Questions for ‘Geometry can shape our world in unexpected but useful ways’ 

a bunch of oranges stacked in a pyramid against a white background

Kepler surmised that for round shapes — be they oranges or cannon balls — the most stable way to stack them is as a pyramid. It would take several centuries, but geometers would ultimately prove this. And that proof would also lead to advances in stacking signals for electronic-data transmission.

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To accompany Geometry can shape our world in unexpected but useful ways’  


Before Reading:

  1. Come up with a one-sentence definition for the word geometry. Based on your definition, describe one type of problem that geometry could help someone solve.
  2. Imagine you are tiling a bathroom with triangle-shaped tiles. Use a blank sheet of paper to sketch these triangle tiles in a pattern that could be used to tile a floor, completely. When you finish, there should be no gaps or partial tiles (except on the edges). Now, try the same thing with pentagon-shaped tiles. (A pentagon is a five-sided shape.) Were you equally successful with both tiles? Explain your answer by contrasting your success with both tile shapes.
  3. Give an example of a three-dimensional (3-D) shape. For the shape you just provided, what would the 2-D equivalent of that shape look like?

During Reading:

  1. What is the most stable arrangement for stacking oranges at the grocery store? How has this technique been used to improve radio communications?
  2. How might a textbook’s two-dimensional (2-D) shadow be used to infer the details about the textbook’s three-dimensional (3-D) shape?
  3. How is the problem studied by Casey Mann similar to the orange-stacking problem? 
  4. What is one way the problem studied by Mann differs from the orange-stacking problem? 
  5. By the early 20th century, how many types of pentagons had been discovered that could completely tile a surface?
  6. What is a potential application of discovering tight-fitting pentagon shapes?
  7. What is one example of geometric tiling strategies found in nature?
  8. An icosahedron is a 3-D shape. How many faces does an icosahedron have? Each of these faces is shaped like an equilateral triangle. What is an equilateral triangle?
  9. Give one example of a common or everyday item shaped like an icosahedron.
  10.  What benefit does a capsid provide to a virus? 

After Reading:

  1. How has geometry assisted the study of viruses? What are some unanswered questions about viruses that geometry may help to address? How might geometry help in the development of new vaccines?
  2. Consider the diverse ways that geometry is used to solve problems. Consider how geometry might appear in your everyday life. What is one way that geometry might show up on a typical day for you?