Geometric shapes

As part of the module for which I am creating patterns, I decided to take a step back and look at mathematicians and philosophers work on this subject before going further with my design creations.
In our lecture we looked at 3 dimensional objects and solids, which is also what I am interested in for my tiling project. The brief is to create 12 patterns tiling a rectangle without gap, 6 of them using 4,6, etc. sided shapes and the other six using unevenly sided shapes. What I am looking to tile at the moment is the irregularly shaped Leeds Carnegie Stadium:

 

To properly be able to tile this complex shape however, I have to understand different shapes; especially when 3 dimensional objects. For this I am looking at polyhedra.

 These polyhedra are known as the 5 Platonic solids, carved stones were found in these shapes dating back around 4ooo years.

The further class of polyhedra consists of 13 solids, known as the Archimedean solids.

I find these shapes fascinating, as they allow to tile even irregular shapes gaplessly. Possibly going further even another step, we also find these shapes in nature. Different geometric shapes which are simple on their own, but through reflection, fractioning and repeating create amazing designs.

The illustrations below were done by the biologist Ernst Haeckel for his book Kunstformen der Natur. These shapes are "complex crystalizations of minerals and appear as the skeletal remains of several species of amoebic sea creatures in the Radiolarian phylum.", being only one example of the variety of geometric forms in nature.

Looking at these forms and drawing them myself I will hopefully get closer to understanding the structures of polyhedra and be able to apply this knowledge to my pattern designs- as a deep understanding of these is what makes a design systematic, structured and hence successful.