It is a Christmas tradition in science and math to examine some aspect of Santa Claus and try to understand it. The most common choice is to compute what sort of speed he would need to visit all the houses he leaves presents at. Today, instead of velocity, Occupy Math will look into a type of mathematical symmetry that might let us detect Santa using his bag to carry the vast quantity of presents he delivers. Since this involves extra dimensions, we need to work up to it bit by bit. Step one is to understand simple symmetries. Occupy Math made a transparent, colored triangle and put it down in six different ways. The first three ways are the original orientation of the triangle and a one-third and two-thirds turn. The second three are the first three, but flipped around their vertical axis.
The interesting thing is that these six ways of placing the triangle are all the ways of placing it. Make your own triangle (remember to color both sides) and check – every way of placing the triangle so that it covers the original triangle is one of the six. Three of these symmetries permit the triangle to move in its original two dimensional space – but the ones where we flip the triangle require motion through the third dimension. This means that the triangle has three two-dimensional symmetries and three 3D symmetries.
Does something similar happen when we use a solid object?
Consider a cube, like a six-sided die. If we leave the cube in three dimensions then we can rotate it around the lines through the middle of opposite faces in quarter turns or around the lines joining opposite corners in one-third turns. When we are done looking at all the ways these can be combined, the cube has 24 symmetries that can be achieved in three dimensions. Look at the diagram below. It shows what happens when opposite faces of the cube are exchanged – follow the letters. This symmetry moves the cube to occupy the same space it did before – but it cannot be done by moving the cube in three dimensions. Try it!
With the triangle we had three symmetries in 2D and three more we could only access through the third dimension. The cube turns out to have 24 symmetries in three dimensions and 24 that can only be accessed through the fourth dimension. These symmetries are abstract – they cannot be realized with a real, solid cube. Now lets get back to Santa’s bag of toys.
Our working hypothesis is that Santa packs the contents of multiple warehouses full of toys in a bag he carries on his back by using extra dimensions.
Imagine that Santa and his elves are packing the toys into the bag. When you pack things you rotate and wiggle stuff to make it fit. This means that Santa, or more likely his elves, may apply a physically impossible higher dimensional symmetry to a toy! Most toys have pictures on the internet and so it might be possible, by comparing a toy brought by Santa to one on a store’s website, to detect the extra-dimensional fingerprints of Santa.
Using a computer, Occupy Math has constructed an example of a 4-space symmetry of a traditional Christmas toy.
Look at the wave of the nutcracker’s beard – there is no way to lift, spin, or turn the toys to make the beards match up. The transformation is a simple mirror image – which tells us what about the number of dimensions a simple mirror operates in? Don’t think about it too hard, you should be enjoying your holiday!
The issue of which symmetries an object has isn’t just a fun exercise in trying to detect Santa Claus. Chemicals often have multiple forms that have exactly the same atoms connected in the same way, but those forms are separated by physically unrealizable four-space symmetries. This chemical phenomenon is called steroisomerism. The different mirror-image forms of a chemical are very similar but can have different properties. The mirror reflection of one type of sugar, for example, can yield a different number of calories from the other. Comment on this blog or tweet Occupy Math if you get a mirror image present or find an interesting topic related to symmetry that you would like to see in a post.
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics