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

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*

Figure 1 – Tessellating regular polygons.


Looking in ~ the other continual polygons as presented in number 2, we have the right to see clearly why the polygons cannot tessellate. The sums of the inner angles room either greater than or less than 360 degrees.


*

Figure 2 – Non-tessellating constant polygons.


In this post, we are going to display algebraically the there are only 3 consistent tessellations. We will use the notation 

*
, similar to what we have actually used in the proof the there room only 5 platonic solids, to represent 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 can be stood for as
*
since a triangle has actually 3 sides and also 6 vertices fulfill at a point.

In the proof, as presented in number 1, we are going to display that the product the the measure of the internal angle the a continual polygon multiply by the number of vertices conference at a point is same to 360 degrees.

Theorem: There are only three constant tessellations: equilateral triangles, squares, and regular hexagons.

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Proof:

The angle amount of a polygon with

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

*

which simplifies to 

*
. Utilizing Simon’s favorite Factoring Trick, we include
*
to both sides offering us 
*
. Factoring and also simplifying, us have 
*
, i m sorry is equivalent to 
*
. Observe that the only feasible values for 
*
are 
*
(squares),
*
(regular hexagons), or
*
(equilateral triangles). This method that these are the only regular tessellations feasible which is what we desire to prove.