Goal: Why do certain shapes fit together while others leave gaps?[br][br]Material: a set of physical regular shapes or digital shapes (triangles, squares, pentagons, hexagons, and circles). Links are provided below.[br][br]Task:[br][list=1][*]Try to cover a whole flat surface with only one shape. You may not overlap and leave no gaps.[/*][*]You may use the digital resource below if you do not have physical / concrete regular shapes.[/*][*]Observe which shapes work easily, and which shape is impossible. Why?[br][/*][/list][br][size=85][i]Concept Focus: For a shape to tessellate, the sum of interior angles meeting at a point must be exactly [/i][math]360^\circ[/math][/size].

By now, you would have noticed that for regular polygons, only equilateral triangle, square, and hexagon work well to create tessellations.[br]Let's explore other kinds of shapes that can tesselate:[br][list=1][*]What types of irregular polygon can tessellate?[/*][*]What types of quadrilateral can tessellate?[/*][*]What types of triangle can tessellate?[/*][*]What types of non-polygonal shapes can tessellate?[/*][/list][br]Use a blank graphing calculator to create tessellations that meet the criteria above.[br]Can you build a flowchart to summarise the types of shapes that can be tessellated by using a single type of shape.

A tessellation is named by choosing a vertex and counting the number of sides of each shape touching the vertex.[br]These numbers are then listed in order (clockwise / counter-clockwise) starting with the polygon with the least number of sides.[br]Since this topic only cover single shapes, the configuration will have same numbers.[br][br]Example:

In summary,[br][list][*][b]Identify the repeating unit (motif)[/b][/*][/list]Look for the smallest individual shape or a cluster of shapes that repeats.[br]To be a tessellation, this motif must be able to fill the entire plane without leaving any gaps or overlapping.[br][br][list][*][b]Search for Isometries (Transformations)[/b][/*][/list]Translation: Does the pattern slide in a straight line to repeat?[br]Rotation: Does the pattern pivot/ rotate around a point?[br]Reflection/Glide Reflection: Does the pattern flip across an axis / line?[br][br]A pattern is only a tessellation if these movements can continue infinitely in all directions to cover the surface.[br][list][*][b]The Vertex Path (The 360° Rule)[/b][/*][/list]Locate a point where the corners of the shapes meet (vertex).[br]Calculate the sum of the interior angles of all shapes meeting at that single point. The sum must be exactly 360°.[br]Note: If the sum is less than 360°, there will be a gap. If it is more than 360°, the shapes will overlap.