A colleague of mine had a really nice backwall light art and I asked him what was it. It turned out to be a design based on nanoleaf light panels

Each triangular panel can be connected to build nice patterns. Each panel also can switch colors and it is all programmable.

I figured I could build the Auth0 Logo using these so I ordered a kit. In this post I cover how the basic flat geometry will not work, and what to do to solve it.

### Auth0 Logo Geometry

The Auth0 “shield” is built around a regular pentagon with all sides of the same length. Or you can also build it with 5 paralelograms around a “concave decagon” = a “star”. And a paralelogram can be decomposed in 2 triangles. The angles of a star are:

### The problem

The problem is that each triangle for the paralelogram we need is not equilateral, otherwise the angles would be exactly 60°. We need a way of turning equilateral triangles into exactly what we need. And cutting is not an option for obvious reasons :-).

### Enter 3D!

If you don’t limit yourself to a plane, you can turn the equilateral triangle into the shape we need through projection:

We need to find the angle β that will make the base 54° (from 60°).

From above:

$cos(\beta)=h/L{}'&space;;&space;\tan(54)&space;=&space;2h/L;&space;L{}'=L\sqrt{3}/2;&space;\tan(54)&space;=&space;\sqrt{3}\cos(\beta);&space;\beta&space;=&space;\arccos&space;(\tan(54)/\sqrt{3})$

Which then yields:

$\inline&space;\large&space;\beta&space;=&space;37.37$

### A prototype

With the above info, I built a simple model using the always awesome TinkerCAD to render an approximation, which looks pretty good:

### Next steps

Now I need to design the support structure for the Nanoleafs, and figure out how to connect them all.

### Credits

The math formulas are rendered with Codecogs