Non-Euclidean Geometry

Ideal triangles in the Poincaré disc.

The parallel postulate

The fifth postulate in Euclid's Elements can be rephrased as

Given a line and a point not on it, exactly one line parallel to the given line can be drawn through the point.

The postulate is not true in 3D, but in 2D it seems to be a valid statement. Considering the importance of postulates however, a seemingly valid statement is not good enough.

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Exactly one line through the red point is parallel to the thick white line.

A postulate (or axiom) is a statement that acts as a starting point for a theory. Since a postulate is a starting point it cannot be proven using previous result. As a statement that cannot be proven, a postulate should be self-evident. The Elements of Euclid is built upon five postulates. For over 2000 years, the fifth postulate has been considered to be less intuitive than the other postulates, and not sufficiently self-evident. Many attempts have been made to prove the fifth postulate using the other four postulates. All these attempts have failed. In the 19th century it was shown that the fifth postulate is independent of the other postulates. It is possible to build a theory of geometry where the fifth postulate is not true. Such geometries are called non-Euclidean. Furthermore it can be shown that non-Euclidean theories can be just as consistent as Euclidean theory. A theory is consistent if it does not contain any contradiction.

The opposite of the parallel postulate, as stated above, is that there are either no lines, or at least two lines "parallel to the given line through the point". It can be shown that if there is at least two lines, there are in fact infinitely many lines "parallel to...".

If the parallel postulate is replaced by:

Given a line and a point not on it, no lines parallel to the given line can be drawn through the point.

you get an elliptic geometry.

If the parallel postulate is replaced by:

Given a line and a point not on it, infinitely many lines parallel to the given line can be drawn through the point.

you get a hyperbolic geometry.

Distances, angles, and lines

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Euclidean, elliptical, and hyperbolic triangle.

Apart from postulates, the theorems in the Elements of Euclid are also built on a number of definitions. Definition number 23 states that two lines are parallel if they never meet. There is nothing in the definition indicating that the distance between two parallel lines is the same everywhere. The parallel postulate is seemingly obvious only if you assume that parallel lines look like railroad tracks. If you redefine what you mean by a line, you may have that two parallel lines either converge towards each other, or diverge from each other. In non-Euclidean geometry, the concept corresponding to a line is a curve called a geodesic. In non-Euclidean geometry a shortest path between two points is along such a geodesic, or "non-Euclidean line".

All theorems in Euclidean geometry that use the fifth postulate, will be altered when you rephrase the parallel postulate. As an example; in Euclidean geometry the sum of the interior angles of a triangle is 180°, in non-Euclidean geometry this is not the case.

Model of elliptical geometry

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One easy way to model elliptical geometry is to consider the geometry on the surface of a sphere. A shortest path between two points on a sphere is along a so called great circle. A great circle is a circle having the same radius, and the same centre, as the sphere. A great circle is a geodesic. If it is clear that the context is elliptical geometry, a great circle can also be called a line.

If two points are directly opposite each other on the sphere (like the north pole and south pole) there are infinitely many shortest paths between the points. If the points are not antipodes, there is only one shortest path.

Two great circles that are not the same, must intersect. Hence, there are no parallel lines on the surface of a sphere.

In GeoGebra 5, a great circle between points A and B can be constructed by creating the plane defined by A, B and the centre of the sphere. Then use the Intersect Two Surfaces tool on the plane and the sphere to construct the geodesic.

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An elliptical triangle is defined by three vertices and the corresponding three great circles through each pair of vertices.

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An interior angle at a vertex of an elliptical triangle can be measured on the tangent plane through that vertex. The sum of interior angles of an elliptical triangle is always > 180°.

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Hyperbolic geometry using the Poincaré disc model

The Poincaré disc (in 2D) is an open disc, i.e the area bounded by a circle not including the circle. When using the Poincaré disc model, only points in the Poincaré disc are considered. The Poincaré disc makes up the entire world. The boundary of the open disc is the circle at infinity, \(C_\infty \).

A geodesic through points A and B is defined as the circular arc through A and B that is perpendicular to \(C_\infty \). If A and B lie on the diameter of \(C_\infty \), that diameter is the geodesic through A and B.

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Red geodesics defined by two red points each.
Given a white geodesic and a red point not on it, there are infinitely many geodesics
through the red point not intersecting the thick white geodesic.

An interior angle of a triangle is measured between the corresponding tangent lines. The sum of interior angles is always < 180°. If the vertices of a triangle move towards circle \(C_\infty \), the angle sum tend to 0. Points that lie at \(C_\infty \) are called ideal points. The limiting triangle as the vertices tend to infinity (\(C_\infty \) ) is called an ideal triangle. An ideal triangle has angle sum 0. Since the vertices of an ideal triangle lie at infinity, the perimeter of an ideal triangle is infinite. Furthermore, the area of an ideal triangle is \(\pi\). Hence, all the triangles in the topmost animation have the same hyperbolic area.

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Hyperbolic triangles.

animated gifs:

Ideal triangles on Ello.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se