Recall from our discussion on
Gauss that
if Playfair's postulate is false then there are two possibilities:
at least 2
parallels.
The following picture shows a model of hyperbolic geometry
called the Poincare disk model.
Imagine yourself at the center of the white disk and imagine this
as a bowl that curves away from you. This visualization is not quite
correct, but it will serve our purposes for now.
The blue circle that encloses the disk is actually supposed to
be infinitely far away from you. Hence, while this model looks like
it is a flat disk, it really is not, and so the geometry is different too.
Given line l and point A not on l (as in the picture) it is possible to construct many lines that do not intersect l. In the picture we see four such (dashed) lines that are parallel to l. Hence Playfair's postulate is false in hyperbolic geometry. You might be concerned about that fact that these "lines" look more like curves. Yet in this geometry, these are shortest distance paths that are intrinsically straight (a stream of water would follow them as the path of least resistance), and so in this manner they are valid lines. |

In this picture, We see three points, G, H and I.
To the left of the model, I've measured the sum of the
angles of the resulting hyperbolic triangle.
We see that this sum is 87.485 degrees!
The following file is an interactive version of the model accessible by clicking on this sheet from the lab directions web page. Drag the points H, G and I around to see what happens to the sum of the angles in the resulting hyperbolic triangle and then answer the following questions. Interactive Poincare disk angle sum |

The following picture shows the Poincare disk model with three points X, Y and Z. I have measured angle XYZ and m[2] shows me that the measure of this angle is about 90 degrees. Hence XYZ forms a right triangle with XZ as the hypotenuse. I then calculated XY

The following file is an interactive version of the model. Interactive Poincare Disk Pythagorean theorem