The five point design found at the centre of an apple has what is referred to as mirror symmetry but also interestingly enough, it has properties of what is called rotational symmetry. What this basically means is that if you were to find a perfect shape in that form, it could be rotated 72 degrees into five different positions and still basically look the same after each rotation. One could say that a big drive for mathematicians is to find or rather their desire to extend ideas and apply them to other contexts thus, the everyday sense of symmetry through reflection and rotation can be extended to encompass a mathematical meaning.
That meaning being – a mathematical object being defined as symmetric with respect to a particular mathematical operation if that operation, when applied to an object, preserves some property of the object.
Now, I can already sense a few eyes glazing over, but bear with me and before you stop reading and move on to someone’s post about their cat, I can explain it in much more interesting terms.
Understanding ‘Mathematical Objects’
Generally speaking this term is used to distinguish and identify objects that exist in the real world from those perhaps a bit more abstract, ideal is the word used, ideal in the sense of them being mathematically ‘perfect’. For example as I said in a post two days ago about symmetry, even the most gorgeous girl in the world will not have a perfectly symmetrical face, in fact rather asymmetrical, even then, it does not detract from their beauty. The same could be said for a butterfly who to the untrained eye may appear to be perfectly symmetrical but upon closer inspection on can see that there are in fact, small differences in even what may appear to be the most symmetrical wing forms. These may seem rather a trifle to you and I, but to the mathematician in search of perfection it is anything but, these mismatches, however minute, ultimately impact their end goal of uniform symmetry, or perfection.
In fact, one of what can be called a lasting legacy of Euclid was that he set out to make this definition apparent, that is to say, between the real and the ideal. Much like one would now between a model and a very beautiful girl.
Geometry deals with mathematical objects such as points, lines, polygons, polyhedra and if you are familiar with this blog, and aspect that Euclid would not have been able to recognize, Fractals. In order to go about setting an extension to the meaning of symmetry we must take a few things into account, in the case of the butterfly for example, the mathematical operation that is applied is reflection and preserves the look of the butterfly to the extent that it would be almost impossible to discern if we were looking at the original butterfly or at its reflection. Similar to the rotational symmetry of the apple we discussed earlier.
From this we are now in a position to extend the meaning of symmetry, if you upscaled a triangle in a uniform fashion or contract it, in the parlance we use everyday, we would not see these different versions of the triangle as being symmetrical however the very mathematical operation of scaling up or down preserves certain properties of the triangle, for instance, the sizes of the angles do not change and the relative proportions of the sides are invariant.
Triangles are symmetric with respect to scaling in terms of properties of angle and ratios of sides. If we were to create an image of a triangle by moving it about a plane, without rotation it – a mathematical operation referred to as translation – we can also count as symmetry because the properties of the triangle remain preserved as it moves about the plane.
Plane Figures & Euclidean Geometry
Taking into account all of the basic symmetries discussed – reflection, rotation, scaling, translation and the combinations they entail we are provided with a base for the study of Euclidean Geometry which roughly translates to – the geometry of plane figures.
Plane figures refer to figures that can be represented in perfectly flat two dimensions as well as be extended to three dimensions through the use of vertical planes. If you have taken these subjects in school, some of these may be familiar to you, certain facts that are taught to us like that angles of a triangle always add up to 180 degrees all have their roots in these forms of symmetry.
I know this is a lot to take in, so I will leave you with this interesting little known fact to wet your appetite for the future pieces – there are only like 14 ACTUAL wallpaper patterns.
Check out the tile design in my new house in a previous post I did, I used the same kind of principle with that tile design.