GORGEOUS 3, 4 AND 6 FOLD SYMMETRIES
THE "FORBIDDEN" 5 FOLD...
A regular pentagonal tiling is impossible
THE “IMPOSSIBLE” 5 AND HOW “NATURE” PUZZLE THE GREATEST MATHEMATICIANS.
“Penta” - five - fold was considered a forbidden symmetry until few years ago. As you can see in the animated gifs above, you can fill a space with squares and triangles there is no space left. Honeybees know to do it with hexagons. It is impossible to do it with pentagons. That is why the pentagonal structure (and the golden ratio strongly related to it), was considered impossible to find in crystals.
There was a hole without rabbit for our ratio in the mineral kingdom. With time, a lot of trying and calculations, medieval Islamic artists, the genial Kepler and lately the mathematician Penrose, in their own ways and with a lot of calculation, found a solution. Look at the 29th minute in the video were Roger Penrose explain why it is “puzzling” that the nature was able to solve the equation.
'IN THE SEVENTIES I COULD NOT IMAGINE HOW NATURE WILL BE ABLE TO DO IT" (to fill the space with a fivefold system ) Roger Penrose
The reason he considered it impossible, is that even if nature was able to find the correct way of tiling, during the growing process it will have to do intelligent choices. And if it made only one wrong option, THERE WILL BE NO CRYSTAL.
Denis Gratias research director for CNRS (France). Explain: More we understand these tiles and less we understand how quasicrystal grows… Serious math are needed to understand the rules of the tiles… For us, to succeed filling the tile means making choices, trying to continue , noticing when the previous choice was wrong , going back to the choice , trying the other option ... etc.… until we find the solution. It is obvious that nature does not do so. At the atomic level, the number of choices become astronomical. He conclude saying that he have absolutely no idea how nature is able to do that.
However, In 1982 THE IMPOSSIBLE CAME TRUE. Dan Shechtman found this tiling in an alloy of aluminium and manganese. The "law of the forbidden symmetry" has been transgressed. It was considered an “heresy” and Dan Shechtman was considered an “heretic” by the scientific community. One scientist considered one of the most important of history says: "There is no such thing as quasicrystals, only quasi-scientists." Very few in the scientific community had the courage to follow him down this rabbit hole. After years of denial and mockery, Dan was finally awarded with the Nobel Prize in Chemistry in 2011 for this discovery and the real definition of what a crystal is has to be changed in the textbooks.
One article was entitled: “Golden Mean Wins Chemistry Nobel Prize”. It is really amazing to see that not only the Golden ratio, but also the Fibonacci sequence are both at the very base of this structure. As if it was not enough, you can even deduce a method of constructing in higher dimensions (higher then three) from this humble “quasi” crystal.