From the "Finite affine planes" section of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$:
- each line contains $n$ points,
- each point is contained in $n+1$ lines,
- there are $n^2$ points in all, and
- there is a total of $n^2+n$ lines.
I've read enough to know that a common notation of an affine plane of order $n$ (or at least one obtained from (removing a line and all points on that line on) a Desarguesian projective plane) is $AG(2,n)$, just as a common notation of a Desarguesian projective plane of order $n$ is $PG(2,n)$. For orders (such as order $2$) in which there is only one Desarguesian projective plane up to isomorphism, there is also only one affine plane under isomorphism. From the above bulleted properties of an affine plane of order $n$, it seems clear to me that in the affine plane of order $2$, $AG(2,2)$:
- each line contains $2$ points,
- each point is contained in $2+1=3$ lines,
- there are $2^2=4$ points in all, and
- there is a total of $2^2+2=6$ lines.
$AG(2,2)$ is clearly... (I'm not sure exactly what the correct word is but equivalent or isomorphic or something like that) to the complete graph $K_4$. With the nodes of that graph displayed as the three vertices and center of an equilateral triangle, $AG(2,2)$ looks like the common representation of the Fano plane, $PG(2,2)$, without the line shown as a circle and the points on that circle, and with the three medians of the triangle, going from the vertices, truncating at the triangle center.
I feel pretty confident in what I've written so far (although confirmation would be appreciated), but my main interest in asking this question here is in the (finite) affine $3$-space of order $2$, $AG(3,2)$, and its tetrahedral representation (akin to the tetrahedral representation of $PG(3,2)$). From the Wikipedia article on $PG(3,2)$, that projective $3$-space has the following properties:
- It has $15$ points, $35$ lines and $15$ planes,
- Each point is contained in $7$ lines and $7$ planes,
- Each line is contained in $3$ planes and contains $3$ points,
- Each plane contains $7$ points and $7$ lines,
- Each plane is isomorphic to the Fano plane, $PG(2,2)$,
- Every pair of distinct planes intersect in $1$ line,
- A line and a plane not containing the line intersect in exactly $1$ point.
The article goes on to describe how $PG(3,2)$ can be represented as a tetrahedron. "The $15$ points correspond to the $4$ vertices + $6$ edge-midpoints + $4$ face-centers + $1$ body-center. The $35$ lines correspond to the $6$ edges + $12$ face-medians + $4$ face-incircles + $4$ altitudes from a face to the opposite vertex + $3$ lines connecting the midpoints of opposite edges + $6$ ellipses connecting each edge midpoint with its two non-neighboring face centers. The $15$ planes consist of the $4$ faces + the $6$ "medial" planes connecting each edge to the midpoint of the opposite edge + $4$ "cones" connecting each vertex to the incircle of the opposite face + $1$ "sphere" with the 6 edge centers and the body center."
Removing the "sphere" from $PG(3,2)$ and all the points (the $6$ edge-midpoints and $1$ body center) and lines (the $4$ face-incircles and $4$ lines connecting the midpoints of opposite edges), and then doing the associated truncating of the face-medians and ellipses to just being lines from the vertices to the body center and lines between face-centers, respectively, the resulting $3$-space would have the following properties:
- It has $8$ points, $28$ lines and $14$ planes,
- Each point is contained in $7$ lines and $7$ planes,
- Each line is contained in $3$ planes and contains $2$ points,
- Each plane contains $4$ points and $6$ lines,
- Each plane is isomorphic to $AG(2,2)$ or $K_4$,
- Every pair of distinct planes intersect in $1$ line,
- A line and a plane not containing the line intersect in $\le1$ point.
In a tetrahedral representation of this $3$-space (akin to the tetrahedral representation of $PG(3,2)$), the $8$ points correspond to the $4$ vertices + $4$ face-centers. The $28$ lines correspond to the $6$ edges + $12$ truncated face-medians (lines from the face-centers to the vertices on that race) + $4$ altitudes from a face to the opposite vertex (these altitudes would cross each other in the center of the tetrahedron if not bowed somehow, but there wouldn't be an actual point in the $3$-space there) + $6$ lines ("truncated ellipses") connecting each pair of face-centers. The $14$ planes consist of the $4$ faces + the $6$ "medial" planes bisecting the dihedral angle at each edge (but no longer extending to the midpoint of the opposite edge, which is no longer a point in the $3$-space) + $4$ "mini-pyramids" connecting each vertex to centers of the adjacent faces.
The vertices are each on the $3$ adjacent faces + $3$ "medial" planes + the $1$ "mini-pyramid" including that vertex, while the face-centers are each on the $1$ face that it's the center of + the $3$ "medial" planes that include the vertex opposite that face + the $3$ "mini-pyramids" that don't include that vertex. The edges are each on the $2$ adjacent faces + $1$ "medial" plane extending from that edge. The truncated face-medians are each on the $1$ face it's a truncated median of + $1$ "medial" plane + $1$ "mini-pyramid". The altitudes are each on the $3$ "medial" planes which extend from an edge of the face that is the base of that altitude. And the lines connecting pairs of face-centers ("truncated ellipses") are each in $1$ "medial" plane and $2$ "mini-pyramids".
The shape described in the last two paragraphs and the group of bullet points immediately before them is, I believe now, one way to represent $AG(3,2)$.
Am I correct?
When I first posted this question, I thought $AG(3,2)$ is basically the complete graph $K_5$ with all $5$ of the component $K_4$s being a plane, with its tetrahedral representation having one "plane" including the entire exterior of the tetrahedron and having either "pinched" triangles or a point at infinity for the other four "planes". But I remembered reading that affine planes were just projective planes minus a single line and all the points on it, and that the relationship was similar for higher-dimensional affine spaces.
Thank you for reading this question and for any attempts to answer it.
[Basically rewritten (apart from the $2$-dimensional part and the description of $PG(3,2)$) to reflect a new hypothesis for what $AG(3,2)$ is and looks like.]