Monday, March 2, 2015

Floral Dodecahedral Space

Floral Dodecahedral Space, an original model by me.  The units come from Meenakshi Mukerji,  specifically, these are units from Flower Dodecahedron 3.  If it seems like I make a lot of stuff by Mukerji, it's probably because I have a lot of her books.

This model was inspired by cosmology.

See, I was reading a bit on the geometry of the universe, and one proposal is that the universe is shaped like Poincaré Dodecahedral Space.  I have a lot of trouble understanding what the hell this geometry is.  I think it's the quotient group of SO(3) and the icosahedral group?  It's been a long time since I've taken group theory.

Anyway, I found this column from the AMS (academic access required) which attempts to explain it.  I'm not sure how successful the article was, but when I saw this image, that's when inspiration struck.

It's got something to do with Poincaré Dodecahedral Space?

My reaction was, waaaait a minute!  You can have 6 regular pentagons symmetrically arranged while sharing a single corner? You can put 4 dodecahedrons together at a single vertex??  How did I not know this!?

As it turns out, you can't really put 6 regular pentagons together.  The angles aren't quite right.  I think this has something to do with the fact that Poincaré Dodecahedral Space is curved space, so the angles don't necessarily all add up.

Anyway, origami sort of exists in curved space too, because you can always fudge the angles a bit.  And that's the inspiration for my model, Floral Dodecahedral Space.