hi!! i’m extremely interested in how you would describe the universe’s three-dimensional euclidean shape to a 5 year old. with the limited knowledge i have right now, it’s pretty hard to picture it and have a full understanding of it, so it would probably be more helpful to pretend that i’m a curious 5 year old. thanks for your time!
In: 1
Walk 3 steps forward. Then another 4 steps forward. Now walk back to where you started, counting the steps. It should take 7 steps.
Walk 3 steps forward, then 4 steps to the left. If you walk straight back to where you started it should take you 5 steps.
It doesn’t matter where you do these experiments, or which way you walk, you should get the same results. And these distances scale with each other – walk twice as many steps each time, and you will be twice as many steps from where you started.
This is how flat space works. The maths is a little annoying, but consistent and fairly straight forward:
> (total number of steps forwards/backwards)^2 + (total number of steps sideways)^2 + (total number of steps up/down)^2 = (number of steps to get back to where you started)^2
Using the first example we get:
> (3 + 4)^2 + 0^2 + 0^2 = 7^2
and for the second we get:
> 3^2 + 4^2 + 0^2 = 5^2
[For bonus points, this is Pythagoras’s Theorem also the Euclidean metric.]
But this doesn’t have to be true for all spaces.
We could have a space in which when you walk 3 steps forwards, then another 4 steps forward you are only 6 steps from where you started! This would be a space that is “curved” inwards (the surface of the Earth is an example of this – walk far enough in any one direction and you get back to where you started!). Similarly, we could have one where you walk 3 steps forwards, then another 4 steps forward, and you are now 8 steps from where you started. This is a space that is “curved” outwards.
We could have a space in which you get a different distance depending on where you are (space “over there” is more squished together, while “over here” it is further apart), when you do it, or which direction you do it in (maybe left/right distances count for double the forward/backward ones!).
We can come up with all sorts of fun different rules for how distances might work in some “non-Euclidean” space.
[A fun and simple one is the “discrete metric,” where the distance between any two different points is always 1 – you are always 1 step away from any other place.]
It turns out the real world is slightly non-Euclidean like this; spacetime gets a bit messed up by various things, so sometimes distances aren’t what they “should” be. Space gets all bunched up around matter, so distances end up being longer than they “should” be (one of the explanations for why light bends around stars – it is a shorter path to go around them than close to them).
You could also use a flat table and a ball. On a table, two toy cards driving next to each other will stay next to each other forever. And they’ll never get back to their starting point. That’s like a “flat” universe: two space ships flying next to each other will always stay like that, and never return to the start.
But on a ball, the cars will start next to each other, but eventually they’ll get closer and collide. (Imagine two people both walking north from the equator. They’ll eventually collide at the north pole.) Also, a car would eventually get back to it’s starting point even though it never turned around. That’s an example of a “closed” universe.
If you have access to a horse’s saddle, you could also illustrate an “open” universe where two cars would get farther apart as they traveled.
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