Recall our discussion about
Escher
Escher's 1960 Circle Limit IV (Heaven and Hell) |
Escher based his work on this model:
Poincare disc model of hyperbolic geometry |

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 above picture we see four such (dashed) lines that are parallel to l through point A. Just as parallels behave differently in perspective geometry (they intersect at a vanishing point), here in Escher's world we have different parallels - many of them through one point. 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 class highlights 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

In the weeks to come, we will see that there are many real-life applications of hyperbolic geometry, such as models of the internet, building crystal structures to store more hydrogen or absorb more toxic metals, mapping the brain, mapping the universe, and modeling Mercury's orbit.