ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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$P G (2, 3)$ has $13$ points and $13$ lines

Example

The finite projective plane $P G (2, 3)$ has $13$ points and $13$ lines, and each line contains $4$ points.

Facts & Assumptions

Given: The projective plane $P G (2, 3)$ .

[L1]

$P G (2, 3)$ is a finite projective plane of order $3$ (For every prime power $q$ , the space $P G (2, q)$ is a projective plane of order $q$ ).

[L2]

A projective plane of order $n$ has $n^{2} + n + 1$ points and the same number of lines (A finite projective plane of order $n$ has $n^{2} + n + 1$ points and the same number of lines).

Verification

technique · direct

1.1

Applying [L2] with $n = 3$ gives $3^{2} + 3 + 1 = 13$ points and $13$ lines.

L1L2algebra

2.1

The line $⟨ e_{1}, e_{2} ⟩$ consists of the one-dimensional subspaces represented by $(1, 0, 0)$ , $(0, 1, 0)$ , $(1, 1, 0)$ , and $(1, 2, 0)$ , so it has $4$ projective points, exactly as order $3$ predicts.

L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Noam D. Elkies, Math 155 notes: Feb. 3 (standard reference, not scraped)

PG(2,3)PG(2,3) has 1313 points and 1313 lines