Symmetry and stick polyhedra

Tetrahedron model on a template with a complete set of edges drawn in.

Net of the tetrahedron showing sticks as solid lines, and virtual edges dotted.

Small model tetrahedron with stick model and template for making sides.

Side view.

 

Tetrahedron is a knot made out of an eulerian circuit.

Hi

In the last few days I have been thinking about, and later making some rather minimalist polyhedra which have interesting properties of symmetry. Here is my abstract on these pictures:

"This work shows the construction, form, symmetry and topology of some equilateral -triangle based polyhedra including triangular octahedra, tetrahedra and icosahedra. The polyhedra use craft sticks, drinking straws, and 3d printed jigs at face centres. The constructions depend on rules of face symmetry. Rules for faces of octahedra  - 4 edges join at vertices - vary from those for tetrahedra and icosahedra where 3 and 5 edges join at vertices. Structures take on different forms depending on configuration (side composition and arrangement of pairs of sides) and can be knots from a single Eulerian circuit. Small models of the polyhedra help with visualisation. Curiosity about symmetry led me to explore these shapes, resulting in interesting forms and topologies. 

 Without further ado, here is a brief summary of the pictures.

Tetrahedron

 This was the first model I made, and only one side configuration is available - configurations consist of pairs of sides joined along edges. Placing the shape on the template used to make sides, and marking in other sides helps to identify the shape. 

 

Net for current icosahedron

All the sides before assembly

Icosehedron side view. This is also an Eulerian circuit and knot.

icosahedron top view with model showing configuration of sides

Another view

 

 

Proposed alternative side configuration

Net of alternative configuration

 

Icosahedron

This was the second shape I attempted, and it came together easily and looks interesting - its another Eulerian circuit. One configuration has been made and I've worked out another. 
 

 

Octahedron with one of its sides and a small model showing arrangement of sides. It is made from asymmetrical sides, but by mirroring them as per the net below, the polygon can be formed. 

Octahedron net

 

I believe this is the only other alternate octahedron. It could be made from symmetrical sides but what about asymmetrical sides. Don't know yet!

Octahedron

 

At least one form of octahedron can be made from asymmetric sides. I don't know about the other one but using 2d cad I should be able to find out.

 

That's all for now!