In Tessellations: The math of Tiling post, we have learned that there are just three continuous polygons that have the right 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 that the consistent polygons conference at a point add up to 360 degrees.

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*

Figure 1 – Tessellating consistent polygons.


Looking in ~ the other consistent polygons as presented in figure 2, we can see plainly why the polygons cannot tessellate. The sums the the inner angles room either greater than or less than 360 degrees.


*

Figure 2 – Non-tessellating constant polygons.


In this post, we space going to display algebraically the there are only 3 regular 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 meeting at a allude where
*
is the number of sides and also
*
is the variety of vertices. Making use of this notation, the triangular tessellation deserve to be represented as
*
due to the fact that a triangle has 3 sides and also 6 vertices fulfill at a point.

In the proof, as shown in figure 1, we room going to present that the product that the measure of the inner angle the a continuous polygon multiply by the variety of vertices conference at a allude is same to 360 degrees.

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

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Proof:

The angle amount of a polygon v

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

*

which simplifies to 

*
. Making use of Simon’s favourite Factoring Trick, we add
*
come both sides providing us 
*
. Factoring and also simplifying, us have 
*
, i m sorry is indistinguishable to 
*
. Observe the the only possible values for 
*
are 
*
(squares),
*
(regular hexagons), or
*
(equilateral triangles). This way that these space the only continuous tessellations possible which is what we desire to prove.