Tiling The Plane

When does a piece tile the plane?
This is the question. This is my work on the tiling problem. It wants to become a systematic study, but it isn't yet. It is open to external contributions. Errors are possible. I appreciate the corrections (also of my English :-). I'm happy when I receive an email.

The equilateral triangle, the square and the regular hexagon tile the plane. This is trivial: All the triangles and all the quadrilaterals tile the plane. This is easy but less trivial: Irregular hexagons tile the plane with conditions. A sufficient condition is the parallele edges, an other one is two equal and parallele opposite edges: Also pentagons tile the plane. For example, at left you see "half hexagons". At right equilateral (not regular!) pentagons with two angles of 90°. A sufficient condition for a pentagon to tile the plane is to have two complementary angles (sum=180°) betwin two pairs of equal edges or to have two complementary consecutive angles: There are pieces witch tile the plane and pieces witch don't tile the plane (Mr. Lapalisse). There are pieces witch tile the plane maintaining the same orientation as the Sexominoes below. There are pieces witch must to rotate as the Xominoes. There are pieces witch must to flip as the SexeHexeS or the PolyTriForms. Many polyforms tile the plane. All the Octiamonds tile the plane. All the Pentominoes. All the PolyTriForms. Also Fractals and a Frogs... If we consider the irregular convex polygon tilings where the vertexes coincide, you could see easily below that only the triangle, the quadrilateral, the pentagon and the hexagon can tile the plane. Demonstration:
In a convex polygon each angle is < Pi. Then each knot is minimum triple. A polygon with n edges confina on n identical polygons. The sum of internal angles and the angles confinanti don't must to superate 2*n*Pi. The risult of disequation is n<=6: We can to deform now each polygon to obtain very irregular figures, or artistic, if you are able:

• Triangle. Each edge P1-P2, P2-P3 and P3-P1 can be deformed with a point of symmetry S1, S2 and S3.

• Pentagon. Deform the P4-P5 edge with the point of symmetry S1. The angle P4-P3-P2 plus the angle P2-P1-P5 must do 180°. The edge P3-P4 must be the mirrow of P1-P2. The edge P2-P3 the mirrow of P1-P5.

• Hexagon. P1-P6 must be equal to P3-P4 and have the same orientation. P1-P2, P2-P3, P4-P5 and P5-P6 must have the points of symmetry S1, S2, S3 and S4.

Does a polygon tile the plane?
 Nr. edges Regular Irregular convex Concave 3 YES ALWAYS - 4 YES ALWAYS ALWAYS 5 NO WITH CONDITIONS WITH CONDITIONS 6 YES WITH CONDITIONS WITH CONDITIONS 7 + NO NEVER WITH CONDITIONS _________________

It isn't trivial!

First edition: Jun 12th, 2000 - Last revision: Oct 20th, 2003 •  Below you'll find a mirror of Livio Zucca pages. Click here for my home