In Tessellations: The mathematics of Tiling post, we have actually learned that over there are just three constant polygons that have the right to tessellate the plane: squares, equilateral triangles, and also regular hexagons. In Figure 1, we deserve to see why this is so. The angle sum of the interior angles of the consistent polygons meeting at a point include up come 360 degrees.

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*

Figure 1 – Tessellating regular polygons.


Looking in ~ the other continuous polygons as shown in number 2, we have the right to see plainly why the polygons cannot tessellate. The sums that the inner angles room either higher than or less than 360 degrees.


*

Figure 2 – Non-tessellating regular polygons.


In this post, we room going to display algebraically the there are just 3 continual tessellations. Us will usage the notation 

*
, similar to what we have used in the proof the there are only 5 platonic solids, to stand for the polygons meeting at a allude where
*
is the variety of sides and also
*
is the variety of vertices. Using this notation, the triangular tessellation can be represented as
*
since a triangle has actually 3 sides and also 6 vertices meet at a point.

In the proof, as shown in number 1, we are going to show that the product of the measure up of the inner angle of a consistent polygon multiply by the variety of vertices meeting at a suggest is same to 360 degrees.

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

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Proof:

The angle amount of a polygon v

*
political parties is
*
. This way that each interior angle that a constant polygon measures 
*
. The number of polygons meeting at a suggest is
*
. The product is therefore

*

which simplifies to 

*
. Using Simon’s favourite Factoring Trick, we include
*
to both sides offering us 
*
. Factoring and also simplifying, us have 
*
, i beg your pardon is indistinguishable to 
*
. Observe that the only feasible values for 
*
are 
*
(squares),
*
(regular hexagons), or
*
(equilateral triangles). This means that these space the only consistent tessellations possible which is what we want to prove.