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Math advice for paranoid aliens

John August crowd-sources answers to some screenwriting issues in his blog comments.

Far away on a distant planet, an intelligent but very paranoid species is constructing a series of terrestrial watchtowers to scan the heavens, making sure no space-traveling enemies sneak up on them.

You can think of these watchtowers as observatories, each one watching a 180-degree (half-sphere) swath of the sky. For this simplest version, you can ignore complications like atmospheric distortion or possible moons.

Question #1: For complete coverage, what is the minimum number of watchtowers they need to build?

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