Can I tessellate the "Golden Section"?

Through reflecting on my research to date, I chose to investigate the concept of spirals  – would it be possible to tessellate a spiral design?  I couldn’t find anyone else who has successfully achieved this idea.

I looked at a fir cone in my studio and focused on the spiraling shapes.  Then I started to research the Golden Section and the Fibonacci sequence.  0,1,1,2,3,5,8,13,21,34,55,89,144….

I drew this pattern in cm on paper and tried to fold along the lines…

I tried different variations of folding “mountains” and “valleys” but could not make it successful because the lines did not meet (as indicated)

But a blog I found another number sequence devised by Padovan (1994) who credited the discovery to Van der Laan(based on mortal rabbits)

http://matheminutes.blogspot.co.uk/2012/02/killing-fibonaccis-rabbits.html

This triangular pattern might “tessellate” more easily?

I began by roughly drawing out the pattern and pushing the creases in different directions to see how it naturally folded.  The result was it folded a bit reluctantly and towards the corners it wanted to bend, but I thought I should test it in a rigid material to see how it might behave.

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