Rocket Project
In this project our goal was to create rockets in both of our math and physics classes. We used information from both classes to gain a better understanding about how physics influence rockets and how that information can be described with math, particularly quadratics. We learned about Newton's laws and kinematic equation in physics. In math we learned about quadratics and the various forms that they take and what they say about the path that they describe.
Project One: Tessellations and Efficiency
A 2d tessellation derived from my 3d shape by projecting it onto a flat plane.
What Makes an Efficient Shape?
To first answer the question we have to first define what it means to 'be an efficient shape. Perhaps one of the first things that comes to mind is a circle. Perhaps the most prominent of the most prominent reason that this could be considered is the fact that it, has no corners, and is the most efficient shape in terms of perimeter to area. In many ways the circle is the most efficient shape, however, if we are taking into account multiple shapes then it loses it's edge. If you were to have a 2d plane and fill it full of circles and fit them together the best you can there will still be gaps, in other words waisted space.
That then leads to the question of if there are shapes that will have no gap, to which the answer is yes. We use a lot of these shapes from squares to rhombuses. But, which of theses are the most efficient in terms of perimeter to area which still maintaining the ability to tesselate? If we know that the circle is the most efficient shape when not put with other shapes then, it stands to reason that a shape really close to it will still work, for example a 100 sided regular polygon. unfortunately, when these are put on a 2d plane there is still gap, so, we just keep moving down the number of sides until a shape that works eventually appears. As it turns out the first shape that it appears is a hexagon. This means that the hexagon is the most efficient shape that can be crammed on to a 2d plane. |
Finding the Area on a N-Sided polygon
During our mathematical exploration of this exhibition we were given the problem of finding the area on a n number of points and a given radius. This basically means that if someone said that they wanted to know the area of a polygon with 13 points and a radius of 23 units. I would be able to put that into my equation and tell them. There are likely a number of ways to approach this problem but, the way that I chose was breaking the polygon up into a lot of different triangle by drawing a line from the center to each point. I then made them right triangles by drawing a line to each mid point on the sides of the polygon. Now that I have created right triangles I can begin to find the area of them. I know that the hypotenuse of this triangle is equal to h and that I can find the angle closest to the center with 360/2n or 180/2 which I labeled as theta. Using this I can then find what I labeled as a and o as shown in the diagram. With this information I can find the area of the right triangle that I created, as shown in the diagram. I also know that I will have two of these triangles per side of the polygon so I can can just multiply the area of the right triangle that I found by 2n to get the total are of the polygon. The final equation is nh^2cos(n/180)sin(n/180).
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Project reflection
All in all this project was a lot of fun for me and I both enjoyed and learned a lot. But, it was not without it's problems. For my project I decided to create a 3d-tesselation of my own. Such things include: cubes, hexagonal prisms, and rhombic dodecahedrons. It was a challenge because I didn't want to do some simple shape, but, I overcame this and created a design of my own. However, I failed to make a repeatable process and wasn't able to make more using the same methodologies I used to create the first. Nevertheless, I was able to do some really cool things with the tessellation that I did make. One of which includes taking my 3d shape, and projecting it onto a 2d plane making 2d tessellations derived from my 3d one. One of these tessellation even had both 4-fold and 6-fold symmetry in different parts of the tessellation. All of this was very exciting and I gained more respect for geometry and it's applications, something that I had loathed previously.