This is a continuation off of something I did for a daily work earlier this past week. For some reason, Venn Diagrams really interest me. It is cool to me how each time you add in a circle, it changes the shape of it entirely. I put together some examples of what a Venn Diagram looks like with 4, 5, 6, and 8 circles. Here's what I came up with.
4
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5
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6
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What I found to be very interesting about this is that working from the outside in, there is a pattern to the number that represents the number of circles that share any given space. For example, on this Venn Diagram for 6 above, there are six spots around the outside that I put a number 1 into. This is because there is only one circle that the space is being owned by. The next level in are the spots being shared by 2 separate circles, the space they intersect. So I put a number 2 in those spots. And so on and so forth throughout the rest of the diagram, I labelled the number of circles that shared the spot.
Here you will see a progression of how I created a venn diagram with 8 circles.
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As you can see, I started with a venn diagram of 4 circles as seen before. I then proceeded to add one circle to each side of the already made venn diagram. I ended up with eight circles total, all converging in the center.
Then I started to look at the patterns in the number of sections created each time a circle is added. (As a little bit of a disclaimer, this table only proves to be true when the circles are arranged in a way that correctly follows the pattern, not when circles are randomly added on the sides.) Looking back to the venn diagrams of 4, 5, 6, and the finished product for 8, I counted the number of sections in the diagram.
Then I came up with an equation that correlated with the pattern found from the table.
So the equation is n(n-1)+1. Using this equation, I was able to fill in the rest of the table.
As I was filling in the table, I realized that there was another pattern to what I was doing. I reworked the equation, and found that another equation that works for the same pattern is n^2-(n-1). Tricky stuff!
Reflection: I found it interesting that after all of the work I did, I realized another equation that did the exact same job just in a different way. It really was neither shorter or longer, but it provided the same results. Also, I realized about halfway through that the rules I was making and the work I was doing only applied in situations where the circles were placed in a certain way. These rules would not work if the circles are not placed in a way that they are connecting to all of the rest at the center. Over all, this was fun to play around with and figure out. I liked having the chance to realize all of these things, and find this equation, just to satisfy my curiosity instead of doing it because it was a problem on a worksheet that I had to answer.
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A few more random experimental patterns I made. Note that these do not follow the rule requirements because they do not all touch the center. However, they all do involve the circles being added in a specific pattern. If the fill tool had not been taken out of Skitch, I would have been able to make some cool designs by adding color to these patterns. Gotta hate that Skitch 2.0 update!
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Also, I wonder what would happen if the circles were different sizes....
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