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! 

 

1d to 3d part 3

 

1. Diagram for compact version

2. All the assemblies shown here were made on this jig

3. Compact tessellation

4. Corner connected tessellation from radial and diverging triangular units.

5. Edge connected tessellation from radial and diverging triangular units.

6. Edge connected octahedron from radial and diverging triangular units. Refer small ovtahedron, red - diverging, blue - radial.

7, Another view = Edge connected octahedron from radial and diverging triangular units. Refer small ovtahedron, red - diverging, blue - radial.

8. Corner connected octahedron from radial and diverging triangular units. Refer small octahedron, red - diverging, blue - radial.

9. Compact Octahedron

10. Compact Octahedron calcs

Hi 

Following on from my last post on this, I have worked on 6 piece sides made with an irregular hexagon jig (pic 2), introducing a new "compact" version where 4 stick ends allign along the edges of an equilateral triangle (pics 1, 3, 9, 10). The work includes making mixed (diverging and radial sided) octahedra of 2 types, drawing tessellations of 3 types, and calculating the geometry of the compact versions.  

I'm proud of this effort of drawing and calculation. When placed on the jig at stick centre position "X", the stick ends (modeled by their centrelines) assume heights H1 and H2 which can be calculated from the assumed stick length L = 172 and the jig / side geometry and the length "X". 

As a first step to getting the ends colinear, X is calculated for the colinear condition H1 = H2 = H.

With expressions for H1 and H2 which are equal, X can be found. Then by putting this X into expressions for either H1 or H2, H can be found. Calculated results for X and H match Cad results almost exactly. For maths purists, all the angles should be in radians, but I don't think it makes much sense here! For example, 10 degrees = PI / 18 radians, much more cumbersome! 

Novelty pencil airlines

 

When my friend and I were on our way home from my cousin's funeral last Friday and, we dropped in on a cemetery, one coffee shop and two Op Shops. We spent a while at the first Op Shop and I had perused the DVDs, CDs and perused and selected several books. My friend was going through all the dresses skirts and shoes, and wasn't done when I had just about seen everything I wanted to. But a bundle of pencils caught my eye they included novelty pencils with two rubbers on one end "HAMMER OUT YOU CLEANING NEEDS".

It wasn't till later when we had convened at the coffee shop later that we found out the bundle also included novelty pencils with a rubber at each end "THIS PENCIL WAS MADE ESPECIALLY FOR OUR COMPETITORS TO WRITE QUOTES” or “THIS PENCIL WAS MADE ESPECIALLY FOR OUR COMPETITORS TO WRITE NEW BUSINESS ". Of course I fiddled around with the pencils at the coffee shop trying to get them to stand up on their ends or balance them in some sort of tower. That night I passed a set of the pencils on to our friend Christine at her place over dinner. The next day I was busy but later while bored watching the TV (this was Saturday night) I started assembling the pencils together with a fold back clip and after a short amount of time came up with and aeroplane on a stand. 

This looked quite good but just needed a tail for the plane – I made one out of sticky tape (pic 1) but wasn’t satisfied. A few days later and a “proper” tail is complete along with a lame joke along the lines of what is written on the pencils. Cad pic made using Bricscad.

Regards Steve Nurse