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A worksheet on incidence geometry and parallel properties.
We recall the axioms of incidence geometry:
(I1) Given two distinct points, there is a unique line incident with both
of them.
(I2) Given a line, there exist two distinct points incident with it.
(I3) There exist three non-collinear points.
Assume only the incidence axioms I1 and I3.
Show that the following statements can already be deduced:
(i)
For every point, there exists at least one line not passing through it.
(ii)
For every point \(P\), there exist at least two distinct lines passing through \(P\).
In each of the following models, determine which of the incidence axioms I1, I2, and
I3 are satisfied and which are not. Then determine whether the model has the
elliptic, Euclidean, or hyperbolic parallel property.
(i)
“Points” are lines in Euclidean three-dimensional space, “lines” are planes
in Euclidean three-dimensional space, and “incidence” is the usual relation
of a line lying in a plane.
(ii)
Same as in (i), except that we restrict ourselves to lines and planes passing
through a fixed ordinary point \(O\).
(iii)
“Points” are points on the unit sphere in \(\mathbb {R}^3\), “lines” are great circles (that
is, intersections of the sphere with planes in \(\mathbb {R}^3\) passing through \(0\)), and
“incidence” is the standard one: a point is incident to a line if it lies on it.
(i1)
Can you find a model with only \(2\) points satisfying I1 and I2?
(i2)
Do you think it is possible to find two such models which are not
isomorphic? Prove your claim.
(ii1)
Can you find a model with only \(3\) points satisfying I1, I2, and I3?
(ii2)
Do you think it is possible to find two such models which are not
isomorphic? Prove your claim.
Show that axioms I1, I2, and I3 are independent. In other words, for each axiom,
find a model satisfying the other two but not that one.
If time permits.
(i)
Find a model of incidence geometry (i.e. satisfying axioms I1, I2, and I3)
which does not satisfy any of the elliptic, Euclidean, or hyperbolic parallel
properties.
Hint. There is such a model with \(5\) points.
(ii)
How many non-isomorphic models of incidence geometry with \(4\) points
can you find? For each model, determine which parallel property (elliptic,
Euclidean, hyperbolic), or lack of parallel property, it satisfies.
