# Pen and paper

At the start of my vacation I closed my laptop and went back to pen and paper. To get at the heart of things, nothing beats an empty sheet of paper and a clear schedule. I spent many weeks drawing objects and scenes, and deriving geometrical formulas with pen and paper. Don’t get me wrong, I am devoted to my computer and the internet. I would be totally lost without them. But each and every day, I spend my energetic and creative hours sketching and scribbling on paper.

My drawing abilities are quite limited but I always figured it was simply due to a lack of practice. I spent my formative years on reading, writing, and arithmetic, not on drawing. So I never appreciated this really simple fact; drawing three-dimensional objects and scenes is extremely difficult. I studied a fair amount of drawing manuals and the best advice I encountered was “In order to draw an object in general attitude, you have to think three dimensionally”. Not very helpful.

I also discovered something about geometry. First of all, I totally love it. I really enjoy the combination of geometric constructions and algebraic equations, that can express the same thing in different, but equally precise, languages. It is very fulfilling to draw a construction of lines and points, to guess at a deeper structure, then prove it in algebra, and go back to the sketch and turn it into an exact drawing. Everything in this little universe works like a charm.

The second thing I discovered is that three-dimensional geometry is extremely difficult. And I have no lack of experience here. I am trained in physics which involves dealing with topics like mechanics and electrodynamics that have 3D vector calculus at their cores. I also spend many years studying the human capabilities of perceiving the 3D world. And still. I get easily confused while trying to solve basic problems in 3D Euclidean geometry.

Why is drawing in three dimensions so hard? Why is geometry in three dimensions so hard? I can safely say I am not the only person with deficiencies in these areas. The main problem is in dealing with 3D rotations. We can perceive rotations of objects without a problem. But imagining these rotations correctly can easily blow our mental fuses. Luckily we have 3D computer graphics to help us.