Platonic Solids ..... Corners

I found some wooden sticks in the supermarked intented for growing plants.

Nice, lets try something different. For another project I needed a circus tent, but everything also would be usable for platonic solics, too. (see wikipedia: https://en.wikipedia.org/wiki/Platonic_solidhttps://en.wikipedia.org/wiki/Platonic_solid) 

My wooden sticks have a diameter of 4mm. So I've designed the corners for it. 

I wanted to “describe” all parts in the same way, so I thought what are the different properties of such a corner.

• Inner diamiter: The wooden stick needs a matching hole?
• outer diamiter: How much larger the does the outer diameter need to be?
• Length of the hole … and the outer lenght towards the center?
• How many spokes does the object have an how far has it to be bended downwards?

A total of 6 parametes.

The first 4 highly depended on the base material, the wooden stick and the intented size

The number of spokes and the angles are defined by what you'd like to achive.

For the platonic solides:

• tetraeder: 3 spokes, … angle 54,736°
• octader: 4 spokes …. angle 45°
• cube/hexaeder: 3 spokes … angle 35,264°
• icosaeder: 5 spokes … angle 31,717°
• dedocaeder: 3 spokes … angle 20,905°

Angle calculation:

If you look from top to one corner, then you have 3 to 5 spokes. We have to calculate the radius of the projected circle of that view.

For a triangle the tangent has to be as long as the spoke. With half the angle (a) of the projection we have to get the half length of the spoke:

radius = 0,5*spoke*sin(120/2) = 0,577*spoke

With the radius and the spoke length we can get angle b or b'.

b = arcsin(radius/spoke) = 54.736°          or        b' = 90-b

Other:

• circuis tent: xx spokes yy angle
• wheel: xx spokes, 0 angle

Enjoy and modify as the basic design is done in OpenSCAD.

Please be not disapointed, the core function/module for only consists of 15 lines for all derived corners ;-).

Note: I've not looked at other models before I've created this. It was a geometry, trigonometry exercise for me.