| The figure shows a fixed d-line and point P,
with a second d-line through P. Rotate the second d-line by moving
the green point in a circular motion around P.
You will see that there are exactly two configurations where the d-lines a parallel. These occur when the two d-lines 'meet' on the boundary of the disc at the points marked 1 and 2. (Of course the points on the disc boundary are not part of the space.) Since there are two possible parallel d-lines to the given d-line, the Euclidean parallel postulate does not hold in this space. As you continue to rotate the second d-line about the point P you will see that not only is there a family of d-lines that intersect the first one but there is also a family of lines that do not intersec and are not parallel. These are the ultra parallel lines. |