How deep is your bowl?


“How Round Is Your Circle” by John Bryant and Chris Sanguin has been delighting my mathematical senses for two nights straight with a collection of counterintuitive, paradoxical, and insightful geometrical toys. Try this: What three-dimensional shape has the same width in all directions (other than a sphere)? My most mathematically-minded friends could not believe that such a thing could exist. Turns out there are many of them.

The authors have created a website illustrating some of the objects from the book, but the book has many more.

There seems to be a category of problems that most people get wrong when they first hear or see them. It takes a second deeper look to get the right answer. Here are a few examples that come to mind:

  1. When the Space Shuttle transfers from a higher orbit into a lower orbit, it fires its engines in reverse to slow down twice on opposite sides of the planet. Will the shuttle be moving faster or slower at the end of the maneuver? (faster)
  2. During descent, commercial airplanes often raise spoilers on top of their wings to increase drag and reduce lift so that they can steepen their approach. Will the airplane slow down or speed up as the result? (speed up)
  3. As the ice melts in Antarctica and Greenland, ocean levels rise across the globe. Will the average ocean depth increase or decrease as the ocean levels rise? (probably decrease)

To illustrate this last problem, can you imagine a bowl that has the following wonderful property: as you add more soup to it, the average depth of the soup in the bowl stays constant? Magic? Not really? I have just whipped up just such a bowl in Matlab:

Once you fill the initial cylindrical portion with water, adding more water will leave the average water depth in the bowl the same: h0. The Great Salt Lake probably has this property: as it gets more water, it gets shallower (on average).


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