Hyperbolic Geometry And The Poincare Disk

Perhaps the most famous example of mathematical art is the work of Escher.

“The six largest (three white angles and three black devils) are arranged about the centre and radiate from it. The disc is divided into six sections in which, turn and turn about, the angles on a black background and then the devils on a white one, gain the upper hand. In this way, heaven and hell change places six times.” M. C. Escher

Speaking about the first work in his Circle Limit Series Escher says:

”The limit is no longer a point, but a line which borders the whole complex and gives it a logical boundary. In this way one creates, as it where, a universe, a geometrical enclosure. If the progressive reduction in size radiates in all directions at an equal rate, then the limit becomes circle.” M. C. Escher

This is a kind tessellation of hyperbolic space which we are going to try to understand.

Hyperbolic space is a kind of Non-Euclidian geometry. So lets start with Euclid. Euclid founded his geometry on five postulates. The first four are quite simple, although they sound little odd form a modern perspective. The idea is that you have a strait edge and a compass. So we can draw a unique strait line between two points (Postulate 1). We can extend that line as far as we want in either direction (Postulate 2). We can also draw a circle centered on any point of any radius (postulate 3). The fourth postulate simply states that all right angles are the same.

But it’s the fifth postulate that causes all the controversy. For two thousand years it had mathematicians scratching there heads in confusion. It’s called the Parallel postulate and a modern formulation goes like this: Given a line and a point on the line, there is a unique line passing though the point parallel to the original line. Lines are considered parallel if they never intersect.

The problem was that everyone felt it should be a theorem. It seemed that it was an essential and unavoidable fact of geometry that must follow from the first four postulates. Euclid, it seemed, had not finished the job.

There is a long and convoluted history of mathematicians trying and failing to prove the parallel postulate. Long story short, Gauss showed that this was impossible. You can have a perfectly valid geometry obeying the first four postulates, but violating the fifth. There are two possibilities. The first is that we have no parallel lines. This gives elliptic geometry, which is the geometry of the surface of a sphere. Strait lines are the great circles, and any two great circles will intersect. The other possibility is that we have have more than one parallel line passing through the point. That gives us hyperbolic geometry.

The Poincare Disk

The Poincare disk is a model of hyperbolic geometry in which the whole infinite space is squashed into a disk. Strait lines in hyperbolic space are modelled by circles on the disk. But not any circle – only circles that meet the boundary at right angles will do. A chord through the center is also hyperbolic line, as we can think of this as a circle with infinite radius.

Reflection in a line in the hyperbolic plane is described by a circle inversion. A circle inversion swaps all the points in the plane outside the circle with all the points inside such that for each point:

\( CP = \frac{r^2}{CQ} \)

where CP is distance from the center to point, CQ is the distance to the inverted point.
The geometry works out like this:

where PQC and CQA form right angle triangles. Its interactive – you can drag the points! Your welcome.

Suppose we have two points in the disk , p and q, and we want to find a line that connects them. We want to find a circle that passes through both points and intersects with the boundary at right angles. Wikipedia gives us a recipe: Find the inverses of the points, p’ and q’. Now find the midpoints (we call them m and n) of pp’ and qq’. Now we shoot out a line from m perpendicular to pp’, and another from q perpendicular to q’. The intersection of theses two lines is the centre of the circle we are trying to find. Wikipedia offers no proof, but its easy to see that this works in special cases, such as points on the boundary. Here’s another interactive diagram:

Drag the points inside the disk.

Distance, Area and Angle
The hyperbolic distance between two points is not the same as the euclidian distance. Its not even the length of the arcs representing the hyperbolic lines. The whole of the infinite hyperbolic plane has been squeezed into this disk. So the boundary of the disk its at infinity. Fortunately there is a fairly simple formula for calculating the distance

\( d(AB) = \log\frac{AQ . BP }{ BQ . AP}\)

where p and q are the points where the arc meets the boundary, and AQ etc are the euclidian distances between points. Note that the distance form any point on the boundary is infinite as expected.

On the other hand angles in hyperbolic space really are just ordinary euclidian angles in the disk. The Poincare disk is also called the conformal picture, which just means angle preserving.

In Euclidian space the three angles of a triangle add up to pi (180 degrees). Thats not true for hyperbolic triangles. The angles of a triangle always sum to less than pi. The difference gives the area of the triangle:
\( Area = \pi – (\alpha + \beta + \gamma)\)

up to a constant, where alpha , beta and gamma are the interior angles of the hyperbolic triangle.

From our formula for distance it is easy to verify that Pythagoras theorem does not hold in hyperbolic space. In a previous post I demonstrated a visual proof of Pythagoras theorem. So why does that fail here? Proofs make assumptions and that proof assumes a lot of common sense facts about euclidean geometry. Its all about squares, and we don’t really have squares in hyperbolic geometry. We don’t have a regular polygon with four right angles.

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