In Tessellations: The math of Tiling post, we have actually learned that there are just three continuous polygons that have the right to tessellate the plane: squares, it is intended triangles, and also regular hexagons. In Figure 1, we have the right to see why this is so. The angle amount of the interior angles that the constant polygons conference at a point include up come 360 degrees.

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*

Figure 1 – Tessellating consistent polygons.


Looking at the other constant polygons as shown in figure 2, we deserve to see clearly why the polygons cannot tessellate. The sums the the interior angles are either greater than or much less than 360 degrees.


*

Figure 2 – Non-tessellating continuous polygons.


In this post, we are going to show algebraically that there are only 3 consistent tessellations. Us will usage the notation 

*
, comparable to what we have actually used in the proof that there room only five platonic solids, to stand for the polygons conference at a suggest where
*
is the number of sides and
*
is the variety of vertices. Making use of this notation, the triangular tessellation can be stood for as
*
because a triangle has 3 sides and 6 vertices satisfy at a point.

In the proof, as presented in figure 1, we room going to present that the product that the measure up of the interior angle that a consistent polygon multiply by the variety of vertices meeting at a point is same to 360 degrees.

Theorem: There are just three consistent tessellations: it is provided triangles, squares, and also regular hexagons.

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Proof:

The angle amount of a polygon through

*
sides is
*
. This method that each interior angle that a continuous polygon measures 
*
. The variety of polygons conference at a point is
*
. The product is therefore

*

which simplifies to 

*
. Making use of Simon’s favorite Factoring Trick, we add
*
come both sides giving us 
*
. Factoring and also simplifying, us have 
*
, which is tantamount to 
*
. Observe that the only feasible values for 
*
are 
*
(squares),
*
(regular hexagons), or
*
(equilateral triangles). This means that these room the only continual tessellations possible which is what we want to prove.