## Drawn to Math

Rowena's perspective drawing of a level 2 Menger Sponge

Lots of people like to draw and many people strive to make more realistic drawings.

That desire can motivate lots of mathematics!

Perspective drawing is one of the keys to making realistic drawings, and, at the same time, offers a wealth of geometry problems that range over many levels of difficulty.

Sadly, there are those who believe that we “dumb down” math because we work with girls at Girls’ Angle. Curiously, the fact that we sometimes work on perspective drawing seems to be to them a kind of confirmation of their mistaken notion. This post doesn’t just defend the use of perspective drawing in math, it is an exhortation to use it to bring math to more people.

Consider cubes, for instance. I’ve met more than a few people, including some mathematicians, who, not having thought carefully about it, think that drawing cubes in perspective is trivial. “You can put the vanishing points anywhere!” some have said.

So here’s a good perspective drawing problem: Consider a perspective drawing of a cube none of whose edges are parallel to the plane of the drawing.  Let A, B, and C be the three vanishing points corresponding to the edges of the cube.  (A cube has 3 sets of 4 mutually parallel edges which give rise to the 3 vanishing points.)  Prove that the triangle ABC is acute.

It’s hard to change people’s minds! Upon realizing this, some respond, “OK, so you’ve got to make sure your vanishing points form an acute triangle.  Well, that’s not hard to do, big deal!”

So here’s a second perspective drawing problem: Prove that the perspective drawing of a cube is not yet determined by the locations of the 3 vanishing points defined by its edges.

And here’s a third perspective drawing problem: Make a proper perspective drawing of a cube!  By “proper,” I mean that the drawing must be accurate and not a rough sketch.

The picture shows such a drawing made by one of our members. It adorns the cover of Volume 3, Number 4 of the Girls’ Angle Bulletin, and inside that issue, you can find an article she wrote explaining how she constructed this drawing. In the process of making this drawing, Rowena attempted to build a kind of Da Vincian 3D protractor. That was an adventure in itself! But with the limited materials she had available to her, the device flopped around too much to be useful. Eventually, she found that she could use mathematics to compute exactly what she needed to complete the drawing. Score one for the power of mathematics!

Here are more perspective drawing problems:

1. Devise a method of constructing a perspective drawing of equally spaced identical telephone poles lining a long straight road that recedes into the distance.

2. Show that perspective drawings of circles always look like ellipses (possibly degenerate).

3. A perspective drawing depicts a circle that is parallel to the ground and not at eye level. From problem 2, we know the drawing of the circle is an ellipse. However, just from the ellipse it cannot be determined whether the eye is above or below the circle. What is the least amount of information you could add to the drawing to disambiguate between these two cases?

4. Construct a perspective drawing of a spiral staircase.

5. Take a look at Vermeer’s The Art of Oil Painting. Given that the painting is 130 cm by 110 cm, determine the ideal distance away from the painting the viewer should stand to get the best illusion of 3D.