# Euclid’s Space

As a warm-up to thinking about spacetime, let’s have a look at just space first. Euclid, the Greek mathematician, was interested in studying the geometry of the space that objects live in; concepts like distances, angles… He came up with a set of five properties that he thought this space satisfied—Euclid’s five postulates 1.

But we all have a natural intuition for the three-dimensional Euclidean space we live in, which mathematicians call $$\mathbb{E}^3$$. We are also quite familiar with the two-dimensional Euclidean plane, $$\mathbb{E}^2$$, like the surface of a table or a wall. So, instead of dealing with axioms and postulates, let’s try visualise the geometry of space. This will set us up to later generalise such visualisations to spacetime.

# Decartes’ coordinates

The first step would be to label the positions of all the points in space in some unique way. René Descartes found a way to do this, giving every point a unique label of numbers—the Cartesian coordinate system.

On a plane, let’s pick an arbitary point and call it O—the origin. Through the origin, draw two lines that are perpendicular (at a 90° angle) to each other. Let’s call these, following traditional convention, the X and Y axes respectively. With these axes as reference we can assign a unique pair of numbers, called coordinates, to every point. Take any point P; to find the x-coordinate of P draw a line parallel to the Y-axis until it intersects the X-axis. The distance from the origin to this point is the x-coordinate of P. In a similar way, we get the y-coordinate. We can think of the x and y-coordinates giving the width and height respectively, of the point P.

For three-dimensions we can just add another Z-axis and repeat the process to assign a unique triplet of coordinates to each point. Note that, the coordinates can also be negative, that is, on ‘other side’ of the axes.

Though they were a revolutionary idea in Decartes’ time, coordinates are actually quite ubiquitous now. Anyone who has worked with a 3D modeling software or even Photoshop/GIMP has seen coordinates. The labelled rows and columns on a chess board, layout design grids, the street and avenue system in some cities, latitude and longitude, numbered aisles in a supermarket, seat numbering in a theatre—all are examples of coordinate systems.

One concept that coordinate systems reveal immediately is the idea of lines of constant height and width. All points on any line parallel to the X-axis have the same height and the other way around. Points on the X-axis itself all have zero height.

# Pythagoras’ distance

Why is this labelling useful for geometry? For instance, given the coordinates of two points how can we find the distance between them? The answer to this was given by Pythagoras, another Greek guy, long ago—the square of the distance is the sum of the squares of the x and y-coordinates.

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

This formula is called a distance function or metric; it holds for any two points in Euclidean space. Similar formulae can be written down for angles and other geometric measurements. In fact all of geometry can now be translated into algebra; or for our purposes the other way around.

Cartesian coordinates provided a very useful dictionary between the visual world of geometry and the powerful, abstract symbols of algebra.

# Transformations

What can we do to our Euclidean space that keeps the distance between any two points the same? If we want to keep the same distance, we should move the space rigidly; no squishing or stretching or distrotions of any kind.

We can also look at this in another way3—instead of moving the plane we can move our coordinate system. After all, we just arbitrarily chose the origin point and the direction of the axes. What if we had chosen another coordinate system with a different origin or a different direction for the axes? We can convert between the new and old coordinate systems.

Moving the origin to a new point, with coordinates $$(a,b)$$, is the same as translation;

\begin{align} x' = x-a \\ y' = y-b \end{align}

We can change the direction of the coordinate axes, keeping the origin fixed, by a rotation by some angle $$\theta$$.

\begin{align} x' = x\cos\theta - y\sin\theta \\ y' = x\sin\theta + y\cos\theta \end{align}

You can convince yourself that translations, rotations and their combinations are the only transformations that preserve the distance on a plane. In fact, the metric or distance formula looks the same in both coordinate systems, even though the individual numbers in it change. Thus the metric is invariant under these transformations 4.

But if we are allowed to change coordinates in this way, how do we pick any one? The answer is, “It doesn’t matter what you pick!”. Any two coordinate systems that are related by such transformations are completely equivalent—they both ‘see’ the same geometry.

All laws of Euclidean geometry are the same in any coordinate system related by translations and rotations!

# Coordinate confusion

There do seem to be some concepts that are actually not the same in both coordinate systems. For example, the notion of two points being at the same height i.e. having the same y-coordinate. If you use a different rotated coordinate system, such points will then have different heights in the new coordinate system.

As an example, suppose you placed a rod of length $$d$$ parallel to the X-axis. The ‘height’ of the rod for you is, obviously 0. But if I were using a rotated coordinate system, the ‘height’ of the rod, as measured by me, would be no longer be zero, but—

$h = d \sin \theta$

Height of objects changes in a rotated coordinate system! It is quite obvious, that this ‘height change’ effect actually occurs, in some sense, but has no intrinsic geometric meaning. Height and width were just convenient labels introduced by the coordinate system we chose—they have no intrinsic geometric existence.

While distance and angles are geometric ideas independent of our choice of coordinates, height and width (as we introduced them) are just artefacts of choosing a nice, convenient coordinate system.

Admittedly, it seems that I’m just trying to be cute but, a lot of the confusion and paradoxes in relativity are of a similar flavour. It will be useful then, to look back at some relatable example like this one; with the lesson being, while coordinates are useful, it is best to take them with a pinch of salt.

# Time to end

Euclidean space is a familiar setting for which we have a lot of visual intuition. Cartesian coordinates help make those visuals more concrete, and will also help us think about spacetime in an analogous way. We’ve seen how to think about coordinates, how geometry is actually independent of the specific coordinates we choose, and some of their uses and misuses.

Next we will add time to the picture. After all, as a now-famous chemist once said—