Non-Euclidean geometry is a field of mathematical study that challenges and expands the postulates established by Euclid in his work "The Elements."
Unlike Euclidean geometry, which is based on five fundamental postulates, non-Euclidean geometries arise by modifying the fifth postulate, known as the parallel postulate.
Fundamentals of Euclidean geometry
Euclidean geometry is based on five fundamental axioms:
- Given any pair of points, there exists a straight line segment that joins them.
- Any straight line segment can be extended indefinitely in both directions.
- Given a point and a radius, a circle can be drawn with that radius.
- All right angles are equal to each other.
- Given a point outside a line, there exists one and only one line parallel to the given line that passes through that point.
The first four postulates are intuitively acceptable and form the basis of classical geometry. However, the fifth postulate has been the subject of debate for centuries, as its formulation is not as obvious as the others. Many mathematicians attempted to prove it as a theorem derived from the other four, but were unsuccessful.
This led to the development of new geometries where this postulate was modified or replaced.
Non-Euclidean geometries
The two main non-Euclidean geometries are hyperbolic geometry and elliptic geometry. Both arise from the negation of Euclid's fifth postulate and exhibit characteristics that defy classical intuition.
Hyperbolic geometry
Hyperbolic geometry, developed independently by Nikolai Lobachevsky and János Bolyai in the 19th century, postulates that an infinite number of parallel lines pass through a point outside a given line. Some key features of this geometry include:
- The angles of a triangle add up to less than 180 degrees.
- Parallel lines can diverge in both directions.
- There are no rectangles in the Euclidean sense.
- Space can be modeled on a surface of negative curvature, such as the Poincaré disk model or the hyperbolic half-plane model.
This geometry has found applications in Einstein's theory of general relativity, as it describes the behavior of space-time in the presence of gravitational masses.
Elliptical geometry
Elliptic geometry, pioneered by Bernhard Riemann in the 19th century, postulates that there are no parallel lines, since all lines eventually intersect. This can be visualized in the geometry of a sphere, where the "lines" are great circles (geodesics), and any pair of these lines intersect. Some properties include:
- The angles of a triangle add up to more than 180 degrees.
- Straight lines are finite but have no edges.
- Rectangles do not exist.
This geometry is relevant in cosmology, where it is used to model positively curved universes.
Applications of non-Euclidean geometry
Non-Euclidean geometry has had a profound impact on various scientific and technological disciplines:
- Physics : In general relativity, space-time is modeled by a geometry of variable curvature, which allows gravitation to be described in a geometric framework.
- Navigation and cartography : Geodetic coordinate systems on Earth use concepts from spheroidal geometry, a form of non-Euclidean geometry.
- Cryptography and number theory : Some mathematical structures that arise in hyperbolic geometry are applied in cryptographic algorithms.
- Art and Design : The representation of non-Euclidean spaces has inspired artists such as MC Escher, whose work explores impossible patterns and unusual perspectives.
Mathematical models
There are several mathematical models that allow us to visualize and work with non-Euclidean geometry:
- Poincaré disk model : Represents hyperbolic geometry on a unit disk where the geodesic lines are circular arcs orthogonal to the edge of the disk.
- Hyperbolic half-plane model : Uses the upper half of the Cartesian plane, where the geodesics are semicircles orthogonal to the horizontal axis.
- Spheroidal geometry : Models elliptical geometry on a sphere, where great circles act as "straight lines."
Philosophical and mathematical impact
The discovery of non-Euclidean geometries had a significant impact on philosophy and mathematics:
- He questioned the uniqueness of mathematical truth , showing that there are multiple equally valid geometric systems.
- He promoted research on the consistency of axiomatic systems , leading to the work of Hilbert and Gödel.
- He redefined the perception of space and reality , influencing modern physics and the theory of relativity.