tessellations
Tessellations are important, interesting and beautiful. One interesting way of thinking of them is as flat polyhedra. For example, I suppose you know how to prove there are only 5 regular 3D polyhedra. How could you use this to prove how many regular tessellations there are? How about semiregular ones? This picture from the Science Museum has some nice ones (click to enlarge). How would you classify a tessellation? A good way is as with polyhedra, by looking at the vertices. Look at these on the right; what shapes surround the vertices for each one? Of course, the angles at each vertex add up to 360 degrees. But what about in 3D shapes  what do the angles round vertices add up to there? What about the degrees left over? 

fivefold symmetry in tessellations
I am fascinated by shapes with fivefold symmetry. The (regular) pentagon, pentagram, dodecahedron and icosahedron all have fivefold symmetry and I believe their beauty is mostly due to this property.
It is impossible to tessellate the plane solely with shapes with fivefold symmetry, but I think this is what makes them interesting. Kepler got very close with the beautiful tessellation below. Looking at the first picture it seems as though it might just work, but further tessellation reveals the need for 'joined decagons' as shown in the second picture, which Kepler called 'monsters'. I made these pieces using plywood and a laser cutter.
tessellating pentagons
As mentioned above, it is not possible to tessellate the plane with regular pentagons. However, it is possible with nonregular pentagons. On the right is a slide show of a tessellation I made (again with a laser cutter) inspired by the design on Ravensbourne College in Greenwich (below), created by architects AZPA. Can you work out how and why it works? Can you work out the angles? Can you create a pentagon tile of your own that tessellates in an interesting way? 

duality in tessellations
I have been thinking about duality in 3D shapes a lot lately, but there is also duality in 2D tessellations (you could think of a tessellation as a flat polyhedron).
Here are some slide shows with some ideas on dual tessellations... (click to play)


I enjoyed investigating the tessellation on the slide show below. I experimented with a ruler and compass and came up with some interesting results...
tilings that are not edgetoedge
square tilings
Following on from tilings that are not edgetoedge, you can clearly tile the plane using two squares.
How can you take this further? Well, you could try tiling the plane in 3 squares... or 4 squares... or more? In fact, this is a really interesting investigation. One possible approach is to consider groups of square tiles that when joined together create other polygons that tessellate (eg pentomino shapes)  see image on the right. Or perhaps you could try creating a tiling using a central square similar to the fibonacci spiral? You can use square tilings to prove Pythagoras' theorem by cutting through the squares using perpendicular diagonals. Watch the video below to see a demo of this! 
tessellation mosaics
Here are a couple of interesting tessellations made by my year 7 and 8 students.