When it comes to painting a wall, I tend to buy too little paint. With a ten liter paint bucket, which weighs quite a bit due to its capacity, you can only cover a wall about 15 meters long. If you have to apply several layers of paint, for example if you want to cover a darker colour, then the paint bucket is no longer sufficient even for a small room. I’m usually not the only one who tends to underestimate the amount of paint needed. This has to do with imagining color as an infinitely thin two-dimensional layer – when in fact it has a finite extent. This fact also underlies the so-called painter’s paradox.
In the 17th century, the Italian physicist and mathematician Evangelista Torricelli (1608–1647) described a geometric object that later became known as “Torricelli’s Trumpet” or “Gabriel’s Horn”. The latter is an allusion to the Archangel Gabriel, who, according to the Bible, blows a very long trumpet on the Day of Judgment. The name hits the nail on the head, because the structure conceived by Torricelli resembles an infinitely long horn whose mouthpiece lies at infinity. And it has amazing properties that some scholars took as an opportunity to think about metaphysical questions: Not surprising at first, the horn has an infinitely large surface – but the capacity is finite.
This finding sparked disputes among experts for years. Many were convinced that an infinitely extended object could never have a finite volume, while others, while accepting this point, could not understand how a finite volume could go hand in hand with an infinite area. And indeed, Torricelli’s trumpet leads to the painter’s paradox: on the one hand, there is not enough paint in the entire universe to paint the horn, since its surface is infinitely large. On the other hand, because of the finite capacity, you can fill it completely with a certain amount of paint. If you do that, however, you have painted the surface at the same time.