Hyperbolic Geometry

Hyperbolic geometry is a type of non-Euclidean geometry that arose historically when mathematicians tried to simplify the axioms of Euclidean geometry, and instead discovered unexpectedly that changing one of the axioms to its negation actually produced a consistent theory. Later, physicists discovered practical applications of these ideas to the theory of special relativity.

Hyperbolic geometry also inspired the art of M. C. Escher, and has various theoretical applications as well, including geometric group theory and the theory of modular forms.

The first four axioms of Euclidean geometry, laid out in Euclid's Elements, are essentially self-evident:

(1) Any two points can be connected by a line.
(2) Any line segment can be extended indefinitely.
(3) Given a line segment, a circle can be drawn with center at one of the endpoints and radius equal to the length of the segment.
(4) Any two right angles are congruent.

These four axioms define what is sometimes called absolute geometry. Euclid's first 28 propositions used only these four axioms, but he was forced to add a fifth axiom which was much less obvious than the first four:

(5) The parallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

This statement is a bit complicated and wordy; many modern accounts of Euclidean geometry use a different fifth axiom known as Playfair's axiom:

(5') Playfair's axiom: Given a line $L$ and a point $P$ not on $L$, there is at most one line that can be drawn through $P$ that is parallel to $L$.

$($In fact, axioms (1)-(4) imply that such a line always exists, by dropping a perpendicular to $L$ through $P$ and then dropping a perpendicular to the perpendicular at $P.)$

It can be shown that (1), (2), (3), (4), (5) $\Leftrightarrow$ (1), (2), (3), (4), (5') logically.

Many mathematicians, going back to the ancient Greeks, attempted to show that (5) $($or (5')$)$ were actually consequences of (1)-(4). One method that many of them used was proof by contradiction; start with the axioms (1), (2), (3), (4), and the negation of (5), and try to produce something that is "wrong." A lot of progress was made along these lines, and the properties of the subsequent geometry that was produced were quite strange and different from the propositions of Euclid's geometry. But there was nothing logically inconsistent about them. Eventually, 19$^\text{th}$-century mathematicians (starting with the Russian mathematician Nikolai Ivanovich Lobachevsky) shifted from trying to prove that non-Euclidean geometries were impossible, and began instead to explore the consequences of changing the fifth axiom.

In fact, the fifth axiom cannot be proved (or disproved) from the first four, a fact that was established in the late $19^\text{th}$ century by the Italian mathematician Eugenio Beltrami and others.

Consider the geometry obtained from replacing Playfair's axiom by its negation:

Given a line $L$ and a point $P$ not on $L$, there are at least two distinct lines that can be drawn through $P$ that are parallel to $L$.

Here are some consequences of these axioms:

(1) The interior angles of a triangle sum to less than $180^\circ.$
(2) The interior angles of a quadrilateral sum to less than $360^\circ.$
(3) There are no rectangles.
(4) Given a line $l$ and a point $P$ not on $l$, there are infinitely many lines through $P$ that are parallel to $l$.
(5) Similar triangles are congruent.

Proofs: The proof of (1) is lengthy and is omitted for now.

Clearly (1) implies (2), since a quadrilateral can be split into two triangles by drawing a diagonal. And (2) implies (3), since a rectangle is a quadrilateral with all right angles.

To see (4), drop a perpendicular from $Q$ to $l$; say it meets $l$ at $Q$. (It is convenient to use lower-case letters for lines, and upper-case letters for points.) Then draw a perpendicular line $m$ to $PQ$ with right angle at $P$. It's impossible for $l$ and $m$ to meet, because if they met at a point $X$ then $PQX$ would be a triangle with two right angles, which would contradict (1). So $m$ is parallel to $l$.

Source: http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5.html

Pick a different point $R$ on $L$, draw a line $n$ perpendicular to $l$ at $R$, and drop a perpendicular $PS$ to $n$. Now $PS$ is parallel to $l$ by the same two-right-angles argument as above, but $PS$ is not the same line as $m$; if it were, then $PQRS$ would be a rectangle. For any $R$ on the line $l$, we thus get a new line $PS$ that is parallel to $l$, and a similar argument shows that different choices of $R$ lead to different lines.

To see (5), suppose the triangles are as pictured. Find a triangle $AB''C''$ congruent to the smaller one lying inside the bigger one. Source: http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5.html But then it is straightforward to see that the two angles on the left side of the quadrilateral $BB''C''C$ sum to $180^\circ$, and similarly for the two angles on the right side, so the angles sum to $360^\circ$, which contradicts (2).

There are two common models used to picture lines and angles in plane (two-dimensional) hyperbolic geometry. They are both due to Poincare.

Poincare half-plane model: The points are the complex numbers in the set ${\mathbb H} = \{ z \in {\mathbb C} : \text{Im}(z) > 0 \}$. This is called the upper half-plane; in Cartesian coordinates it consists of points $(x,y)$ such that $y$ is positive.

Lines in this model are of two types: straight vertical lines, and half-circles whose centers lie on the $x$-axis. Angles are measured as one would expect (the angle between the two curves at the point of intersection), but distances are trickier. Generally speaking, one thinks of distances between points near the $x$-axis as "blowing up"; one way to represent this is that the infinitesimal unit of distance $ds$ satisfies the formula

$$(ds)^2 = \frac{(dx)^2+(dy)^2}{y^2},$$

rather than the usual $(ds)^2 = (dx)^2+(dy)^2$ in standard analysis. There is a formula for the distance between two points $z$ and $w$ that uses the inverse hyperbolic trigonometric functions, similar to the one in the Poincare disk model (see below), but it is unwieldy to work with. (One way to think of this is that it is the price one pays for keeping angles at their normal values; there is another model due to Beltrami and Klein with a nicer distance function, and lines which are straight, but angle measures in this model are distorted.)

CD and EF are parallel, EF and GH are parallel, but CD and GH are not.

Poincare disk model: Using a conformal mapping that takes the $x$-axis to the unit circle gives a model of hyperbolic geometry contained inside the unit disk. In this model, lines are either diameters of the disk or the intersection of a circle $C$ with the disk, where $C$ is perpendicular to the unit circle at its two points of intersection. Angles continue to be measured as expected.

The distance formula for two complex numbers $z,w$ inside the disk becomes

$$d(z,w) = \text{arccosh}\left( 1+2\frac{|z-w|^2}{\big(1-|z|^2\big)\big(1-|w|^2\big)} \right),$$

where $\text{arccosh}(x) = \ln\big(x+\sqrt{x^2-1}\big)$ is the inverse function of the hyperbolic cosine.

Note that as $|z|$ approaches $1$, the distance between $z$ and another point goes to $\infty$.

This is a tiling of the hyperbolic plane by congruent triangles.

Modular forms are fundamental objects in modern number theory; they were famously used in Wiles' proof of Fermat's last theorem. They are functions defined on the upper half-plane which are invariant under isometry, which is a (hyperbolic-) distance-preserving map from the upper half-plane to itself. So properties of hyperbolic geometry become important when studying modular forms.

There is an explicit connection between special relativity and hyperbolic geometry via Minkowski space, which is a common setting for both of them.

Hyperbolic geometry graphs have been suggested as a promising model for social networks where the hyperbolicity appears through a competition between similarity and popularity of an individual.

The artist M.C. Escher created many beautiful artworks based on tessellations of the Poincare unit disk. Both "Circle Limit III" and "Circle Limit IV" are famous examples.

HyperRogue is a computer game that lets you experience hyperbolic geometry. It plays similar to a single-player boardgame, but where the chessboard is randomly generated by the computer. In case of HyperRogue, the chessboard is an infinite hyperbolic plane. Many of strategies, locations, and navigation puzzles in HyperRogue are based on the properties of the hyperbolic plane.