I saw this topic and was facinated with the 'snail' graphic animation . The topic related to calculating the positions of the triangles.

I thought that it should be easy to avoid the complex calculations by taking advantage of the six-fold symetry and by using the matrix transformation operations available in Processing.

It is only now that I realise the post is originally 9 months ago but since I have spent some time on the problem I am going to share the resulting sketch here anyway because others might find it useful.

In the sketch (you can see / download the code from Open Processing) I have created a class called Snail. Objects of the class are created with a specific triangle size. The class has a single method draw which will draw a single triangle within the snail based on a triangle number. For instance triangles 0-5 are in the first ring and 6-23 are in the next ring and so on.