We’ve all learned “parallel lines never intersect” in high school geometry.
But what your teachers failed to explain is that “parallel lines never intersect” is only true in Euclidean geometry, which dates back to Euclid, a Greek mathematician living in Alexandria during the 4th century CE.
In the 1800’s, leading mathematicians developed non-Euclidean geometry – which changed everything that people thought they knew about geometry.
In Euclidean geometry parallel lines always remain at a the same distance from each other no matter how far you extend those lines.
But in non-Euclidean geometry, parallel lines can either curve away from each other (hyperbolic), or curve towards each other (elliptic). This looks like this:
But even though they are curving we see that at one point the lines are parallel at a point. If you have forgotten your basic geometry, parallel lines are represented by the little squares you see above, which signify 90-degree angles.
Here’s the catch, non-Euclidean geometry does not operate on traditional geometric planes. The easiest way to understand non-Euclidean geometry is to use a sphere.
Here’s where the basketball comes in, since basketballs are real life spheres. Let’s focus on the four lines in the center.
If we zoom in, we see:
We see that the lines form 90-degree angles, just like in the first black and white sketch. These lines are parallel at the zoomed in point.
But if we zoom out again, we see that the black lines are curving away from each other, meaning that the black lines we are seeing look like they are hyperbolic.
But these lines are not actually following the curve of the ball. Whoever designed a basketball probably didn’t study non-Euclidean geometry. If he did, he’d know that a sphere is not hyperbolic, but rather it is elliptic.
So, what happens if we followed the natural elliptic curvature of the ball and continued the parallel lines down the side of the ball?
When you extend those lines, they do not curve away from each other. In fact, they actually curve toward each other as they approach the bottom of the ball. This is what this looks like:
You see that these three lines, which are all parallel to each other at the top (the part we zoomed in on), all meet at the common point at the bottom of the basketball when you extend those lines along the natural curvature of the ball.
And if you have a basketball at home, you can try and trace this with your finger – it will feel weird if you do not follow the natural curvature.
And voila! We’ve successfully disproven “parallel lines never intersect” using just a basketball.
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