Hilbert's 3rd Problem

Area is a quantity that expresses the extent of surface or shape, in the plane. More practically it is amount of paint necessary to cover the shape. When we want to calculate the area of some figure, we place the figure over a graph paper and count number of squares inside it. That number is related to area of the figure. So in the end we know how to calculate the area of square and use it to calculate area of any other figure. Mathematically, what this says is given that we know the area of a square, we know the area of any object, and that is done by simply dividing the object into pieces of square and counting them. Now Gauss asked the question weather one can cut and rearrange a polygon to any other polygon of same area. Its not hard to see that the answer is yes. The algorithm is very simple:

• Triangulate the polygons

• rearrange each triangle into rectangles of correct side length, so that when all the triangles are joined we get a square.

So when we start with two polygon with same same area we will end up with same square. With this question answered we can ask the same question for higher dimensions, that is given a polytope can we cut and glue it to get a cube of same volume. This was unanswered as was third problem proposed in the list of Hilbert's problem. It was answered by Max Dehn one of Hilbert's student in same year it was asked. The answer is no, it cannot be done, in fact a tetrahedron cannot be cut and rearranged into cube. He gave a proof that was geometric and he gave an invariant (called dehn's Invariant) which classify this problem. That is two elements of same volume and same dehn invariant can be cut and rearranged into one another. Basically for plane there is only one dehn invariant, but for higher dimensions dehn invariant can take a set of values.

There are many proof of this, but one way is by using homological algebra, this project is done using it. One of the reason for this approach is because this is first time I have seen the use of algebra to solve some non-trivial real life question.