 # Tiling/packing problems

9411216.pdf. Co.combinatorics - Tiling A Rectangle With A Hint of Magic. Here's a a famous problem: There are a number of proofs of this result (14 proofs in this particular paper). One would think this problem is a tedious exercise in combinatorics, but the broad range of solutions which do not rely on combinatorial methods makes me wonder what deeper principles are at work here. In particular, my question is about the proof using double integrals which I sketch out below: Suppose the given rectangle has dimensions and without loss of generality suppose has a corner at coordinate . Notice that iff is an integer. If we sum over all tile rectangles , we get that the area integral over is also zero: Since the cornor of the rectangle is at , it follows that either or must be an integer. My question is as follows: where exactly does such a proof come from and how does it generalize to other questions concerning tiling? One can pick other functions to integrate over such as and the result will follow. Tilings. 82_04_tiling.pdf. Easier Fibonacci Number puzzles. All these puzzles except one (which??) Have the Fibonacci numbers as their answers. So now you have the puzzle and the answer - so what's left? Just the explanation of why the Fibonacci numbers are the answer - that's the real puzzle!! Puzzles on this page have fairly straight-forward and simple explanations as to why their solution involves the Fibonacci numbers;. Puzzles on the next page are harder to explain but they still have the Fibonacci Numbers as their solutions. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More.. Fibonacci numbers and Brick Wall Patterns If we want to build a brick wall out of the usual size of brick which has a length twice as long as its height, and if our wall is to be two units tall, we can make our wall in a number of patterns, depending on how long we want it: There's just one wall pattern which is 1 unit wide - made by putting the brick on its end.

Penrose tiling. A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right. A Penrose tiling has many remarkable properties, most notably: It is non-periodic, which means that it lacks any translational symmetry.It is self-similar, so the same patterns occur at larger and larger scales. Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules, cut and project schemes and coverings. Background and history Periodic and aperiodic tilings Figure 1.  