over Irish coffee in Café Einstein.
I explained how I made it, and that the structure of the hat is sort of in between a regular dodecahedron (12 pentagons) and a icosahedron (with 20 triangular sides). Then Sylvie made me a big compliment by remarking that I could have been an experimentalist! :-)
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 ridiculouscomplicatedmessy 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…
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