Regular Covering of The Plane, 5th Spanish Task

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REGULAR COVERING OF THE PLANE The plane can be regularly covered by only three regular polygons: an equilateral triangle, a square or a hexagon. Now you are going to see it and you will know the reason. 1. The screen is divided in three parts. We are going to cover each of them with three polygons that you will find in the toolbar. ( We must show the points of the polygons on another polygon constructed previously, an anticlockwise direction) 2. Do you know how long the interior angles of each figure are? Interior angle of the equilateral triangle: Interior angle of the square: Interior angle of the hexagon: If you don´t know it you can measure them with the tool angle. 3. Look at the different mosaics: a) In the mosaic of equilateral triangles: How many triangles match up the same vertex? b) In the mosaic of squares: How many squares match up the same vertex? c) In the mosaic of hexagons: How many hexagons match up the same vertex? 4. Are the angles of paragraph 2 related with the angle of 360 degrees? Help: Look at the result of paragraph 3. 5. Find out if there is another regular polygon that achieves this. Why? Yolanda Baena Muñoz, Creación realizada con GeoGebra