What is Symmetry? Polygons, polyhedra, kaleidoscopes and honeycombs.

The idea that patterns and symmetry lie at the heart of the structure of matter dates all the way back to classical Greece. The philosopher Empedocles suggested that matter is formed of four elements: Earth, Air, Fire and Water, and in the fourth century BC the philosopher Plato took this proposal and produced a possible geometrical theory of matter based around it. In his dialogue the Timaeus, Plato proposes that each of the elements: Earth, Air, Fire and Water has fundamental components that are in the shape of the regular polyhedra. And this is why the regular polyhedra are now known as the Platonic solids. In Plato's scheme each of the elements was identified with tiny components in the shape of one of the regular polyhedra. So Fire was composed of tiny tetrahedra, Air was composed of tiny octahedra, Water was composed of tiny icosahedra and Earth was composed of tiny cubes. And the fifth regular polyhedron, the dodecahedron, Plato identified with the structure of the cosmos as a whole. The idea seems to have been that the heavens were formed of a fifth ethereal element unlike the four terrestrial elements. This idea was later taken on by Aristotle. The fifth element or quintessence has a sort of spiritual dimension. This idea of Plato and later Aristotle doesn't really bear any relationship to the true structure of matter but it was very influential down the centuries. It influenced many philosophers and early scientists most notably Johannes Kepler. Now this is quite a remarkable idea that symmetry and patterns lie at the heart of nature, because when we look around us there's not much sign that this is actually the case. Nevertheless, symmetry lies at the heart of modern theoretical physics. The whole of physics is really understood in terms of symmetry and patterns.
So what is symmetry and what do we mean when we say that an object has a certain symmetry. Well the simplest illustration is with a simple object such as the regular hexagon. If we rotate a regular hexagon around its centre through 60 degrees, the regular hexagon looks exactly the same after the rotation as it did when we started. So we say this is a symmetry of the regular hexagon. And similarly if we rotate the hexagon through multiples of 60 degrees we again find that the hexagon is exactly the same. So these are again symmetries of the regular hexagon. Another example is a reflection symmetry. If we place a mirror connecting two opposite vertices of the regular hexagon and reflect the hexagon in the mirror, then one side of the hexagon is swapped over for the other side of the hexagon and vice-versa. So the regular hexagon looks exactly the same, so this is again a symmetry of the regular hexagon. My first book Higgs Force includes all the historical and scientific background leading up to the discovery of the Higgs boson at the Large Hadron Collider. It includes a survey of how we've come to understand the structure of matter and particle physics and our search for patterns and symmetry in the laws of nature. In the first chapter, to illustrate the idea of symmetry I used the kaleidoscope which is a toy that was invented by the Scottish physicist David Brewster in the 19th century. It was an instant massive success and loads of kaleidoscopes were sold. But unfortunately Brewster himself didn't make much money because his idea was copied by lots of other people who are much more businesslike than he was.
Brewster's original kaleidoscope was a tube containing two rectangular mirrors set at a 60 degree angle so that beads at the end of the tube were reflected to produce a pattern with hexagonal symmetry. I've created this animation using a virtual kaleidoscope with the same symmetry as the original kaleidoscope of Brewster.
Many kaleidoscopes such as those that you might buy in a toy shop today include a third rectangular mirror with the three mirrors arranged so that they have a cross-section that is an equilateral triangle. These kaleidoscopes produce patterns with the same symmetry as a tessellation of regular hexagons or equilateral triangles.
It's possible to develop the theme of kaleidoscopes further. I've produced a series of three ray-traced animations using virtual kaleidoscopes formed of four triangular mirrors and these mirrors are arranged in the form of a tetrahedron with the virtual camera actually inside the tetrahedron.
There's more information about kaleidoscopes and how these computer-generated animations were produced on the interactive companion to Higgs Force. This is a CD-ROM called Higgs Force Interactive and it includes a wealth of other material as well including other animations and puzzles and text that goes a bit beyond what is in the book itself.

[@evangelineahad5537]

It is a beautiful representation of what a kaleidoscope can do, a toy I knew as a child after WW II in 1947, I was 5yrs old. Everything was in short supply in our post war destitution, and this toy was one of the few entertainment we had. We had no idea why those shapes converged and changed by turning the carton tube around. It is your magnificent kindness to enlarge my memory of our simple joys at the time.

[@hindkhatib8692]

Dear Nicholas . It is very beautiful animation and it is very clear and beautiful .Thanks for this great job wish you all the best