Presentation on theme: "Tilings and Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore"— Presentation transcript:
Tilings and Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore firstname.lastname@example.org www.math.nus.edu.sg/aslaksen/polyhedra/
They look nice! They teach us mathematics. Mathematics is the abstract study of patterns. Be conscious of shapes, structure and symmetry around you! Why are we interested in this?
What is a polygon? Sides and corners. Regular polygon: Equal sides and equal angles. For n greater than 3, we need both.
A quick course in Greek 34567 TriTetraPentaHexaHepta 89101220 OctaEnneaDecaDodecaIcosa
More about polygons The vertex angle in a regular n-gon is 180 (n-2)/n. To see this, divide the polygon into n triangles. 3: 60 4: 90 5: 108 6: 120
What is a tiling? Tilings or tessellations are coverings of the plane with tiles.
Assumptions about tilings 1 The tiles are regular polygons. The tiling is edge-to-edge. This means that two tiles intersect along a common edge, only at a common vertex or not at all.
Assumptions about tilings 2 All the vertices are of the same type. This means that the same types of polygons meet in the same order (ignoring orientation) at each vertex.
Regular or Platonic tilings A tiling is called Platonic if it uses only one type of polygons. Only three types of Platonic tilings. There must be at least three polygons at each vertex. There cannot be more than six. There cannot be five.
Archimedean or semiregular tilings There are eight tilings that use more than one type of tiles. They are called Archimedean or semiregular tilings.