It should only occur in the direction of travel of the object.

EDIT: Actually, length contraction occurs in a direction parallel to the direction of motion, not just the direction of motion. In other words, even if the object was going the exact opposite way as its original direction of motion, a stationary observer would still see the spaceship as being contracted. The only difference between moving toward a stationary observer and moving away from a stationary observer is that when moving toward the observer, the spaceship would be blue shifted and when moving away, the spaceship would be red shifted (relativistic doppler effect).

Thank you for your kind words :)

Incidentally, Einstein's theory of special and general relativity are also a different kind of geometry! Just like how Euclidean geometry takes place in Euclidean space, special relativity takes place on what's known as "Minkowski" space or spacetime, whereas general relativity takes place on what's known as a pseudo-Riemannian manifold.

These words sound complex but just remember that they are merely names for some kind of foreign geometry that are derived from a completely different set of axioms than the normal Euclidean geometry that we studied in high school. This article is very nice at introducing the history and context behind non-euclidean geometries!

But you don't even have to think about something as abstract as relativity to realize that non-euclidean geometries are everywhere. Even the surface of the earth has a non-euclidean geometry (the geometry that takes place on the surface of the earth is called "spherical geometry"). In this sort of geometry, you can have triangles which can have two right angles!

The reason I am introducing all these different types of geometries is because in the end, relativity theory is a geometric theory of spacetime. If you can understand the context behind some of the different kinds of geometries, then you can understand the context behind relativity theory as well! In this way, relativity won't seem as mysterious anymore. It's still very counter-intuitive, but at least you can understand that relativity is just a consequence of choosing some set of axioms, and drawing conclusions from there, just like any other sort of non-euclidean geometry.

Sorry for all these other links haha. This topic is extremely interesting to me and I wanted to share some resources to hopefully get you excited about it too! :)

Another good question :)

So basically, Euclidean geometry is founded on five posulates, the last of which is called the "parallel postulate". It turns out that the statement that describes this parallel postulate is ambiguous, and so you can have multiple descriptions/variations of this postulate (you can even omit the 5th postulate altogether!), all of which allow you to derive entirely new geometries.

Those 90 degree angles in different shapes are a consequence of the different statements of the parallel postulate. Those figures shown in the article are all examples of parallel lines, which at first seem preposterous but are a natural consequence of accepting a different version of Euclid's parallel postulate.

No problem! :)

Good question, sorry for not clarifying in my original post. An axiom is a statement that we take to be self-evident. I.e. it's a statement that we don't need to prove because we just ASSUME that it's true. Then, based on that assumption (i.e. axiom), we try to deduce certain conclusions.

This is also how we do things in geometry (and indeed every single branch of math)! You may have heard of Euclid's five postulates (another word for axiom). All of Euclidean geometry can be derived by using the five axioms that Euclid laid out some 2000 years ago.

However, it turns out that if you refuse to accept the fifth axiom of Euclid, then you can deduce/derive other kinds of geometries. These are called non-euclidean geometries.

You're asking about relativity, which isn't exactly a subject that can easily be explained in simple terms. But I will try my best.

To understand why time slows down with increased velocity, you must first accept that the universe conspires so as to keep the speed of light the same for ALL observers, regardless of their frame of reference. This axiom of the constancy of the speed of light is directly responsible for time passing at different rates for different observers. Let's see how.

Suppose that you have a friend who is stationary (with respect to, say, the Earth). Suppose also that you're in a spaceship travelling at, say, 0.5c with respect to your friend's frame of reference. In other words, if your friend measures your speed, they will see that you're moving at 0.5c. (c = speed of light, so 0.5c means "half the speed of light").

Now, let's perform a physics experiment. Actually, let's perform two experiments -- you perform one experiment, and your friend performs the other experiment.

Inside your spaceship, you try to measure the speed of light. How do you do that? Well, c = d/t and so you measure the distance that light travels in a certain time period. Suppose that you measure how long it takes light to reach from one end of your spaceship to the other end. You know what d is because you can easily measure the length of your spaceship. It is important to note that your clock and your measuring stick retain their length. 1 meter is exactly equal to 1 meter, and 1 second is exactly equal to 1 second in your frame of reference. This sounds like a really dumb (and obvious) thing to say, but keep it in mind. So, you measure what t must be. Then, when you perform the calculations, you get that c = 299,792,458 m/s.

Likewise, your friend, who is not in your frame of reference, also performs the same experiment. He also notes that 1 meter is exactly equal to 1 meter, and that 1 second is exactly equal to 1 second in HIS frame of reference (again, a seemingly dumb observation). He measures the speed of light by measuring how long it takes light to reach from one end of your spaceship to the other end. When he does the calculations, he too gets that c = 299,792,458 m/s.

How is this possible?

It's because when your friend measures distances, he finds that your spaceship is actually SHORTER than what YOU measured. Even though 1 meter = 1 meter for him in HIS reference frame, and 1 meter = 1 meter for you in YOUR reference frame, when you compare the length of a meter from one reference frame to another, 1 meter in one frame of reference is no longer equal to 1 meter in the other frame of reference: your friend has just discovered the phenomenon of length contraction.

Now, c = d/t, and your friend measured d to be shorter than what YOU measured it to be. Yet, c must always equal 299,792,458 m/s for both you and your friend. How is this possible? Well, if d is different for your friend, then t must ALSO be different. However, the RATIO, d/t MUST equal the same: c. Hence, if d is smaller, then t must be bigger so as to keep the ratio, the speed of light, the same: your friend has just discovered time dilation.

This makes sense -- the word "contraction" in "length contraction" means to shorten. The word "dilation" in "time dilation" means to lengthen. So, if length contracts (i.e. d is shorter) then time must dilate (i.e. t is bigger) so as to exactly compensate.

Now I hope you can appreciate "relativ"ity. In your reference frame, time and space act the same -- 1 meter = 1 meter, and 1 second = 1 second. Likewise, in your friend's frame of reference, 1 meter = 1 meter and 1 second = 1 second. However, 1 meter in your friend's frame of reference, WITH RESPECT TO (i.e. RELATIVE TO) your frame of reference is no longer 1 meter. Similarly, 1 second in your friend's frame of reference RELATIVE TO your frame of reference is no longer 1 second.

Weird stuff starts happening only when we start measuring things RELATIVE TO other frames of references. Otherwise, in their own individual frames of references, everything appears to be normal.

Once you have understood the above, then the next natural question to ask is "why does the universe force the speed of light to remain constant for all observers?" And unfortunately, physics doesn't have the answer to this question. It's just how the universe seems to work. Perhaps a deeper theory will answer this question.