Math is Beautiful

My senior honor's thesis in college was on the relationship between math and art. I find the parallels between the two fields fascinating. Just think of the various math classes you might take in college: Abstract Algebra, Modern Algebra one can't help but think of art movements. Now I might have a difficult time relating Abstract Algebra to Abstract Art.  I would have a pretty easy time relating the progress math was making in the Renaissance to the progress seen in paintings.

For instance, thanks to Filippo Brunelleschi in the 1400's we have linear perspective! This is basically what happens when you take the last postulate in geometry (parallel lines never intersect) and allow parallel lines to intersect at infinity. Then go about building a new geometry based on those assumptions. It's both mathematically valid and an amazing tool to depict distance.

Hopefully, you can picture a set of railroad tracks and how someone with no knowledge of perspective would draw them. They would most-likely draw them with parallel lines never intersecting and with evenly spaced ties. Which makes perfect sense because the don't intersect and they are evenly spaced. But in the real world they don't look that way.

Once we apply this new postulate (parallel lines intersect) we get a perspective drawing. We can see in the first picture that the two parallel lines intersect at the horizon (infinity or vanishing point). Using another set of parallel lines (in red) that connect diagonals you can approximate accurate spacing.  Usually art teachers neglect to incorporate a how to measure spacing when teaching perspective drawing. In art class it's usually good enough that there is a vanishing point and that stuff of the same height is in line. But in true perspective as things recede into the distance they get smaller and they also get closer together.


We can do one better. Instead of the diagonals following the old rules of parallel lines (intersection bad). We can have those lines also intersect at infinity. To do this we need our horizon line (Our line at infinity.)

If we continue to extend this idea, we can get a accurate tiling of a floor.  See below.  Look at all of those lovely parallel lines intersecting! The red lines are only in there as diagonals, used to measure spacing. In art they should be erased. In math, we would call that showing your work.