# Questions related to the definition of the smallest affine plane

Here I have two questions related to affine planes.

1. The smallest affine plane has four points and six lines where

$$\mathcal{P}=\{A, B, C, D\}$$

and

$$\mathcal{L}=\{\{AB\}, \{AC\}, \{AD\}, \{BC\}, \{BD\}, \{CD\}\}$$

as illustrated in the following picture.

However, looking at the figure, I observe that $l(A,D)$ does not contain the point $C$, but there is no line passing through $C$ and parallel to $l(A,D)$. Then, how can this figure describe the smallest four-points affine plane?

1. How can I show that the axioms for an affine plane hold in case of considering the following four-points set

$$\mathcal{P}=\{(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)\}$$

and the following set of lines

$$\mathcal{L}=\{x=0, y=0, z=0, x=y, x=z, y=z\}$$

?

## 1 Answer

1. The line $l(B,C)$ passes through $C$ and is parallel to $l(A, D)$. Recall that two lines in an affine plane are parallel if they do not intersect, as is the case here.

For (2) you'll have to be more specific about what you want to know.

• In Fig it looks like that the lines $l(B,C)$ and $l(A,D)$ intersect each other – Noor Aslam Sep 13 '18 at 4:49
• That's just a feature of the illustration---there is no point on both lines. – Travis Sep 13 '18 at 17:13
• If this is just illustration and there is no point in common then it means that all four points not lies in a same plane? – Noor Aslam Sep 18 '18 at 10:34
• I don't understand the question---by definition the plane contains all four points. – Travis Sep 18 '18 at 11:39