Number of Regions in Plane Defined by Given Number of Lines

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Theorem

The maximum number $L_n$ of regions in the plane that can be defined by $n$ straight lines in the plane is:

$L_n = \dfrac {n \paren {n + 1} } 2 + 1$

This sequence is A000124 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Setting up a Recurrence Rule

First we consider the plane with no lines at all.

This has one region, so $L_0 = 1$.

Now when we have one line, we divide the plane into two regions, so $L_1 = 2$.


Now consider the $n$th line.

This increase the number of regions by $k$ iff it splits $k$ of the old regions.

It can split $k$ of the old regions if and only if it hits the existing lines on the plane in $k-1$ places.

Two straight lines can intersect in at most one point.

So the new line can intersect the $n - 1$ old ones in at most $n - 1$ different points.

Therefore $k \le n$.

So we see that $L_n \le L_{n - 1} + n$.


Now, it is always possible to place the $n$th line so that:

$(1): \quad$ It is not parallel to any of the others, and therefore intersects all the other $n - 1$ lines
$(2): \quad$ It does not go through any of the existing intersection points (so intersects them all in different places).

Thus we see that $L_n \ge L_{n - 1} + n$.


Hence the recurrence:

$L_n = L_{n - 1} + n$


Solution of Recurrence

Using induction, we show that:

$L_n = \dfrac {n \paren {n + 1} } 2 + 1$

The base case is straightforward:

$L_0 = 1 = \dfrac {0 \paren {0 + 1} } 2 + 1$
$L_1 = 2 = \dfrac {1 \paren {1 + 1} } 2 + 1$


Now assume the induction hypothesis:

$L_k = \dfrac {k \paren {k + 1} } 2 + 1$

and try to show:

$L_{k + 1} = \dfrac {\paren {k + 1} \paren {k + 2} } 2 + 1$


Hence the induction step:

\(\displaystyle L_{k + 1}\) \(=\) \(\displaystyle L_k + k + 1\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {k \paren {k + 1} } 2 + 1 + k + 1\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {k + 1} \paren {k + 2} } 2 + 1\) after algebra

Hence the result by induction.

$\blacksquare$


Examples

Number of Regions in Plane Defined by $6$ of Lines

With $6$ lines, the plane can be divided into a maximum of $22$ regions:


LinesDissectingPlane-6.png


Historical Note

This result was first published in $1826$ by Jakob Steiner, in one of the first issues of Journal für die reine und angewandte Mathematik.


Sources