In Tessellations: The math of Tiling post, we have learned that over there are just three continuous polygons that deserve to tessellate the plane: squares, it is intended triangles, and regular hexagons. In Figure 1, we can see why this is so. The angle sum of the internal angles of the consistent polygons conference at a point add up to 360 degrees.

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*

Figure 1 – Tessellating regular polygons.


Looking at the other constant polygons as displayed in figure 2, we have the right to see clearly why the polygons can not tessellate. The sums that the inner angles are either greater than or less than 360 degrees.


*

Figure 2 – Non-tessellating constant polygons.


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

*
, comparable to what we have used in the proof the there room only 5 platonic solids, to stand for the polygons conference at a suggest where
*
is the variety of sides and also
*
is the number of vertices. Making use of this notation, the triangle tessellation deserve to be stood for as
*
since a triangle has actually 3 sides and also 6 vertices accomplish at a point.

In the proof, as shown in figure 1, we are going to display that the product of the measure up of the internal angle that a consistent polygon multiplied by the variety of vertices conference at a point is equal to 360 degrees.

Theorem: There are only three constant tessellations: it is intended triangles, squares, and also regular hexagons.

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Proof:

The angle amount of a polygon through

*
sides is
*
. This way that each inner angle the a constant polygon measures 
*
. The number of polygons meeting at a suggest is
*
. The product is therefore

*

which simplifies to 

*
. Using Simon’s favorite Factoring Trick, we add
*
to both sides giving us 
*
. Factoring and simplifying, we have 
*
, i m sorry is indistinguishable to 
*
. Observe the the only possible values for 
*
are 
*
(squares),
*
(regular hexagons), or
*
(equilateral triangles). This means that these are the only continual tessellations possible which is what we want to prove.