One of the powers of mathematics is its ability to tell us, usually in the abstract, that a certain ‘thing’ must exist. It may not tell us where to find this thing, or necessarily what its properties are, but proof of existence alone can be very useful and reassuring. One of the most famous existence theorems in the field of topology was devised in 1912 by Dutch mathematician Luitzen Brouwer. This ‘fixed point’ theorem has many important applications in fields like differential equations, economic theory and even John Nash’s famous equilibrium result. But the theorem also has some interesting reality based consequences which we’ll try to explore here. Let’s start with a formal statement and work from there:
The [mathematical] construction itself is an art, its application to the world an evil parasite. – L. E. J. Brouwer.
Theorem – Brouwer’s Fixed Point (1912): Every continuous function f from a convex, compact subset K of Euclidean space has a fixed point, f(x) = x.
What this theorem says is if we take a well-defined space without holes (think of a regular continuous surface or volume), and we rotate, stretch, bend or deform it, then at least one point in the space will be unchanged by this action and remain in the same position. In other words there will be at least one ‘fixed point’. Mathematically our deformation or re-imaging of the space K is represented by a function f(K) that maps all points x in the original space to points y in the new version. The fixed point is then represented as f(x) = x. Moulding a piece of plasticine with your hands is a reasonable analogy to think of when considering the theorem in normal three-dimensional Euclidean space. Of course, one of the powers of this theorem is its generality – we can go as high as we like in dimensionality and still be confident in the assertion.
The theorem leads to some fairly interesting and amusing truths: 1. Make a banana smoothie and stir it with a spoon. No matter how long or thorough your stirring action at least one point of the smoothie will still be in the same position when you’re done. 2. Take a map of Scotland and make a copy of it. Shrink it, enlarge it, rotate it or even crumple it. Place one map so it rests within the boundary of the other. Then we know with certainty that a particular location on one map will be resting on-top of the same location on the other.
3. Two DJs are in a nightclub. Both share the same favourite track. One DJ plays the tune first at its native speed of 140 bpm (beats per minute). The other DJ decided to play the tune faster, so pitch shifts it to 150 bpm, adding some reverb and echo for good measure. He hits play so both tracks are now playing. Brouwer’s theorem says that as long as the faster track starts and finishes whilst the slower version is still playing there will always be one moment of musical harmony when both tracks are in perfect sync – a musical fixed point!
4. Go for a walk in the countryside and pull out your ordinance survey map. Because your map is a smooth and continuously scaled representation of the real terrain you’re in – there will always be a ‘You are here’ point on your map to help you navigate.
Explore the mathematics
Let’s look at a less general form of Brouwer’s theorem in the plane . We’ll then explore homeomorphisms and a trivial example in one dimension.
Theorem – Brouwer’s Fixed Point on the Plane: Every continuous function from a closed unit disc D to itself has a fixed point.
This form of the theorem talks about the unit disc, rather than an abstract subset. That doesn’t restrict us to considering only discs because homeomorphism is assumed. Roughly speaking a homeomorphism is a bending or stretching of a space into a new one. Any remoulding of a disc on the plane is therefore included.
Definition – Homeomorphic: Two spaces X and Y are homeomorphic to one another (‘topologically the same’) if there exists a function f: X -> Y with the following properties:
f is continuous.
f is a bijection.
The inverse of f exists and is continuous.
Example: Any open interval (a,b) is homeomorphic to the whole real number line . One such homeomorphism is the function f(x) = 1/(x – a) + 1/(x – b). Have a think about it. Finally a simple example. A trivial example in one dimension: Consider the closed interval [-1,1] and the function f(x) = (x+1)/2. Where does f send [-1,1] and can we find the fixed point? Answer: The interval [-1,1] is closed and f(x) is certainly continuous. f([-1,1]) = [0,1] with f(1) = 1. Therefore x = 1 is the fixed point.