Friday, 21 January 2011

Symmetrical and chiral units of origami polyhedral model Ext 1

Recently I'm making a new polyhedral model by using a new unit, I don't know its name so just call it "polyhedral unit".
Some basic properties about the pol unit:
1) length-width ratio: 1 : rt3/2 (it's determined by (i) folding along center y-axis (ii) flip right-bottom corner to the center axis along the line from left-bottom corner)
2) 2 plugs, sockets in rhombus form
3) Representation: two triangular real, flat face on the polyhedral
4) Curvature: flat for single triangle and 6-ring
When we compare this with the Sonobe unit, we can find that pol unit has multi use: either form a triangular face itself, or forming polygon faces with positive curvature (this is the only function of Sonobe unit).

Then we'll have the following question:
Why can't Sonobe unit forms a single face itself? Both Sonobe and pol unit are chiral, but mirror unit of Sonobe unit is not needed to compose a polyhedral, why pol unit needs mirror units? What's the curvature limits and are there any other l-w ratio to compse other curvature, say making a flat pentagon?

1) Sonobe can't represent single face.
This is because polyhedral model requires regular polygons are faces. If irregular, it's easy to prove that the polyhedral can't close up without little help from elasticity of origami paper.
Single socket of Sonobe unit is right-angled icos. triangle, which don't form a regular faces except square. Therefore cube (dual of tetrahedron) and extended version of cube (all 4-ring, which is flat)
In divide-dual analysis, polyhedrals can be represented by triangles only, where each polygon is divided intro triangles, adjusted to a certain curavture (or flat) and this is the polyhedral that Sonobe model can form.
For 3-ring, they form triangle face which like the pol face, for 4-ring, it's flat. For 5-ring and 6-ring, They forms "external ring", which represents the vertex, or the base of a polygon. It's impossible for Sonobe unit to form 7 or higher rings with a strong bonding unless forces are given to hold the shape.

2) Sonobe model doesn't need mirror unit but pol unit needs.
We should identify a fact that, when Sonobe unit is made up to a shape of pol unit (by 5 Sonobe unit), we can see that the Sonobe unit are symmetrical while pol unit aren't. Therefore we need mirror unit of pol unit is needed.

3) Curvature
It forms an equilateral triangle, so by the formula as shown in by project,
C = (cos ^2 30 - sin ^2 30)^0.5 / 2sin30cos 30
    = (2/3)^0.5
and cumulation value can tends to infinity.

to be cont.

In order to extend my project done before I will try to ellaborate more on cumulation pattern, mirror rings pattern and flat pattern.

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