Vanishing Point

Here are the first three stages of the process:


In the first picture, the blue area is $\frac12$.
In the second picture, the blue has been partly covered by a purple square with area $\frac14$ so the blue area is $\frac12-\frac14$.
In the third picture, a blue square with area $\frac18$ has been added. As the pattern continues, the blue area will be $\frac12-\frac14+\frac18-\frac{1}{16}+\frac{1}{32}-...$

Instead of looking at how the pattern builds up, we could divide the completed pattern into the nested sections:

This shows the outer ring of the pattern, in which there are four larger purple triangles and four smaller blue triangles with half the area. So the ring is made up of twice as much purple as blue.

The same will be true for each ring added inside, therefore the overall pattern has twice as much purple as blue.




Here's another way to look at the completed pattern:

$\frac23$ of the highlighted shape's area is purple, and $\frac13$ is blue.

The whole pattern can be subdivided into pieces like this, so that means $\frac23$ of the total area of the pattern must be purple, and $\frac13$ of the total area must be blue.