This subject is interesting to me for a few reasons. One is the fact that this is a fairly new concept only being about 50 years old and with an estimation method only being about 30 years old. With these newer concepts I believe that there is a lot more to study. Another reason why this subject is appealing to me is that it is a way to gather usable data without invading a persons privacy. For example, if a survey was to go out with questions such as “Have you used illegal drugs?” it is a possibility that some may not answer honestly. This is where the randomized response can help. It can give a more accurate answers for sensitive questions such as those.
As stated in the previous post, there are five regular polyhedra. The tetrahedron, icosahedron, cube, dodecahedron, and octahedron. The Greek believed these to be the only regular polyhedra. After analyzing these polyhedra, multiple characteristics can be found that might help describe their uniqueness.
It helps to look at the basic 2D shapes that make up each of these polyhedra: the regular triangle for the tetrahedron, octahedron, and icosahedron; the square for the cube; and the pentagon for the dodecahedron. The regular triangle has 60 degree interior angles at all three of its vertices, The cube has 90 degree interior angles at all four of its vertices, and the pentagon has 108 degree interior angles at all five of its vertices.
It also helps to analyze how these shapes move from their two dimensional forms to their three dimensional forms. When looking at a cube, each of its vertices have three squares connected them.
When singling out a single vertex on a cube, we can analyze this in 2D by folding the 3 squares flat.
When thinking of this in another 2D scenario with 2 squares, folding this back into 3D would not be possible be because the two squares would simply fold flat onto each other. This helps show that, in order to make a regular polyhedron, three or more shapes must connect at a single vertex. This is seen from all of the five regular polyhedra listed. The tetrahedron, cube, and dodecahedron all have three of their respective shapes at each of their vertices while the icosohedron and octahedron have more of their respective shapes at their vertices.
So it is clear that the vertices of the regular polyhedra as well as the angles of the shapes surrounding the vertices contribute to the creation of the regular polyhedra. In further research, we will attempt to prove these to be the only regular polyhedra based on these characteristics they share.
This lab is all about polyhedrons and investigating their properties. This is the only chapter that does not use the computer to view data. This instead requires you to use models to help you visualize and and think about the observations that are made.
Some of the questions that are explored include finding the diameter of each of the regular solids if the edge length is one and, “what is the diameter of each of the regular solids if the faces each have area 1?.”
A reason why this topic is of interest to me because it does involve geometry which is a subject that I enjoy. Also the fact that this lab does not use the computer means to me that it will be somewhat hands on which interests me a well.
My name is Jordan and a sub-topic of mathematics that I enjoy would be Euclidean Geometry. I like this topic because with just 5 axioms and 5 common notions, many theorems can be proved. When looking through Euclid’s first book of elements there are many postulates with proofs from these axioms and notions. While the first 4 axioms seem to be very simple and almost common sense, the 5th axiom is more complicated and has had many people try to disprove it. Continue reading