Editing Projective plane (section)

===A finite example===

This example has just thirteen points and thirteen lines. We label the points P<sub>1</sub>, ..., P<sub>13</sub> and the lines m<sub>1</sub>, ..., m<sub>13</sub>. The [[incidence relation]] (which points are on which lines) can be given by the following [[incidence matrix]]. The rows are labelled by the points and the columns are labelled by the lines. A 1 in row ''i'' and column ''j'' means that the point P<sub>''i''</sub> is on the line m<sub>''j''</sub>, while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige–Wexler normal form.
:::{| class="wikitable" style="text-align:center;"
|-
! {{diagonal split header|Points|Lines}}
! m<sub>1</sub>
! m<sub>2</sub> !! m<sub>3</sub> !! m<sub>4</sub>
! m<sub>5</sub> !! m<sub>6</sub> !! m<sub>7</sub>
! m<sub>8</sub> !! m<sub>9</sub> !! m<sub>10</sub>
! m<sub>11</sub>!! m<sub>12</sub>!! m<sub>13</sub>
|- style="border-bottom:2px solid #999;"
! P<sub>1</sub>
| bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp;
|-
! P<sub>2</sub>
| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp;
|-
! P<sub>3</sub>
| bgcolor="#9cf"|1 || &nbsp; ||&nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp;
|- style="border-bottom:2px solid #999;"
! P<sub>4</sub>
| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;|| &nbsp;||&nbsp;||&nbsp; || &nbsp; || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1
|-
! P<sub>5</sub>
| &nbsp; || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || bgcolor="#9cf"|1 ||&nbsp;|| &nbsp;|| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || &nbsp; || &nbsp;
|-
! P<sub>6</sub>
| &nbsp; || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || &nbsp;||&nbsp;|| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || &nbsp;
|- style="border-bottom:2px solid #999;"
! P<sub>7</sub>
|   || bgcolor="#9cf"|1 ||   ||   ||   || || bgcolor="#9cf"|1 || ||   || bgcolor="#9cf"|1 ||   ||   || bgcolor="#9cf"|1
|-
! P<sub>8</sub>
|   ||   || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 || ||  || || bgcolor="#9cf"|1 ||   ||   ||   || bgcolor="#9cf"|1
|-
! P<sub>9</sub>
|   ||   || bgcolor="#9cf"|1 ||   ||   || bgcolor="#9cf"|1 ||  || ||   || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 ||   ||
|- style="border-bottom:2px solid #999;"
! P<sub>10</sub>
|   ||   || bgcolor="#9cf"|1 ||   ||   || || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 ||   ||   ||   || bgcolor="#9cf"|1 ||
|-
! P<sub>11</sub>
|   ||   ||   || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || ||  || ||   || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 ||
|-
! P<sub>12</sub>
|   ||   ||   || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 ||  || bgcolor="#9cf"|1 ||   ||   ||   ||   || bgcolor="#9cf"|1
|-
! P<sub>13</sub>
|   ||   ||   || bgcolor="#9cf"|1 ||   || || bgcolor="#9cf"|1 || || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 ||   ||
|}

To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1s appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1s appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P<sub>1</sub>, P<sub>4</sub>, P<sub>5</sub>, and P<sub>8</sub>, for example, will satisfy the third condition. This example is known as the '''projective plane of order three'''.