## Thursday, 27 January 2011

### Symmetrical and chiral units of origami polyhedral model Ext 2

Proposition 4
When we are using a unique origami unit (mirror unit is permitted), it has a unique cumulation value, hence it's impossible to make a totally flat non-regular polyhedron, not even a semi-regular polyhedral (i.e., unique vertex composition expression).

The proposition is correct, and is incorrect, depends on which type of model you are making.
For a normal model, all units has 3 fold lines and are used up in the model. For non-regular polyhedron, each vertex polygon expression has more than one numbers, e.g. {6,6,5} or {6,5,5}, For a specified w-h ratio it has a specified cumulative value for a given n-gon. The cumulation value behaves monotonically so we can ensure that there's no repeated roots, hence for each verteices, flat polygon can't appear in all three sides of the vertex.

It's not correct when you are making an enlarged model.
Take Sonobe unit as example on cube. Cube, is of the same nature with regular tetrahedron by giving a specified cumulation value (6^-0.5), regular tetrahedron has configulration {3,3,3} while cube has {4.4,4}.
In enlarged cube, the verteices are 3-rings while the center part is (flat) 4-rings. however 3-4 ring pattern is flat.
Also in pol unit, 5-6 rings pattern are flat, using the pattern {6,6,5}, a 27-times enlarged icosahedron is produced instead of a turncated dodecahedron.

Proposition 5
We can use unit with different w-h ratio to produce all flat polyhedral.

I haven't do any experiment on this one (since the calculation work is too much), but I believe this is yes, regardless how you put different plugs into sockets. Also This violates the spirits of origami, right?