# 1.2: Completeness

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Definition: Fano Geometry

Use the following axioms and definitions of intersection and parallel as a definition of the Fano geometry.

more stuff

1. There exists at least one line.
2. There are exactly three points on every line.
3. Not all points are on the same line.

Explore the Fano geometry as follows.

1. Draw a line using Geogebra.
2. Add a third point to the line.
3. Note that Axiom 3 requires one more point. Draw one.
4. Must any more lines be added? If so, do so.
5. How many lines are in this geometry?
6. Add the axiom: every point is on at least one line.
7. To your line with three points, and one point not on that line add any lines required by this new axiom.
8. Make sure these lines satisfy Axiom 2.
9. How many lines are in this geometry?
10. May any more lines be added? If so, do so, and be sure the axioms are satisfied.
11. How many lines are in this geometry?