# What’s non-Euclidean geometry?

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I never got beyond calculus in school, and I’ve heard this term thrown around by smart math and science people bit have no clue what it means or why it’s special.

In: Mathematics

Euclid dealt with flat planes. Everything could be figured out with a straight edge and a piece of string.

What if the surface were curved?

Seems obvious now, but wasn’t then.

Non-Euclidean geometry is the geometry for non-flat surfaces. Most of the rules we learn in geometry only apply to Euclidean geometry. Parallel lines never intersect, the sum of the angles in a triangle always add up to 180 degrees, etc.

These rules do not hold true for non-Euclidean geometry. Look at the surface of the earth. Despite what a flat earther would tell you, the Earth is a globe. Lines of longitude are parallel lines. In a flat map representation of the Earth, you can see these are parallel. But those parallel lines all intersect at both the north and south poles. Similarly, if you were to pick three points far enough apart, say New York City, São Paulo in Brazil, and Tokyo in Japan, the triangle formed by these three would be greater than 180 degrees.

Similarly, you could think of the inside of a bowl. This would be another non Euclidean surface. A triangle between three points inside of a big enough bowl would have the angles of these lines be less than 180 degrees.

Euclidean geometry is the geometry that you’re most familiar with in your every day life. It’s the geometry you learned about it school. It deals with flat spaces. (Flat in this context means not curved, not flat as in a piece of paper.) In Euclidean geometry, the sum of angles in a triangle is always 180 degrees and parallel lines will never meet. You probably remember learning those axioms in school.

Non-Euclidean geometry is any kind of geometry other than that. So any kind of curved space or shape is non-Euclidean. For example, the surface of a sphere is non-Euclidean because it’s curved – the sum of the angles in a triangle on the surface of a sphere will not always be 180 degrees and parallel lines will meet.

Euclid came up with a bunch of rules he noticed geometry follows. Parallel lines will never cross. If lines are not parallel and go on forever in both directions they will cross. Things you would take for granted and just intuitively understand, for the most part.

It turns out some break down if you do not work on a flat plane. For example on a globe, lines of longitude are parallel at the equator and cross at the north and south pole, and you can draw a triangle with corners that add up to 270 degrees instead of 180 degress.

So geometry on a curved surface is considered non-euclidean geometry.

Excellent video on that.

You can also run into more exotic non-euclidean geometry in fiction or video games. Designing a room that is bigger on the inside than the outside, or a doorway that opens up to the other side of the planet is easy in a game, and it violates the rules of Euclidean geometry.

Euclid had 5 postulates, things we assume to be true because they are obvious

1. A line can be drawn between two points

2. A line segment can be extended infinitely

3. A circle can be drawn with any radius

4. All right angles are equal

5. Any two lines with a 3rd line crossing them both such that the sum of the interior angles on one side is less than 180°, the lines will eventually intersect on that side.

The first 4 are all pretty straight forward, we can see why those are true because it’s obvious. The problem is the 5th postulate.

Euclidean geometry is when we assume the 5th postulate to be true. This just happens to be when the space we are working in is flat.

If we have space that is curved in some way, the other 4 postulates hold true, but the 5th doesn’t. This is noneuclidean space.

An example of noneuclidean space would be the surface of a sphere. This would be what we call positive curvature. Take the Earth, for example, at the equator, the lines of longitude are parallel, but at the poles, they intersect. In Euclidean space, parallel lines never intersect (which is provable with the use of the 5th postulate).

Mathematicians, including Euclid, hated the 5th postulate because it was so much more complicated than the other 4, so many mathematicians spent years of their lives trying to either prove it or resolve it in some other way.

Writers like HP Lovecraft also use the term “non-Euclidean geometry” to describe the bizarre sanity-twisting environments their alien monsters live in. But mostly it means the not-on-a-flat plane stuff the other folks have said. 🙂

Normal Euclidean geometry is on a flat surface, think a piece of paper. Where things like angles in a triangle always add up to 180 degrees.

Non-euclidean geometry is just geometry on a curved surface. Usual examples are a sphere or a horse saddle shaped plane. For example you can make a triangle with 3 90° corners, breaking the previously mentioned triangle rule.

Because of this you do things you otherwise couldn’t do in Euclidean geometry.

The geometry of Euclid is flat. Parallel lines remain constant separation.

Classic counter example is spherical geometry. Longitude lines are parallel but meet at the poles. This can be a bit confusing because it’s possible to treat a sphere in uncurved space.

It takes a bit of imagination. Some people are so used to flat space geometry that anything else is unimaginable.

Euclidean geometry is what we experience in this world. It has rules such as walking east 1 mile then north 1 mile is the same as walking sqrt(2) miles on the diagonal. It’s just life.

Well, mathematically all sorts of other “worlds” can exist and still be internally consistent. For example, the diagonal above could be equal to 2 miles. What? That’s not how diagonals work. Yep, you would have to be in non-Euclidean space for that weird diagonal to work, but it turns out, that mathematically this new world actually makes sense.

Non-Euclidean geometry is simply all mathematics that does not apply to the world that we live in.

Geometry in space that is not uniformly distributed.

Imagine a 3D graph with two diagonal parallel lines. Both lines increment the same amount of Y for every X. Let’s say that Z is constant but different from one another, i.e.; one line is further away from the other but they have the same angle.

Keep that in mind.

Now, if we imagine the grid of such a graph, it would be composed of cubes.

In Euclidean space we treat all these cubes as uniform; they all share the same width, height and depth. But what if they didn’t?

In non-Euclidian space those cubes can have different dimensions. Some could have more width than others for example. If the width of the cubes increases with Z, the line in the back would appear (from the perspective of the line in the front) to have a steeper angle. It has more “X” per X in the front which means that it moves more in the Y direction. The parallel lines appear to cross. However both lines still increase the same amount of Y per X!

If we would look from the line in the back to the one in the front it would appear to decrease in angle. It has less “X” per X in the back which means that it moves less in the Y direction.

Space itself is non-euclidian in nature. Gravitational lensing is a phenomena that describes how light bends around large masses due to them distorting spacetime, i.e.; mass influences the size of the “grid cubes” in space.