# Numerical Integration: Summary and Conclusions

In this lab we looked at some various methods to compute integrals.
Those methods were, Riemann Sums: left rectangle, midpoint, and right rectangle, Trapezoidal, and Simpsons Rule.

We tested various methods and got results that showed that Simpsons Rule was very accurate, in many cases exact for cubic functions. We explored this more saw that Simpsons rule and the actual integration yielded the same result. This was also a proof of our conjectures about this method Finally we showed where the Simpsons Rule came from.

Now with all of that there are many other things that can be explored. Exploring more functions than cubics, (trig functions, logarithmic, exponential, etc.) and seeing which methods work best for those. Other routes one might take would be to explore higher dimensions. Some questions that could be explored would be how to test the accuracy for functions that don’t have an antiderivative? What methods would seem to work best? How small of an interval would be needed?

# Polyhedra: Conclusion

In this lab, we set out to explore the five regular polyhedra including their qualities as well as their exclusive nature. Through analyzing these shapes, it is clear that regular polyhedra are only created through three basic 2D shapes: the equilateral triangle, the square, and the pentagon. It is also clear that the vertices on a regular polyhedra require at least three of these shapes to obtain their 3D nature. Another requirement for their 3D nature is that each vertex can only be surrounded by interior angle equaling less than 360 degrees in sum. This is because 3D shapes must have a height and this comes from bending the shapes at their connected edges.

These findings helped to prove that there are only five regular polyhedra. It was found that there are only five regular polyhedra because the limits on the the total amount of degrees that can surround a vertex in the third dimension as well as the minimum amount of shapes that must share the vertices. Because equilateral triangles have small interior angles of 60 degrees, the triangle is capable of forming three regular polyhedra with 3, 4, and 5 triangles around a vertex. The 90 degree angles of the square allow for the cube to form, and the 108 degree angles allow for the dodecahedron to form. It is clear that no other regular polyhedra exist because the interior angles of regular shapes only grow larger the more edges they have. Therefore, no sum of three interior angles is less than 360 after the pentagon and never will be. Another experiment that helped to further show the exclusivity of regular polyhedra was the truncating of their vertices which resulted in the shapes to form into one-another. It is clear that the truncating of a shape creates a new shape with the number of faces equal to the number of vertices from the previous shape. These new faces have the same number of edges as the number of old faces shared at the cut vertex.

In further studying regular polyhedra, it would be interesting to analyze the incomplete truncation of these solids. When truncating, polygons are seen at the corners that eventually become the faces of the new shape when truncation is complete. When truncation is incomplete, each corner is inhabited by these miniature polygons while the old faces shrink in size and gain an extra edge at each of their vertices. However, are these shapes still symmetrical throughout their entire transformation? If so, how many of these solids can be created. For example, a cube with slight truncation would have triangles on its vertices and the squares would have four new edges, making them octagons. How many of these shapes exist? Is there a finite amount? These are the questions that would be explored in further experimentation.