Welcome to an exploration of non-Euclidean geometry. Non-Euclidean geometry is any geometry that differs from Euclidean geometry by rejecting or modifying Euclid's fifth postulate, also known as the parallel postulate. Euclidean geometry is the familiar geometry taught in schools, based on Euclid's Elements, a mathematical treatise consisting of 13 books written around 300 BCE. In Euclidean geometry, through a point not on a given line, exactly one line can be drawn parallel to the given line. This seemingly simple statement leads to the familiar flat geometry we use every day.
Let's examine Euclid's Fifth Postulate, commonly known as the Parallel Postulate. In its original form, it states that if a line crosses two other lines, and the interior angles on one side sum to less than 180 degrees, then the two lines will eventually intersect on that side. The modern equivalent is simpler to understand: through a point not on a given line, exactly one line can be drawn parallel to the given line. This postulate is what distinguishes Euclidean geometry from non-Euclidean geometries. For centuries, mathematicians tried to prove this postulate from the other axioms, but all attempts failed. Eventually, they realized that different, consistent geometries could be created by modifying this postulate.
Now let's explore hyperbolic geometry, one of the two main types of non-Euclidean geometry. In hyperbolic geometry, through a point not on a given line, infinitely many lines can be drawn parallel to the given line. This creates a negatively curved space, often visualized as a saddle shape. The Poincaré disk model shown here is one way to represent hyperbolic space in two dimensions. In hyperbolic geometry, the sum of angles in a triangle is always less than 180 degrees, and the area of a triangle depends on its angles. Parallel lines diverge from each other, moving away in both directions. Hyperbolic geometry has important applications in Einstein's theory of relativity, network science, and certain models of the universe.
Let's now explore elliptic geometry, the second major type of non-Euclidean geometry. In elliptic geometry, through a point not on a given line, no lines can be drawn parallel to the given line. All lines eventually intersect. This creates a positively curved space, similar to the surface of a sphere. On a sphere, 'lines' are represented by great circles, which are circles with the same center as the sphere and with the sphere's radius. Any two great circles always intersect at exactly two points. In elliptic geometry, the sum of angles in a triangle is always greater than 180 degrees, and the total area of the space is finite. Elliptic geometry has practical applications in navigation on Earth's surface, astronomy, and Einstein's general theory of relativity.
To summarize what we've learned about non-Euclidean geometry: Non-Euclidean geometry differs from Euclidean geometry by modifying or rejecting the parallel postulate. The two main types are hyperbolic and elliptic geometry. In hyperbolic geometry, through a point not on a given line, infinitely many parallel lines can be drawn, creating a negatively curved space like a saddle. In elliptic geometry, no parallel lines exist, as all lines eventually intersect, creating a positively curved space like a sphere. The sum of angles in a triangle equals 180 degrees in Euclidean geometry, is less than 180 degrees in hyperbolic geometry, and is greater than 180 degrees in elliptic geometry. These non-Euclidean geometries have important applications in Einstein's theory of relativity, navigation, astronomy, network science, and models of the universe.