When you think of a straight line, I’m sure you think of using a ruler or going between two points on a flat surface, but how do you think of a straight line in space? Straight lines are typically thought of as the shortest distance between two points. Take a piece of paper, make two dots, then use a ruler or another piece of paper and connect the dots with the least amount of drawing. This is called a straight line to most people and they don’t know any different. What they don’t realize is that everywhere in the universe is on a curved or elliptical plane. Newton states that the presence of mass causes curvature based on gravity, but Einstein had theories as well. In Newton’s theory, gravity makes particles leave their straight paths causing them to come together and not follow a straight line. In Einstein’s theory of general relativity, gravity is a distortion of space-time. Particles still follow the straightest possible paths in that space-time. But because space-time is now distorted, even on those straightest paths, particles accelerate as if they were under the influence of what Newton called the gravitational force, (Physics, 2015) Basically, Newton’s theory says in space when you have two objects close together the gravitational pull they exert on each other causes curvature, making a straight line actually be curved, but when you’re drawing a straight line you don’t imagine that the line would end up back where you started, even though that is exactly the “straight line” planets take in their orbits.

Just like drawing on a sphere, start on one side; continue the line and you end up right back where you started; this is what Einstein’s theory was stating. A straight line can really be curved- the shortest distance doesn’t always have to be straight in the sense that most people look at. Einstein viewed the world from a bigger picture and saw that outside forces reacting to each other could change the way straight lines are perceived. It somewhat boggles the mind, but when you think about it for a bit, it all seems to fall into place and makes sense.

When we discussed the parallel postulate in class it seemed as if it could be true in every situation, until you thought about it and applied it in the non-Euclidean space, which is one type of 2-dimensional geometry. In Euclidean geometry it is very cut and dry on how lines look and can or cannot intersect, but as you look into hyperbolic geometry or elliptic geometry things start to look a bit different. In hyperbolic geometry your parallel lines seem to curve away from each other and in elliptic geometry they curve towards each other.

When non-Euclidean geometry first came to fruition, Lobachevsky helped show how lines could curve and still be straight by negating the parallel postulate. The parallel postulate states that 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. α+β<180°, therefore those two line will intersect at some point, (wikipedia, 2015). This is all fine on a flat surface. I know that it’s a bit complicated to think that there really are not many flat surfaces though.

Think of the bigger picture: look down on Earth as if you were on the moon. No matter which way you look the path curves around. When you walk to your car it seems as if you are walking on a flat surface and the further in you look at the ground it appears to seem as if there’s undeniable proof that you are on a flat surface. What most people don’t see is that it’s not flat.

Imagine you’re a ladybug; you are living a very happy life. You have friends and loved ones that are impossible to replace and your home is amazing, but you need to go on a trek to get food every day. You have to go many inches across the treacherous terrain to find the best aphids to bring home for dinner. You start your journey each morning and it seems like you cross many flat surfaces with only a few bumps along the road and you’re successful every day. Now jump back out and look at this journey from a human perspective. This ladybug thought the journey was long and flat in areas where you see that it was short and somewhat curved slightly in spots. Crossing an almost flat leaf to the lady bug looks differently to you. You see the slight curve that it had to take and realize that to the ladybug the leaf is somewhat like the earth.

It’s a bit mind boggling to realize that depending on the scale you look from it changes the ideas of lines and what straight is drastically. It makes you wonder if there is any one way to describe straight lines or if you have to look at it differently within the parameters you set. The idea that space is curved has changed how many have approached the mysteries of the universe. It has brought about many discoveries like gravitational lensing.

Gravitational lensing takes a look at the whole picture of space and looks at the way light bends from a star to show other stars and planets that you can’t see. A gravitational lens looks at a distribution of matter, such as a cluster of galaxies, between a distant source and an observer, which is capable of bending the light from the source, as it travels towards the observer. This is only one way that a curved, but straight line, in space can show things we couldn’t see before. Which has also helped in the exploration of space with probes and satellites, we can’t just plan to launch satellites in a straight line off of the planet and hope that they don’t hit something or they don’t go off of course. It’s the same with missions further into outer space, you have to take in account the curvature of space and the gravitational pull of everything that the ship or craft could come close to, making straight lines that you need to go along curved. The gravitational pull could also be used to make the journey faster and more energy efficient if you can use the pull of the object you fly by to somewhat slingshot yourself around and back onto the direction you intended to go. These discoveries are wonderful and have further advanced our knowledge of space and the way we view lines all around us.

# Works Cited

Physics, M. P. (2015). *Elementary Einstein General Relativity *. Retrieved from Einstein online: http://www.einstein-online.info/elementary/generalRT/GeomGravity

wikipedia. (2015, March 7). *Non-Euclidean geometry*. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Non-Euclidean_geometry#External_links