Firstly In this coursework I will draw six or more lines which will cross each other and while doing this I hope to get as much crossover points as I can, as well as I will try to get the maximum regions. I will try to avoid any sort of double intersecting i. e. intersecting over a ready made crossover point. I will try and keep all my lines as consistent (placement). To make sure I do this I will draw 1 line and then copy past it and then add another line this will help me keep my lines the same i. e. consistent. But I will vary the number of lines I use.
I will draw six line models (each having one more line than the previous one) the first model that I will create will have only one line the second will have two lines the third will have three lines etc. Secondly I will try to figure out the formulas that will find the nth term for the crossover points, open regions, closed regions and the total regions. I also plan to find the relationships / sequence between any two of these characteristics of the line(s). I will accomplish this by using the following formula: A+B (n+1) +0. 5(n-1) (n-2) C. A = 1st term of the sequence.
B = the difference between the first two terms. C = the 2nd difference (the difference of the 1st difference). Thirdly I will put all my findings onto to a table consisting of total regions, closed regions, open regions, crossover points and all of the formulas. Lastly I will try to summarize all my findings by writing up a conclusion and evaluation. Prediction I am going to predict that every time the number of lines increases by one the number of open regions will increase by two. For e. g. if there are two lines then the number of open regions will be four.
In other words the number of open regions will double the number of closed lines. I predict that the relationship between open regions and closed regions will be a sequence that has a 1st difference and a 2nd difference. This will have to be the case, if I am to work out the nth term in the way I have described above. I also predict that there will always be a greater number of closed regions in comparison to open regions and finally that the number of total regions will be greater than the number of crossover points.
Diagrams showing my investigation Summarizing of the 6 lines that I drew with the formulas Patterns I managed to spot while investigating While undergoing the investigation I found numerous amount of patterns e. g. the relationships between open & closed regions. The following information will give you insight information about what I managed to find i. e. the relationship between open and closed regions. From my results I can gather that my hypothesis were correct. For any particular number of lines the open region was double it.
This had to be the case because if a line was to be drawn then it will form a barrier between two regions on either side, i. e. there will be two regions for any one line. Similarly if you were to have two line than the two line will make up four open regions on either side of each of the two line present. Therefore, when two lines were present then the number of open regions was four. I also predicted that the relationship between open regions and closed regions will be a sequence that has a 1st difference and a 2nd difference and it was.
There was also always a greater number of closed regions in comparison to open and finally the number of total regions was also always greater than the number of crossover points. Evaluation In my opinion my investigation went quite well. I was able to prove my hypothesis correct and was able to fulfil my aim of working out the nth term for many regions. It took me a while to draw the diagrams as I wanted each crossover to give me the maximum number of open and closed regions. But once I had the diagrams, I simply put the results in a table and then looked for any obvious or underlying sequence.