Drawing rectangular grids. Rectangular arrays. Drawing and reading number lines. Number lines and repeating patterns. Drawing an analogue clock face. Number sequences. Introducing number sequences. Let's have a party! Find a rule. Triangles tessellate. Making tessellations. Growing patterns can tessellate. Familiar situations. A pattern hunt. Patterns in school activities.
Repeating patterns in school activities. Triangles Squares Hexagons. Large grid of triangles Large grid of squares Large grid of hexagons. Squares, triangles Hexagons, triangles Hexagons, squares, triangles Octagons, squares Dodecagons, triangles Dodecagons, hexagons, squares. Large grid of squares and triangles Large grid of hexagons and triangles Large grid of hexagons, squares and triangles Large grid of octagons and squares Large grid of dodecagons and triangles Large grid of dodecagons, hexagons and squares.
Irregular pentagons Waffle pattern Fish patterns. Looking for other tessellating polygons is a complex problem, so we will organize the question by the number of sides in the polygon. The simplest polygons have three sides, so we begin with triangles:.
To see this, take an arbitrary triangle and rotate it about the midpoint of one of its sides. The resulting parallelogram tessellates:.
This property of triangles will be the foundation of our study of polygon tessellations, so we state it here:. Moving up from triangles, we turn to four sided polygons, the quadrilaterals. Before continuing, try the Quadrilateral Tessellation Exploration. Taking a little more care with the argument, we have:. The point of all the letters is that the angles of the triangles make the angles of the quadrilateral, which would not work if the quadrilateral was divided as shown on the right.
Begin with an arbitrary quadrilateral ABCD. The angles around each vertex are exactly the four angles of the original quadrilateral.
Recall from Fundamental Concepts that a convex shape has no dents. All triangles are convex, but there are non-convex quadrilaterals. The technique for tessellating with quadrilaterals works just as well for non-convex quadrilaterals:. It is worth noting that the general quadrilateral tessellation results in a wallpaper pattern with p2 symmetry group. Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane.
In both cases, the angle sum of the shape plays a key role. The next simplest shape after the three and four sided polygon is the five sided polygon: the pentagon. Rather than repeat the angle sum calculation for every possible number of sides, we look for a pattern.
In fact, there are pentagons which do not tessellate the plane. Attempting to fit regular polygons together leads to one of the two pictures below:. Can you make them fit together to cover the paper without any gaps between them? This is called 'tessellating'. What about triangles with two equal sides? These are isosceles triangles. Can you tessellate all isosceles triangles? Now try with right angled triangles. These have one right angle or 90 degree angle.
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