Real Mathematics – Strange Worlds #6

Morning Ritual

Famous French philosopher Descartes had a rough childhood as he dealt with various illnesses including tuberculosis. His health condition effected his early school life. He was always late for school. After some time Descartes developed a habit of laying in his bed until noon. It is now known that he spent his mornings in bed thinking about life, nature, mathematics and so on. (Yes, Descartes was in fact a brilliant mathematician.)

René_Descartes_i_samtal_med_Sveriges_drottning,_Kristina
Rene Descartes (right) and Queen of Sweden Christina.

This habit of Descartes kept going until he started working for the queen of Sweden. Apparently she wanted to learn mathematics every day at 6 o’clock in the morning. I try to avoid making speculations in these articles, but I wasn’t surprised that Descartes passed away right after he moved to Sweden! (Warning: Pure speculation.)

I feel for Descartes as I have my own morning rituals too. After waking up I need some time until my senses come back to me completely. Hence I keep my breakfasts as short as possible. Cornflakes are ideal for people like me, although I get sick of getting cornflakes every morning. This is why once in a while I take a week off of cornflakes and declare such weeks “toast & coffee” weeks.

IMG_6229

Equality

I am not sure when this happened but every move I make is completely automatic during my toast & coffee mornings: Switch the kettle and the toaster on, prepare the bread slices and cheese (some days ham), cut the toast diagonally so that there will be two equal right-angled triangles, finish making the coffee.

Some mornings my involuntary moves during my breakfast preparation cause a huge problem: I can’t cut the right-angled triangles equally. Uneven triangle toasts? That sounds disturbing, doesn’t it?

Pancake Theorem

  1. Assume that there is one pancake and two of you. There is always a single knife-cut that will give two equal pieces of pancake.
    images (1)
  2. In case there are two pancakes with two different flavors, again there will always be a single knife-cut which will give equal pieces of both pancakes.

Pancake theorem is a good example for cutting two equal parts in a 2 dimensional plane. In case we try that in a 3 dimensional plane, another theorem called the ham-sandwich theorem will come to our rescue.

Ham-Sandwich Theorem: Imagine you have a ham sandwich. Mathematics says that it is always possible to slice the sandwich with one cut so that the ham and both slices of bread are each divided into equal halves.

It sounds obvious and simple. However there are deeper meanings inside this theorem. Let’s assume that you are a careless person and you prepared the sandwich haphazardly: Ham is on top of the bread slices. Ham-sandwich theorem says that it doesn’t change anything; there will be such cut that will give us two equal pieces of each ingredient.

Suppose that you prepared your sandwich perfectly but you dropped it on the floor. Now, bread slices and ham are far away from each other. Theorem says that this won’t change anything. Even if the ingredients are far away from each other, there is always a single cut which will give us equal pieces of them.

One wonders…

  • Sandwich’s ingredients, not even its shape is important. Sounds like topology spirit, huh?!
  • Imagine you have five equal circles as shown:
    cemberler.jpg
    Is it possible to have a cut that will result two equal parts of circles?

M. Serkan Kalaycıoğlu

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