Something that has fascinate me for many years is the fact that, if a disordered systems is larges enough, it will tend to contain pockets of unavoidable regularity. Order will emerge from disorder.
For example, imagine we colour every point of the 2D plane in one of two colours, red or blue say. Then there will inevitably be equilateral triangles all three of whose vertices are the same colour; either all red or all blue. Here's how we can prove this.
Imagine we set out to colour the plane in a way that avoids a monochromatic equilateral triangle. We shall find that this is impossible. Consider the following steps in our attempt.
We start with two points which are coloured red (Step 1).This must be possible; if not, there would no more than one red point in the entire plane, so either all equilateral triangles would be blue monochromatic, or all the ones that avoided the single red point would be. Either way our attempt to avoid monochromatic equilateral triangles would have failed spectacularly.
At Step 2 both the ringed points will have to be coloured blue, otherwise one or both the triangles marked will be red monochromatic.
At Step 3 the ringed point will have to be coloured red, otherwise the triangle marked will be blue monochromatic.
At Step 4 the ringed point will have to be coloured blue, otherwise the triangle marked will be red monochromatic.
At Step 5 a monochromatic triangle can no longer be avoided; whether we choose red or blue for the uncoloured point a monochromatic triangle with be created.
Colouring the Plane 🔴 🔵
Something that has fascinate me for many years is the fact that, if a disordered systems is larges enough, it will tend to contain pockets of unavoidable regularity. Order will emerge from disorder.
For example, imagine we colour every point of the 2D plane in one of two colours, red or blue say. Then there will inevitably be equilateral triangles all three of whose vertices are the same colour; either all red or all blue. Here's how we can prove this.
Imagine we set out to colour the plane in a way that avoids a monochromatic equilateral triangle. We shall find that this is impossible. Consider the following steps in our attempt.
We start with two points which are coloured red (Step 1).This must be possible; if not, there would no more than one red point in the entire plane, so either all equilateral triangles would be blue monochromatic, or all the ones that avoided the single red point would be. Either way our attempt to avoid monochromatic equilateral triangles would have failed spectacularly.
At Step 2 both the ringed points will have to be coloured blue, otherwise one or both the triangles marked will be red monochromatic.
At Step 3 the ringed point will have to be coloured red, otherwise the triangle marked will be blue monochromatic.
At Step 4 the ringed point will have to be coloured blue, otherwise the triangle marked will be red monochromatic.
At Step 5 a monochromatic triangle can no longer be avoided; whether we choose red or blue for the uncoloured point a monochromatic triangle with be created.