Three Regular Tessellations
Theorem
There exist exactly $3$ regular tessellations of the plane.
Triangles
Equilateral triangles form a regular tessellation:
Squares
Squares form a regular tessellation:
Hexagons
Regular hexagons form a regular tessellation:
Proof
Let $m$ be the number of sides of each of the regular polygons that form the regular tessellation.
Let $n$ be the number of those regular polygons which meet at each vertex.
From Internal Angles of Regular Polygon, the internal angles of each polygon measure $\dfrac {\paren {m - 2} 180^\circ} m$.
The sum of the internal angles at a point is equal to $360^\circ$ by Sum of Angles between Straight Lines at Point form Four Right Angles
So:
\(\ds n \paren {\dfrac {\paren {m - 2} 180^\circ} m}\) | \(=\) | \(\ds 360^\circ\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {m - 2} m\) | \(=\) | \(\ds \dfrac 2 n\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1 - \dfrac 2 m\) | \(=\) | \(\ds \dfrac 2 n\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 m + \dfrac 1 n\) | \(=\) | \(\ds \dfrac 1 2\) |
But $m$ and $n$ are both greater than $2$.
So:
- if $m = 3$, $n = 6$.
- if $m = 4$, $n = 4$.
- if $m = 5$, $n = \dfrac {10} 3$, which is not an integer.
- if $m = 6$, $n = 3$.
Now suppose $m > 6$.
We have:
\(\ds \dfrac 1 m + \dfrac 1 n\) | \(=\) | \(\ds \dfrac 1 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 6 + \dfrac 1 n\) | \(>\) | \(\ds \dfrac 1 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 n\) | \(>\) | \(\ds \dfrac 1 3\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3\) | \(>\) | \(\ds n\) |
But there are no integers between $2$ and $3$, so $m \not > 6$.
There are $3$ possibilities in all.
Therefore all regular tessellations have been accounted for.
$\blacksquare$
Also see
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): tessellation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tessellation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): tessellation