The sun and the moon

by Fred Ross Last updated: January 3, 2012

I once saw the sun in the moon in the sky, with the moon's dark side shadowed just so. The way they lay, though I couldn't perceive the distances involved, I could grasp the shape of the triangle of sun, moon, and earth.

The distance from earth to sun is roughly 1 AU. From the Earth to the moon is about 2.5 × 10-3 AU. How often do you deal with an aspect ratio of three orders of magnitude in your life?

The tallest building in the world, the Burj Khalifa in Dubai, is 828m. What is the smallest object you can see if you have the entirety of the tower in view? The vertical field of view of the human eye is about 0.75π radians. 10-4π radians. If we had the entire Burj Khalifa in view, the smallest object we could see next to it is theoretically 10cm, but to get any effect we would have to perceive it as an object, not a speck, so it would actually have to be several meters in size. That's at most two orders of magnitude.

The great cathedrals, the Grand Canyon, the pyramids, don't offer anything like two orders of magnitude of visible difference. You cannot draw a triangle with an aspect ratio of 1000 on a piece of normal paper with a pencil that you don't perceive as a sloppy line. A4 paper is 210 x 297 mm, standard pencil leads are 0.7mm in diameter. The width of the a triangle 297mm long with the same aspect ratio as the sun, moon, earth triangle I saw would be 0.7mm.

We live in a world of very limited aspect ratio. What happens when we are suddenly faced with something as far beyond our experience as the Burj Khalifa is beyond a house in the suburb?