 So we first saw that space was homogenous, there was no special place that I could point to and say everyone must agree that this is the origin. We next saw that speed isn't special either, you might see someone is going very quickly or very slowly depending on your reference frame and either point of view is equally valid. To wrap things up, let's see how we can transform between two reference frames moving at different speed. So here we have Bob's reference frame like this and in describing this situation we're going to make a few simplifying assumptions. The first of these is that the initial time t equals zero, Alice's origin coincides with Bob's origin so at t equals zero Alice's origin is right here. Now we're also going to assume that Alice's and Bob's axes initially line up and lie on top of each other like this and finally we're going to assume that Alice moves along Bob's x-axis and it's going to be at some constant velocity v so no acceleration, no turning. Now the no acceleration and the no turning are important but the other three assumptions aren't so we could have for example at t equals zero Alice's origin is somewhere else maybe their axes are rotated with respect to each other and Alice moves in some random direction. Now doing this would make the maths a bit more complicated but it won't teach us any new physics so we're going to stick to this simple system. So at some later time t Alice's axes have moved over here and let's say some event happens right here. What coordinates do Alice and Bob give this event? Before we start though let's write things a little bit differently. So previously when I've written x, b and y, b I've meant Bob's x and y axes now what I'm going to do is let x, b and y, b be the coordinates of the event in Bob's reference frame. Similarly let x, a and y, a be the coordinates of the event in Alice's reference frame. So now what we want is an expression for x, b and y, b in terms of x, a, y, a, v and t.