A Greek in an Italian restaurant

 
In the past, we have reported on Pythagoras's appearances in Melbourne. We wrote about the façade of Federation Square, consisting of special right-angled triangles and a pinwheel tiling. In another column, we discussed the buried sculpture on Swanston Street, based upon a 3-4-5 triangle. They were great illustrations of Pythagoras at work, but neither compares to the geometric wizardry we just discovered in our local Italian restaurant.

Entering the restaurant, we looked straight past the menu to the unusual tiling on the floor. It consists of small grey squares and larger red squares. This may not seem so Pythagorean, but the triangles are there, just hidden.

 
In the diagram above we've employed the sides of the grey and red squares as the smaller sides of a right-angled triangle; we've also drawn an empty square off of the hypotenuse. Pythagoras's Theorem then says that the sum of the areas of the grey and red squares is equal to the area of the empty square.

What is remarkable is that hidden in the floor tiling is a PROOF of Pythagoras's Theorem. The left diagram is a patch of the floor tiling consisting of 16 squares of each colour. The right diagram is comprised of 16 of the large empty squares in a grid. The two diagrams can now be naturally superimposed, but with some bits of the coloured squares sticking out.

 
Here is the punch line: all the coloured bits sticking out up top and to the right can be cut off and exactly refitted into the spaces down below and on the left. This means that the 16 empty squares and the 32 coloured squares sum to exactly the same area. This is exactly Pythagoras's Theorem. Q.E.D.

But have we really proved Pythagoras's Theorem? What about other right-angled triangles? Or what if we had used a tiling with squares of different sizes? Well, the final question answers the first two.

Starting with any right-angled triangle we can use the two short sides to make a restaurant tiling with coloured squares. Then we create the grid of empty squares and they fit as hypotenuse squares just as before. We argue just as above, and Pythagoras's Theorem holds true again.

By the way, the red squares in our restaurant are exactly twice the size of the grey squares. So, the hidden triangles have exactly the same proportions as the triangles in the façade of Federation Square. Definitely the most mathematical restaurant in Melbourne.