A Single Line That Opened Hidden Universes

 

Mathematics has always been about discovering the rules of the universe, and sometimes, a single line in an ancient book can lead to revolutionary ideas. One such line, buried in the oldest known math book, puzzled mathematicians for centuries. This line, Euclid’s fifth postulate, seemed almost unnecessary so much so that for hundreds of years, experts tried to prove it using simpler principles. But when they stopped trying to prove it and instead questioned it, they uncovered entire new universes.

Euclid’s Controversial Fifth Postulate

Euclid, the father of geometry, wrote a book called Elements over 2,300 years ago. In it, he outlined five basic rules, or postulates, for geometry. The first four postulates were simple and intuitive, like “A straight line can be drawn between any two points.” But the fifth postulate, also known as the parallel postulate, was different. It stated:

“If a straight line falls on two straight lines in such a manner that the interior angles on one side add up to less than 180 degrees, the two lines will eventually meet on that side when extended.”

This rule essentially says that if you have a line and a point not on that line, only one unique line can be drawn parallel to the first one. The problem? Unlike the other four postulates, this one wasn’t obvious. Mathematicians suspected that maybe, just maybe, it could be derived from the other four.

For over 2,000 years, they tried and failed.

Parallel Lines on a Curved Surface?

What happens if we remove or modify Euclid’s fifth postulate? This is where things got weird. It turns out that if you assume more than one parallel line can exist through a point, or that no parallel lines exist, you end up with entirely new types of geometry.

One of these is hyperbolic geometry, which describes a space where parallel lines diverge and never stay the same distance apart. Imagine trying to lay a flat sheet of paper on a curved surface—it crumples, right? That’s the hyperbolic plane. Mathematician János Bolyai created a model to describe this, called the Poincaré disk model. In this model, straight lines appear as arcs of circles that intersect the boundary of a disk at right angles.

On the other hand, if you assume no parallel lines exist, you get spherical geometry—think of lines on a globe, where any two “straight” lines eventually meet.

Gauss, Bolyai, and the Birth of Non-Euclidean Geometry

The idea of non-Euclidean geometry fascinated Carl Friedrich Gauss, one of history’s greatest mathematicians. He even used spherical geometry to measure the shape of the Earth! While Gauss kept his work private, János Bolyai and Nikolai Lobachevsky independently developed hyperbolic geometry, forever changing mathematics.

By tweaking one assumption, they showed that Euclidean geometry wasn’t the only way to describe the universe. This realization had enormous consequences, especially for physics.

Einstein and the Geometry of SpaceTime

Fast forward to the early 20th century. Albert Einstein was struggling to understand gravity. Isaac Newton described gravity as an invisible force, but Einstein suspected that gravity was actually the effect of massive objects warping the fabric of space and time, a concept called general relativity.

Imagine placing a bowling ball on a trampoline. The ball creates a dip in the fabric, and if you roll a marble nearby, it naturally moves toward the ball, just like how planets orbit stars. This bending of space was the missing piece of the puzzle—straight lines in the universe are actually curved due to gravity.

Einstein’s insight led to an entirely new way of thinking about space. In fact, it turned out that our universe isn’t perfectly Euclidean! The large-scale structure of the universe is determined by its mass-energy content, which affects its curvature.

What Shape Is Our Universe?

Astronomers use a clever trick to measure the universe’s shape: they look at giant cosmic triangles. When we observe the cosmic microwave background—the faint glow left over from the Big Bang—we can measure the angles of large-scale cosmic structures.

• If the sum of the angles in these triangles is exactly 180°, space is flat, meaning it follows Euclidean rules.

• If the sum is more than 180°, space is curved like a sphere.

• If the sum is less than 180°, space is curved like a hyperbolic plane.

So far, all observations suggest that the universe is remarkably flat. But here’s the mystery: the mass-energy density required to make the universe flat is extremely specific. If the universe had just a little more or less mass energy, it would be curved. We don’t fully understand why the universe has precisely the mass-energy density it does.

Why Does This Matter?

At first glance, geometry might seem like a topic for school textbooks. But as we’ve seen, questioning an old assumption led to the discovery of new geometries, which ultimately helped Einstein describe gravity, which in turn shaped our understanding of the entire cosmos.

From Euclid’s stubborn postulate to Einstein’s warped SpaceTime, geometry has been at the heart of some of humanity’s greatest discoveries. And who knows? Maybe modifying another assumption will reveal yet another hidden universe.

Sources & Further Reading:

• Euclid’s Elements, translated by T.L. Heath

• János Bolyai’s The Science Absolute of Space

• Albert Einstein’s Relativity: The Special and General Theory

• NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) data on the curvature of the universe

• “The Shape of Space” by Jeffrey R. Weeks

• Carl Friedrich Gauss’ geodetic studies on Earth’s curvature