# What Are the Only Regular Figures That Fill the Plane?

• mprm86
In summary, the only shapes that can fill the plane in a regular tesselation are the triangle, square, and hexagon. This can be proven using algebra, and the only possible values for n are 3, 4, and 6. Any other regular n-gons would not be able to fit together without leaving gaps.
mprm86
Show that the only regular figures that fills the plane are the triangle, the square and the hexagon.

This can be done with algebra after the problem has been restated.

Note that the angle of a corner of a regular n-gon equals (n-2).Pi/n.

To fit the plane, k copies of a regular n-gon must be able to touch with their corners and therefore k angles should make up for a total arc of 2.Pi.
So, k should be chosen such that

k(n-2)Pi/n = 2Pi eq.
k(n-2) = 2n (*) eq.
k = 2n/(n-2).

So, we must choose n such that n-2 | 2n.
This looks already as if there are only few possibities. First note that n must be larger than 2. (Otherwise we don't even have a polygon).

n=3 gives 1|6 which is true, and k = 6/1 = 6
n=4 gives 2|8 which is true, and k = 8/2 = 4
n=5 gives 3|10 which is NOT true
n=6 gives 4|12 which is true, and k=12/4 = 3

How does it go on?

Well, the next quotient will be smaller than 3, so it must be 2, but then this would mean that just two corners n-gon fill an arc of 2Pi and this corner should be Pi, but this does not happen for a finite n.
Therefore, there are no other regular n-gons that tesselate the plane.

---

You can also look at solving the same equation (*) for n, we get:
n = 2k/(k-2)

So k should satisfy k-2 | 2k
So either
(i) k-2 = 1 or
(ii) k-2 = 2 or
(iii) k-2 is odd and k-2|k
(iv) k-2 is even and (k-2)/2 | k

Ad (iii) k=2i+1 and 2i-1 | 2i+1. It is clear that if i>1 then (2i+1)/(2i-1) < 2, so this leaves no solutions
Ad (iv) k=2i and (i-1)|2i eq. i=2 OR i=3 only. i=2 gives k=4 and we had this already. i=3 gives k=6 and n=3.

This approach gives the same solutions.

QED

Thanks alot.

## What are regular tessellations?

Regular tessellations are patterns made up of repeated shapes that cover a flat surface without any gaps or overlaps. These shapes are all the same size and shape, and the pattern can continue infinitely in all directions.

## What are the three types of regular tessellations?

The three types of regular tessellations are called regular triangles, regular hexagons, and regular squares. These shapes can fill a plane without leaving any gaps or overlaps.

## What is the difference between regular and irregular tessellations?

Regular tessellations have repeating shapes that are all the same size and shape, while irregular tessellations have different shapes and sizes that fit together without any gaps or overlaps.

## What is the significance of regular tessellations in mathematics?

Regular tessellations are important in mathematics because they demonstrate symmetry, geometry, and the concept of infinity. They also have real-world applications in architecture, art, and design.

## Can any shape be used to create a regular tessellation?

No, not all shapes can be used to create a regular tessellation. For a shape to be able to form a regular tessellation, the angles of the shape must be able to fit together without any gaps or overlaps.

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