Euclid’s Eulogy: The Death of the Fifth Postulate

Some time ago, Euclid came up with five rules.

The last rule is the Parallel Postulate. It’s just a fancy way of saying parallel lines.

These five rules make up all of geometry.

If we follow all five rules, then we can say we’re in Euclidean geometry.

However, not everything is in Euclidean geometry.

Take this coin funnel.

If you dropped two coins into the funnel, they wouldn’t go straight (compared to if you rolled them onto a flat sheet of paper). Instead, they will converge to the center. This is called Elliptic gemeotry.

Hyperbolic is the opposite. The coins would diverge.

So basically, in Euclidean geometry, lines are straight, ie. two lines that start off straight will always be straight.

In Elliptic geometry, lines that start off straight will eventually converage. In Hyperbolic geometry, they will diverage.

Here is a diagram.

In Elliptic and Hyberbolic geometry, lines are no longer parallel.

Which breaks Euclid’s fifth rule. Thus, they are known as non-Euclidean geometries.

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