Pattern

An investigation into the effect the size of a square has on the symmetrical patterns

"A square grid is coloured in so that each square is the same colour as all the symmetrically placed squares and a different colour from the rest of the squares. This means that the grid will look the same if it is turned around or turned over." Considering the parameters of the problem, we will be coming up with a series of hypotheses to investigate the number of colours you need to colour in each size of square, what these numbers mean and how they can be derived algebraically, and what happens when polygons with different numbers of sides are used. 1. Squares 1.1 Definitions A square is a regular polygon, with 4 sides, whereby all the sides are the same length and all the angles have the same value. If the grid is coloured in so that it looks the same turned over, then this implies that the square has reflectional symmetry, and there are 4 lines of symmetry. If a grid is coloured in so that it looks the same turned around, then this implies that the square has rotational symmetry of order 4. We can say this because a square itself, whether coloured in or not, has these properties. To make the algebra in this booklet easy to follow, we will only be using squares with sides of length ? 1. We will also only be using natural numbers, which are defined as positive integers. We will also be defining the length of the side of each square in units, and shown algebraically by n. There will be no formal measurement, as the length of the sides is irrelevant to the investigation. We will be defining the number of colours used by c. 1.2 How many colours does it take to colour squares of different sizes? We will start with the smallest grid allowed, a grid of size 1x1. As there is only one square within this grid, only 1 colour may be used, and c has a value of 1. Figure 1.1 A 1x1 Grid Next, we use a grid with sides of length 2. There is only one way of colouring in the square so that it has reflectional symmetry, and rotational symmetry of order 4, and this is using only 1 colour again, therefore c is still 1. Figure 1.2 A 2x2 Grid Next, we use a grid with sides of length 3. In this grid, we find that there is a way of colouring in the squares so that we may use 3 colours. This is the first time the number of colours used has increased, and the value of c has increased from 1 to 3. Figure 1.3 A 3x3 Grid Next, we use a grid with sides of length 4. In this grid, we find that there is a way of colouring in the squares so that we may use 3 colours. Figure 1.4 A 4x4 Grid From these figures so far, we can see that there is a pattern developing. It appears that every time we increase the size of the grid from an even number to an odd number, c increases, where c is defined as the number of colours used. However, when we increase the size of the grid from an odd number to an even number, the value of c stays the same. From this, we can predict that the pattern will continue for higher values of n, where n is defined as the length of the side of a square grid. Predict what the value of c will be when n is 5 c= . Now we will investigate what happens when n is increased to 5. Therefore we are now looking at a 5x5 grid. Figure 1.5 A 5x5 Grid We can see from the grid that there are actually 6 colours that can be used to fill in a 5x5 grid. Compare this value of c with your guess from the previous page. Is it the same? What pattern can you see emerging from these values? Now that we have the value for a grid of odd value of n, can you predict what will happen to c when we increase n to an even number? Figure 1.6 A 6x6 Grid As we can see from this grid, the number of colours that have been used has not increased, and c remains at a value of 6. This is a continuation of the pattern that had emerged with the lower values of n, whereby when n goes from having an odd value to having an even value, c does not increase. We appear to be seeing a pattern beyond just the fact that the number of colours increases. When we take a closer look at the values of c we are achieving for each value of n, we can see that they are all a certain type of number. value of n 1 2 3 4 5 6 value of c 1 1 3 3 6 6 Table 1.1 Values of c for each value of n up to 6 Looking closely at Table 1.1, what can we say about the values of c? From the pattern of the first 6 values of c, we can say that they represent what looks like triangular numbers. If we ignore the repeated values, then the pattern looks like: value of n 1, 2 3, 4 5, 6 value of c 1 3 6 Table 1.2 Values of c for each value of n up to 6, ignoring repeated values Now we can truly see the pattern. We can now look at what happens when n=7 to see if the pattern is likely to continue. If the values of c are going up in accordance with triangular numbers, then what is the value of n=7 or 8 likely to be? . Figure 1.7 A 7x7 Grid Figure 1.8 A 8x8 grid Because of this, we can start making predictions for much larger values of n. As stated (remind me this formula needs to be in the proof), the formula for finding out the value of c for any given value of n is: where (0 d.p.) Therefore, for finding out the number of colours that is needed for a grid where n=99, we can just put the values into the formula like so: This means that when the grid is of the size 99x99, then you can colour in the squares in 1275 different colours, and the grid will still have rotational symmetry of order 4, as well as having reflectional symmetry. This can also be represented graphically with all the lines of symmetry drawn onto the grid. For this we can use a 4x4 grid. When all the lines are drawn on, this grid shows that it can be split into 8 separate right-angled triangles. Each of these triangles contains a whole square, and 2 half squares. Within these 3 shapes, every colour that is used on the entire triangle can be found. In this case, the number of colours used on a 4x4 grid is 3, and the number of colours within each of the right-angled triangles is also 3. This can be represented graphically like so: Figure 1.9 8 right-angled triangles cut by the lines of symmetry of the grid.