Why polygons tessellate

Active 7 years, 10 months ago. Viewed 27k times. Nick Nick 6, 9 9 gold badges 39 39 silver badges 67 67 bronze badges. You've probably changed my entire future with that. My entire future. I wouldn't still be in mathematics if not for this. Add a comment. Active Oldest Votes. Zubin Mukerjee Zubin Mukerjee I hope you like this answer.

Community Bot 1. By the way, if you want me to add something more, let me know it. Yes, in the 3D as well please. Just enough detail for any person's curiosity with uni-polygonal tessellation to be satisfied.

Also, I'd be pleased if someone put a diagram into Zubin Mukerjee's answer. This page has an interesting version. And the question is related with billiards! Later, I'll add some explanation. I love them.

Show 2 more comments. In hyperbolic geometry you can tesselate by regular polygons with any number of edges. Brian Rushton Brian Rushton Sign up or log in Sign up using Google. For those who have access to The Geometer's Sketchpad the following explorations are available. Recall that a polygon is a closed plane figure made by joining line segments. You might want to review the relevant material in Fundamental Concepts concerning polygons before reading this section.

More precisely, which polygons can be used as the only tile in a monohedral tessellation of the plane? Before moving on, you may want to do the Tessellation Exploration: The Basics. Stacks of these strips cover a rectangular region and the pattern can clearly be extended to cover the entire plane. The same technique works with parallelograms, and so:. 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. What shapes meet here? Three hexagons meet at this vertex, and a hexagon has 6 sides. So this is called a "6. And always start at the polygon with the least number of sides, so "3. Question 2: One of those patterns becomes different when we make a mirror-image of it Reading, MA: Addison-Wesley, Weisstein, Eric W.

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