The real plane

Whenever we look at a computer screen or at pictures taken with phones or digital cameras we are looking at a digital image file. Digital image files usually have the tags jpg, png, bmp or tiff and they can be regarded as simple mathematical objects. For example my digital camera takes pictures which consist of a grid of 5152 × 3864 pixels where each pixel has a colour which, in RGB format, is of the form (r, g, b). Here r, g and b are the red, green and blue components of the colour and they take the values of whole numbers in the range 0 to 255, with (0, 0, 0) being black and (255, 255, 255) being white. A quintuple is an ordered set of five numbers and so a digital picture is just a set of quintuples, (i, j, r, g, b) where (i, j) are a pixels coordinates and (r, g, b) is its colour. For example (0, 0, 0, 0, 0) means the pixel located at the top left of the picture is black and (5152, 3864, 255, 255, 255) indicates that the pixel at the bottom right is white. It is not hard to change the pixel coordinates (i, j) to coordinates (x, y) with x and y real numbers and where we are free to choose the position of the left, right, top and bottom of the picture. Specifically, for a digital picture taken with my digital camera, if we put

then the digital picture becomes a set of quintuples (x, y, r, g, b) where each pixel has a coordinate (x, y) in the real plane and the colour (r, g, b). In this way the pixels of any digital picture can be assigned coordinates. This means that any digital picture can be regarded as a rectangular grid of coloured points embedded in the real plane.

The real plane is the set of all points with coordinates (x, y) where x and y are real numbers. It is also known as the Cartesian plane and 2-dimensional Euclidean space. The notation ℝ² is a shorthand for the real plane where ℝ is the set of real numbers also known as the real line. Any real number is either a fraction, with a finite or recurring decimal expansion, in which case it is called a rational number, or it can only be represented by an infinite non-recurring decimal expansion, in which case it is called an irrational number. The square root of 2 and π are irrational numbers with infinite non-recurring decimal expansions and so they only really exist as abstract mathematical constructions. There are many more irrational numbers than there are rationals. In fact there are two different sizes of infinity which show this. The rationals are only countably infinite which means they can be listed one after another in an infinite sequence, whereas the irrationals are uncountably infinite which means there are far too many of them to be listed in any sequence.

We can't store any irrational number on a computer because an infinite (non-recurring) decimal expansion requires infinite memory, all we can do is approximate them with rational numbers with a finite number of decimal places. This means that any computer model of a rectangle in ℝ² necessarily consists of a subgrid of points (x, y) where x and y are rational numbers with finite length decimal expansions. It's worth pointing out that our computer model of a rectangle in the plane is still of enormously greater resolution than any digital image can be, where the digital image can also be regarded as a rectangular subgrid of points embedded in the plane as we described above. A digital image needs to be stored in its entirety in memory whereas we don't need to do this for a computer model of a rectangle (and we wouldn't have enough memory to do so even if we wanted to). Instead we often work on one pixel of a digital image at a time. The general method we use is to give a pixel in a digital image some coordinates, as described above, and then do calculations at the highest accuracy available on the computer, where the calculations are determined by the particular dynamical system being modelled, and then use the results in some way to map a colour back to the original pixel.

Mathematically the real plane has a richer structure than just being a collection of points. For example we can measure areas of sets of points and distances between points. One way of measuring distances is to use the Euclidean metric |  |. This gives the distance between two points (x, y) and (v, w) in ℝ² as

The real plane is also a real linear (vector) space, where we can regard points as vectors. A point (x, y) can be thought of as a vector which connects the origin (0, 0) to (x, y). The length of the vector is then

Vectors can also be added together and scaled in length by multiplying by real numbers using the following equations

All of our pictures are constructed using dynamical systems (X, f) where X is either the real plane or a very close relative of it.