Ants marching

I decide to go to a picnic with my friends. I bring to the picnic my lunch, which consists of the following food items:

  1. An orange (sphere)
  2. A can of soda (cylinder)
  3. A peanut butter and jelly sandwich (rectangular prism)
  4. A doughnut (torus)

I put my food down on the ground at the picnic and go play football. When I come back, I notice that 100 ants are crawling on the surface of one of my food items. The ants are apparently very mathematical ants, as their paths satisfy the following properties:

  1. The ants always crawl straight ahead, never turning left or right, at a speed of 0.25 cm/s0.25~\mbox{cm/s}.
  2. The ants' paths do not intersect each other.
  3. If an ant looks to its right or left as it crawls along its path it always sees another ant within 4 mm4~\mbox{mm} of itself.
  4. Each ant's path is closed and the period of an ant on its path is 20 seconds20~\mbox{seconds}.

What is the maximum volume in cm3\mbox{cm}^3 of the piece of food the ants are crawling on?

Details and assumptions

  • All linear dimensions of the food are much bigger than 2 mm, i.e. the ants aren't marching on an orange with a radius of 1 mm. That would be silly.
  • The ants crawl all over the food. You do not have to think about one side of the food touching the ground and the ants crawling on the ground, etc. Just imagine the food is suspended slightly off the ground.

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