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Symmetries and Group Theory – Science4All Symmetries and Group Theory Written by Lê Nguyên Hoang in Article How can scientists define theories describing the immensity of the Universe? I’d say that the key component is symmetries . In fact, the mere concept of science requires the consideration of a symmetry: If we do the same experiments in different times or places with the same conditions, then results should match. The symmetry of space and time lies in this simple sentence, as explained by Henry Reich on MinutePhysics: Particle physics takes this concept of symmetry even further: It consider that particles are symmetric in some sense. But reality isn’t the only field where the virtues of symmetries highly contribute. Art is full of them. Egypt’s ancient pyramids are geometrical wonders, musics follow remarkable patterns, Jackson Pollock’s paintings display fractal symmetries and even movies and series apply symmetry to actors’ lines, like in this extract from friends . It’s also a major concepts in our society, which tries to consider its inhabitants symmetrically, to ensure fairness . Symmetry often lies behind beauty. As a mathematician, I am fascinated by the beauty of symmetries. Symmetries in Geometry The symmetries I’ve learnt about at school have nothing to do with the one in the video… I know. Yet, symmetries are not restricted to geometry. But let’s start with these symmetries. After all, they are the first ones in the History of mathematics. Reflection In particular, the first one that would come to most people’s mind is the symmetry induced by a mirror. This is known as the reflection symmetry , as displayed in the following picture, where the mirror is the Okarito lagoon on the West Coast of New Zealand’s South Island. This symmetry is actually a matching of any element of the picture with an image . The image of an element is obtained by turning the element around the axis of symmetry. The following figure shows how the image of a star-shaped element is obtained: Reflection symmetry has plenty of nice properties, but I won’t get into this now, as I want to talk more globally about all symmetries. But if you can, please write an article on the topic. That’s nice and simple… Let’s get into more complicated symmetries! Translation I actually find the reflection symmetry rather complicated, at least compared to translations . What are translations? Translations consist in moving elements. It’s a symmetry that appears a lot when several products are made out of the same models and put near each other, like these statues of Angkor: Notice that all elements of the picture must have the same motion. This motion is characterized by a direction and distance. These are captured by the concept of a vector . Vectors are a fundamental concept of mathematics. If you can, please write about them. Point Reflection Now, the first symmetry we’ve discussed was a reflection symmetry with respect to an axis. Following this idea, we can consider reflection symmetries with respect to a point, called… point reflection . I could have guessed the name of this symmetry! So what is it? Point reflection is a symmetry that is used by eyes and cameras to capture an image. Let’s see how it works with this picture taken inside the amazing Cloud Gate of Chicago, also known as The Bean : Notice that point reflection can also be obtained by turning upside down around the point of symmetry. Can we stop turning before being upside down? Great question, whose answer is: Sure, why not? Rotation If we stop before (or after) being upside down, then, we’ll be doing what’s known as a rotation . Rotations appear in circular motions, like the Earth around the Sun, but also on waves, where a central disturbance implies motions away from a central point, such as these chickens running away their mother: Just like for point reflection, the central point here plays an important role. But the even more important variable is the angle of rotation. The central point and the angle define the rotation. The four operations we have seen so far, that is, reflection, central reflection, translation and rotation, guarantee that the image of an element will have the exact same dimensions as the element. This property is known as isometry , which, derived from ancient greek, means same (iso) measure (metron). Figures which have a lot of isometries form beautiful tessellation . These can be found in mathematics and arts, but also in nature and constructions: Isometries are largely studied in mathematics, as they can be defined for any metric space . This includes space with many more dimensions! In particular, in linear algebra, all isometries leaving the origin still are regrouped in the set of orthogonal matrices , which play a central role. If you can, please write about orthogonal matrices. Moreover, as explained by Scott, the study of isometries is essential to map-making . Newton’s mechanics consider that space was isometric, when measured by different observers. Yet, this has been proven wrong by Einstein (see my article on his spacetime ). So Einstein’s world isn’t symmetric for all observers? In fact, it is, in some sense, even thought it’s not isometric. These leads us to homotheties . Homothety What’s a homothety? Consider our first description of point reflection. If we didn’t stop until we arrived on the opposite of the point, we would have done a homothety. Simple, right? Now, this property is the key element of perspective, which was a great breakthrough in art during the European Renaissance. It’s how we observe elements translated into the background, such as in the case of the following picture of the Golden Gate Bridge: The homethety is defined by the central point and a ratio. This ratio says by how much distances will be multiplied. It can even be negative, in which case we would carry on the motion into the opposite side of the central point. In fact, you may notice that a ratio of -1 makes the homethety equivalent to point reflection. Is this how space is modified in Einstein’s theory? Not exactly. In special relativity, only one direction is contracted. This corresponds to a homothety according to an axis instead of a point (or, more accurately in Einstein’s theory, according to a plane). This gives us the following contraction: The plane or axis, and the ratio define this specific homothety. To learn more, read my article on spacetime of special relativity . Can we think of some sort of tessellation for homotheties? Yes we can. And this yields some of the most beautiful wonders of mathematics: fractals . Fractals correspond to object which appear to remain similar once you zoom in. They occur in plenty of natural phenomena, and is applied in certain fields, such as virtual image creation. Here are some of these wonders: To learn more, read Thomas’ article on fractals . The list of symmetries we have described so far is not exhaustive. But they are the best-known and most visual symmetries. So let’s go further in their study! Groups of Symmetries Let’s focus on translations and rotations. In the following, whenever I refer to symmetries, I mean translations and rotations only. The reasonings would be similar, although a little more complicated, if you add other symmetries, but the fundamental concept of group would remain. Before getting to the definition of what a group is, there are two major remarks which need to be made regarding translations and rotations we have defined. What are they? First, these symmetries are bijective and the inverse of a symmetry is a symmetry. What does that mean? Bijectivity means that symmetries correspond to one-to-one matching between points of the initial image and points of the obtained image. More precisely, given any symmetry, every point has a unique image through this symmetry, but, also and more importantly, every point is the image of a unique other point, called fiber . Bijectivity is a major concept in mathematics, which is, for instance, involved in the continuum hypothesis, one of mathematics’ weirdest result. If you can, please write about it. OK… Symmetries can therefore be inverted. The inversion of a symmetry correspond to a transformation. Well, the second part of the remark is that this inverting transformation, called the inverse , is also a symmetry. Indeed, inverting a translation is equivalent to a translation towards the opposite direction. Similarly, inverting a rotation is equivalent to a rotation of the opposite angle with the same center. Indeed. Now, what’s the second remark? The composition of symmetries is a symmetry. What do you mean? What’s a composition? A composition of several symmetries is the operation of successively applying each of the symmetries. For instance, the composition of two translations corresponds to applying the second translation, and then the first translation. Now, it can be shown that composing any 2 symmetries yield a symmetry. If you compose 2 translations, you get a translation whose motion is the sum of the motions of the 2 translations. You can see this in the following figure: And this is true if you take any 2 symmetries? Yes! If you composed two rotations, you’d obtain a rotation whose angle is the sum of the angles of the two rotations, unless the sum of the angles is nil, in which case you’d obtain a translation. Determining the center point or the translation motion is more complicated though and I’m not going to describe how it’s obtained. Similarly, composing a translation and a rotation makes a rotation of the angle of rotation, but whose center is complicated to describe. But you should try to find these on your own. That’s an interesting exercise! What if you compose more than 2 symmetries? Composing more symmetries is equivalent to composing all of them but two successive ones, with the composition of these two symmetries. For instance, if you need to compose a first rotation, a translation and a second rotation, then it’s eq