Tessellating
The Plane With Regular Polygons
Yesterday you found a complete list of combinations of regular polygons that fit without gaps or overlaps around a single point.
1. Which of the arrangements of regular polygons that will fit around a point ( a local solution) can be extended to cover the entire plane (a global solution)? To keep this problem manageable I am going to impose the severe restriction that ant two vertices (point where three or more tiles meet) are the same. More formally, there must be an isometry of the entire tessellation that takes any vertex to any other vertex. There are eleven solutions. Note that in this investigation you must take into account the order of the tiles. For example, two 6-gons and two 3-gons is a different permutation from a 6-gon, a 3-gon, a 6-gon and a 3-gon. You must check both of these arrangements.
2. You will have noticed at once that the patterns on all these tilings repeat. You would have no difficulty telling someone over the phone how to continue each design. Can you find in each tiling a parallelogram that contains all the information necessary to reproduce the tiling? In other words, find a parallelogram that you could email to someone who could then simply translate copies of your parallelogram and thus reproduce the tiling.
3. A repeat of a tessellation is a region that will generate the tiling just by translating that region. A parallelogram repeat will often cut across the individual tiles. Can you find a repeat of each of your tessellations that leaves the individual tiles intact? In general, the repeat will not be a parallelogram. Find a repeat in each of the MC Escher posters around the room.
4. If you were to make a patty paper copy of each of these tessellations and turn it over, you will find that the copy can be made to match up with the original tessellation in every case except one. Which is the odd man out?
5. You will find that it is possible to place a mirror perpendicular to the plane of one of your tessellations in such a way that the reflection in the mirror is the same as the part of the tessellation behind the mirror. The tessellations has mirror or reflection symmetry. This is not true of one of the tilings. Which?
6. For each tiling draw some of the lines of reflection symmetry and investigate the regions into which the plane is divided by the mirror lines. In this exercise you are ignoring the actual tiling and just concentrating on the underlying grid of mirror lines. What are some of the interior angles of these regions?
7. You will find that it is possible to find points in every one of your tilings that have the following property. Make a patty paper copy of a fragment of the tiling. Place the copy directly over the original. If you place the point of your pencil directly on one of these special points you will be able to rotate the patty paper in such a way that after a rotation of 60, 90, 120 or 180 degrees the patty paper copy will match up with the original tiling. These points are called 6-fold, 4-fold, 3-fold and 2-fold rotocentres respectively. A rotocentre is a centre of rotational symmetry. You will find them at the centre of a polygon, at a vertex or at the midpoint of an edge shared by two tiles of the same type. For each tiling find the pattern formed by the rotocentres of the same type.
8. You will have noticed from a previous investigation that a regular 5-gon will not tile the plane on its own. They form gaps that cannot be filled. But you can pack the plane with pentagons. A packing is an arrangement that leaves gaps. Ideally the gaps are as small as possible. Investigate the particularly perplexing, but peculiarly pleasing, packing the plane with pentagons problem.
9. A problem like the packing problem is the covering problem. In a covering of the plane gaps are not allowed but tiles may overlap. Can you find a covering of the plane with regular pentagons in which the overlaps are as small as possible?