![\"escher-tess.jpg\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_292.jpg?resize=250%2C261\")
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As I said in the last post, in group theory, you strip things down to a simple collection of values and one operation, with four required properties. The result is a simple structure, which completely captures the concept of symmetry<\/em>. But mathematically, what is<\/em> symmetry? And how can something as simple and abstract as a group have anything to do with it?<\/p>\n <\/p>\n Let’s look at a simple, familiar example: integers and addition. What does symmetry mean in terms of the set of integers and the addition operation?<\/p>\n Suppose I were to invent a strange way of writing integers. You know nothing about how I’m writing them. But you know how addition works. So, can you figure out which number is which? Let’s be concrete Well, you can figure out what zero is. With a bit of experimentation, you can see that adding What else can you figure out? With enough experimentation, you can find an ordering. You can find two So – which symbol represents one?<\/p>\n You can’t tell. “@” might be -5, in which case “#” is 1. Or “@” might be 5, in which case “#” is -1. You can tell that “*” must be either +2 or -2, but you can’t tell which. You can’t tell which numbers are the positives, and which are the negatives. If all you have available to you is the group operation of addition, then there is no way<\/em> for you to distinguish between the positive and the negative numbers.<\/p>\n Addition of the integers is symmetric<\/em>: you can change<\/em> the signs of the numbers That’s a simple of example of what symmetry means. Symmetry is an immunity to transformation<\/em>. If something is symmetric, then that means that there is something you can do to it, some transformation you can apply to it, and after the transformation, you won’t be able to tell that any transformation was applied. When something is a group, there is a transformation associated with the group operator which is Think of the intuitive notion of symmetry: mirroring. What mirror symmetry means is that you can draw a line through an image, and swap what’s on the left-hand side of it with what’s on the right-hand side of it – and the end result will be indistinguishable from the original image. Addition based groups of numbers captures the fundamental notion of mirror symmetry: it defines a central division (0), and For an example of how that can get more interesting that just the integers without changing the fundamental concept, you can look at the following example: given a hexagon, you can reflect it along many different dividing lines without recognizing any difference.<\/p>\n When some object is symmetrical with respect to a particular transformation, that means that you cannot distinguish between that object before the transformation, and that object after. There are, of course, many different kinds of symmetry beyond the basic mirroring ones that we’re all familiar with. A few examples of basic geometric symmetries:<\/p>\n Scale<\/b>: scale symmetry means that you can change the size of something without altering it. Translation<\/b>: translational symmetry means you can move<\/em> an object without detecting any change. If you have a square grid, like graph paper, drawn on an infinite canvas, you can move it the Rotation<\/b>: rotational symmetry means you can rotate something without creating a detectable change. For example, as illustrated in the diagram below, if you rotate a hexagon by 60 degrees, without any external markings, you can’t tell that it’s rotated. <\/p>\n There are, of course, many more, and we’ll talk about some of them in later posts.<\/p>\n For a fun exercise, look at the Escher image at the top of this post. It contains numerous kinds of symmetries; several different kinds of reflective symmetries, translational symmetries, rotational symmetries, color-shift symmetries, and more. See how many you can find. I’ve been able to figure out at least 16, but I’m sure I’ve missed something.<\/p>\n","protected":false},"excerpt":{"rendered":" As I said in the last post, in group theory, you strip things down to a simple collection of values and one operation, with four required properties. The result is a simple structure, which completely captures the concept of symmetry. But mathematically, what is symmetry? 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\nabout this: here’s the set of numbers from -5 to 5, out of order, in my strange notation. {#, @, *, !, , ^, &, %, $ }. If you give me two of them, I’ll tell you their sum. What can you find out?<\/p>\n
\n“&” and “>” gives you “>”, and adding “&” and “~” gives you “~”; the only integer for which that could
\nbe true is zero. So you can tell that “&” is how I’m writing zero.<\/p>\n
\nvalues, “@” and “#”, where “@” plus “#” = “>”; “>” plus “#” equals “%”, and so on – each time you add “#” to something, you get another value, and if you start with “@”, and repeatedly add “>”, you’ll get a
\ncomplete enumeration of them: “@”, “>”, “%”, “$”, “!”, “&”, “#”, “*”, “~”, “^”, “>”. <\/p>\n
\nbut by everything you can do with the group operator, the change is invisible. I can write an equation in my representation: “%” plus “*” equals “!”, where I know that “%” is +3, “*” is -2, and “!” is +1. I can then switch the positive and the negative numbers, so that “%” is -3, “*” is +2, and “!” is -1, and the equation – in fact, any equation which relies on nothing more than the group operator of addition – can tell the difference. If you don’t know what the symbols represent, you wouldn’t even be able to tell that I’d changed my mind about what the symbols meant!<\/p>\n
\nundetectable within the structure of the group. That “within the group” part is important: with the integers, if you have multiplication, you can distinguish between 1 and -1; 1 is the identity for multiplication, -1 is not. But if all you have is addition, that the transformation is invisible.<\/p>\n
\nthe fact that swapping the objects on opposite sides of that division has no discernable effect.<\/p>\n![\"reflect.png\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_293.png?resize=276%2C99\")
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\nThink of geometry, where you’re interested in the fundamental properties of a shape – the number
\nof sides, the angles between them, the relative sizes of the sides. If you don’t have any way of
\nmeasuring size on an absolute basis, then an equilateral triangle with sides 3 inches long and an equilateral triangle with sides 1 inch long can’t be distinguished. You can change the scale of
\nthings without creating a detectable difference.<\/p>\n
\ndistance between adjacent lines, and there will be no way to tell that you changed anything.<\/p>\n![\"rotation.png\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_294.png?resize=198%2C96\")
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