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December 18, 2018
Dudeney Puzzle : A Tale from Pythagoras to Dehn
" Take care of yourself, you're not made of steel. The fire has almost gone out and it is winter. It kept me busy all night. Excuse me, I will explain it to you. You play this game, which is said to hail from China. And I tell you that what Paris needs right now is to welcome that which comes from far away. "
et's travel back 2 Millenia to the Land of the Greeks where Plato, Pythagoras, Archimedes devoted their life to Math and Science to understand and enjoy nature. Math was a Puzzle to them and they enjoyed doing it as we do it now. Pythagoras gave birth to the beautiful proof of Pythagoras Theorem just using pictures.
This approach of dissection and rearrangement was not new to him as the tradition of these sort of puzzles can be traced back to Plato and also continued by Archimedes.
Square trisection: Three squares to One square
Two equal squares are turned into one square in fourteen pieces by subdivisions of the previous four pieces by Archimedes
The Puzzle: Cut an equilateral triangle into four pieces that can be rearranged to make a square [Cut means by scissors so the result will be a set of polygons]
An Obvious Question:
Why four? To understand this, let’s go back to the early-mid 19th century when there was a solution to the general form of this puzzle. The truth is encoded in the following theorem:
Wallace–Bolyai–Gerwien theorem: A Polygon can be cut into a finite number of pieces and then by rotations, reflections and translations(isometric transformations) can be reassembled into another Polygon iff both the polygons have the same area. We call this equidecomposability of two polygons of the same area. [Actually, it is called equidissectabilty, but then it is equidecomposability]
Well, this seems to be a beautiful piece of truth!
Let’s think about it a bit more!
Suppose you are given two polygons of the same area made of paper and a pair of scissors. You have in some sense no limit of cuts. How will you start cutting one of them?
First, observe that the intersection of two polygons is a polygon. Also, Union of polygons along the edges is also a polygon.
Now, you take two of them, place on above another, cut out the common portion by scissors and then reassemble the remaining pieces again and assemble them to form two different polygons of the same area and the problem reduces to the same problem, now if you are sure this process is going to end after some time, then we are done. This sort of congruence has a name “scissor congruence”.
Pause here for a moment!
Let’s snip the scissor on our thought and try to understand what’s the matter in reality.
[A and B are scissors congruent if A can be cut into finitely pieces–each of which is homeomorphic to a disc and bounded by a curve of finite length–which can be rearranged to form B (ignoring boundaries).]
We are essentially cutting out congruent pieces of a polygon from both the polygons each time, right! Will this process ever end? Is so, why? Or how? Also, observe if a polygon P can be transformed into Q in this way denote it by P Q to avoid too much clumsiness. Now,
P \(\rightarrow\) Q means Q \(\rightarrow\) P
P \(\rightarrow\) Q, Q \(\rightarrow\) R means P \(\rightarrow\) R
P \(\rightarrow\) P
It is easy right
Now, this sort of relationship is called Equivalence Relation which has a nice use that is if we can show that any polygon can be transformed a certain central figure of the same area then the theorem will be proved true. We want that central figure to have a minimum of parameters(why) defining it. Hence an easy choice is any regular polygon with one parameter as it is always fixed with a given area.
Ok, before proceeding let’s adventure through the basic and simple examples of dissection of basic figures: Triangles, Squares, and Rectangles to get some ideas and maybe it can be reduced to these cases only.
Triangle \(\rightarrow\) A Rectangle (Not Predefined):
Rectangle \(\rightarrow\) A Square (Always fixed given an area):
2 Rectangles \(\rightarrow\) A Square (Always fixed given an area):
Ok, due to the rigidity of the square structure, let us consider converting the two rectangles to their corresponding squares by the method described above.
Now observe that we need to change the sum of two square areas to a single square area, Hey that is exactly the Pythagoras Theorem type. Do we know a dissection proof Pythagoras Theorem? Let’s do it in a separate way.
2 Squares \(\rightarrow\) A Square (Always fixed given an area):
n rectangles \(\rightarrow\) A Square (Always fixed given an area):
Simple like induction, add a rectangle every time after transforming two rectangles into a square and follow the previous steps.
Have you seen the usefulness of the Rigidity of the Square Structure and also the hunch of the theorem?
Steps of Proof:
Triangulate the Polygon
Each Triangle -> Rectangle
Set of Rectangles -> A Square by the method described above.
Now given that every polygon with the same area can be transformed into a square with the same area as those of the triangle. QED!
Yaaayy, we proved it! It wasn’t too difficult right!
Now, let’s return to the original question!
Let's keep this suspense till the next post, while in the meantime you get hands-on experience with scissors and paper and enjoy your own journey and tickle your brain to think about "Why Four?" and also "How Four?".
Please share your experiences and thoughts in the comments, which will make the math become alive with Paper, Scissors and You.
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