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An IBL Introduction to Geometries

Mark A. Fitch

Section 1.2 Completeness

Definition 1.2.1. Fano Geometry.

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

Checkpoint 1.2.2.

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?
  12. What do the answers to Item 1.2.2.5, Item 1.2.2.9, and Item 1.2.2.11 say about this attempt at constructing a geometry?
  13. What is needed to fix the difficulty noted in the previous question?
Solution.
  1. Draw line AB.
  2. Add point C on line AB.
  3. Add point D not on line AB.
  4. No additional lines are needed.
  5. There is one line in this geometry.
  6. N/A
  7. A line is required through point D. There are two ways to do this. Firstly we can draw a straight line AD, and secondly we can draw a curved line that goes through points A, D, and C. Let's go forward the second way with the curved line.
  8. This satisfies axiom 2. One line goes through A, B, and C. The other lines goes through A, B, and C.
  9. There are two lines in this geometry.
  10. We can add as many more lines as we would like, we just need to be mindful that there are three points on every line that we add, and if there aren't we must add points.
  11. At least 2 lines are in this geometry.
  12. The answers for numbers 5, 9, and 11 show that if we are careful with how we add lines and points, we can make a geometry with only two lines and 4 points. If we constructed the geometry differently, we would have needed to add more points.
  13. The difficulty in needed to add more and more to satisfy the axioms can be fixed by adding carefully. An example is in number 7, we added a curved line specifically so that it could go through 3 existing points and we would not need to add another point in another step.
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