02- Tilings & Symmetries - Paver un sol - 02

1 Tiling techniques

Can we cover a floor with tiles of any shape without gaps or overlaps?
Many shapes work, but not all as, for example, a regular pentagon. Tiling patterns, which repeat periodically by translations, are well understood and their symmetries allow 17 different types of patterns.

The study of these types and their symmetries is based on the Group Theory devised by Evariste Galois. If we want to tile more freely -not periodically- the study is far from being finished.
So, is it possible to tile using only one shape? It's a mystery!
Tiling patterns find applications in mathematics, crystallography, codes, particle physics...

• Evariste Galois (1811-1832)
• Sir Roger Penrose (born in 1931 in Colchester)

2 Is nature symmetric?

Why does the double helix of DNA always turn in the same direction?
Why is a human face and its reflection in a mirror not superimposable?

From the infinitesimal to the infinitely large, symmetries are shown in many mathematical models. But nature rarely presents perfect symmetries. Some elude us, others are convenient to assume they are perfect.
Living forms which turn to the right are much more frequent. This bias towards asymmetry may be explained by chance or by the asymmetry of physical forces: the question remains open.

3 Where am I ?

How many satellites, in orbit around the earth, are needed for us to know where we are all the time ?
Three are enough: they measure their distance to the tracked object (a fourth gives a time correction which improves precision).

The object to be located is fitted with a portable receiver which communicates with the satellites by electro-magnetic waves. Its position is at the intersection of 3 spheres each centered on one of the satellites and their radius is the distance from the object.
The GPS (Global Positioning System), the Russian systems -and soon the European Galileo system- allow us to know where we are all the time.

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