Dr. Sarah's Introduction to Hyperbolic Geometry
Imagine yourself at the center of the 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 boundary 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.
Recall our discussion about
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.
Move the points around (in the form of a triangle)
so that the sum of the angles
is as small as you can make it.
How small is this number?
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.
Poincare disk angle sum
Question 2 Move the points around (in the form of a triangle)
so that the sum of the angles
is as large as you can make it. How big is this number?
Question 3 Is your last triangle large or small?
The following picture shows the Poincare disk model with three points
X, Y and Z. I have measured angle XYZ and m 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 XY2 + YZ2 and compared it to XZ
2 to see whether the
Pythagorean theorem holds in this model. We see that for this
triangle XY2 + YZ2 - XZ2
= -.335 and so we see that
XY2 + YZ2 < XZ2 by .335.
Hence the Pythagorean theorem does not
hold for this triangle in this model.
The following file is an interactive version of the model.
Interactive Poincare Disk
Drag the points to make a small right triangle.
Be sure that the points don't
touch and be sure that the angle (m)
is as close to 90 degrees as you can make it without going under (ie
make sure the angle is not 89.8 degrees).
XY2 + YZ2 - XZ2
(which is listed as
Drag the points to make a large right triangle and be sure that the angle
(m) is close to 90 degrees without going under. What is
XY2 + YZ2 - XZ2?
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.