Plato liked to say that ideas exist outside us. They do not need us. We do not create them. We just discover them and wonder at their sight. It is probably good sense paying attention to old Plato since many thinkers hold the view that most western philosophy could be described by adding short footnotes in Plato’s texts. This might be an extreme way to emphasize the relevance of Plato’s philosophy, but it is easy to hear similar comments nowadays since Neo-Platonism is more in fashion now than ever. Mathematicians and theoretical physicists are starting to advocate for neo-platonic conceptions of nature and science. The extremely popular writer and extraordinary scientist Roger Penrose is a clear exponent of this trend. The strongest experience of my childhood came when my teacher proved Pythagoras theorem to me. I suddenly realized that it was true in an absolute different way of all truths I was aware of at this age. It was true, but not because my teacher said it, nor because my father said it, not even because Pythagoras himself said it. It was true because I, a poor child, was saying it. I was terrified with the implied responsibility and could not fully enjoy the amazing beauty of my first encounter with an idea. Later on looking for ideas to admire has been the fun fair activity of my life. I even try to share this personal fascination with youngsters when I am given the opportunity to interact with them. Some times I try to produce a first encounter with an idea as a game. Look kid this is a funny puzzle. Take twenty equilateral triangles and glue them in 3D to get a closed thing. (Follow this link to get started on this). We build it playing and joking until the kid wonders at the miracle of the symmetry of these twenty faces matching perfectly in space. After the puzzle is “solved”, they usually ask who invented it or where I have found or bought it. I immediately grab the opportunity and answer something like: Its name is icosahedron, one of the five platonic solids. However it was not invented by Plato. In fact, nobody invented it so it has no owner and we cannot buy it. It has been out there long before you and I were born. In fact it is much older than mankind. It has no age. It is an idea or as Plato would have it, the shadow of an idea.
Next is another remarkable shadow.
A two dimensional tube using the third dimension to embrace itself in a topologically unbreakable kiss.
It is called the trefoil knot. It is the simplest example of a non trivial knot, a knot that is not topologically equivalent to a doughnut. To prove this in mathematical terms takes a while. It is necessary to deal with 2D projections of knots handling the unavoidable overlaps as under or over crossings. One must learn about Reidemeister moves as the legal transformations of these projections that preserve topology. Then one must realize that the trefoil is tricolorable, i.e. it is necessary to use at least three colours to fill its 2D projection with no two strands, segments between crossings, using the same colour. Since the trivial knot can be obviously filled with a single colour and it is easy to prove that tricolorability is preserved by Reidemeister moves, the trefoil knot cannot be topologically equivalent to the “unknot”.
Here is link to find information about a highly recommendable book to learn more on this). But in spite of this mathematical rewording, the true thing stands in front of us in the beautiful Wikipedia’s image inserted before. Without any words at all, every normal human being, after admiring it for a while, would bet his life on the fact that this thing cannot be untied. It cannot be made trivial without destroying it. It is the photograph of an idea, or rather of its shadow.
Toni Trias. Doctor in Physics. Vice-President and R&D director Grupo AIA