Author Archives: carriesmith

Carrie, Chapter 6: Randomized Response Surveys

A randomized response survey is a method used in research to help researchers find the answers to controversial questions while keeping the participants’ identity anonymous. The theory is that if the participants know that there is no risk involved because of the anonymity, they are more likely to answer honestly, therefore producing a more reliable outcome. Researchers found that when participants are asked a direct question, they are less likely to answer honestly and the results of the studies were skewed because of this. So S. L. Warner first proposed the idea of randomized responses in 1965 and B. G. Greenberg modified the method in 1969.

Human psychology, as I have said in my earlier post on randomized response, is very interesting to me. I appreciate the difficulty in conducting an accurate survey in order to study human behavior. In this lab we are asked to explore different aspects in randomized response such as the margin of error and probabilities. I think that it would be incredibly interesting to conduct our own survey using the randomized response method and use that to study the different aspects of the method that the lab talks about. I would also just like to better understand the process of conducting a randomized response survey in order to be able to utilize it in the future.

Polyhedra: Proof Of The Five

While the perfect structure of the regular polyhedra are fascinating, there are only five that are currently known: the tetrahedron consisting of four triangular faces; the octahedron with eight triangular faces; the icosahedron with twenty triangular faces; the cube with six square faces, and the dodecahedron with twelve pentagonal faces. These five unique polyhedra have been recognized throughout the ages dating all the way back to the ancient Greeks. The Greeks believed that these five shapes were the only in existence and none others exist naturally. Through analyzing these five polyhedra, we have come to the conclusion that the Greeks were right in their theory.

Weinberg-1-102711_jpg_630x171_crop_q85

Why is this, however? First, it helps to understand the differences between 2D and 3D. Think of the vertices of the faces on a 2D polygon. When connecting polygons at their vertices, each vertex has a 360 degree radius for which shapes can be placed. Once all of the degrees are occupied, the vertex can no longer hold any other shapes. The angles that center around a vertex are the interior angles of the polygons attached. Regular polygons have equal interior angles, and regular polygons make up regular polyhedra.

Now, what makes 2D different from 3D? In order for an object to shift from the second dimension to the third dimension, it must have a height. An object that has no height is simply flat, a 2D polygon. Therefore, in order to be a 3D object, the object must also have faces. As a result, a regular polygon cannot enter the third dimension on its own. There must be more polygons. In the case of the regular polyhedra, there must be more regular and equal polygons. These polygons are attached at their vertices and line up along their edges. Also notice that two regular polygons are not enough to create an enclosed object because the two polygons would simply fold onto one-another. A third polygon is needed, and as a result, a vertex must connect a minimum of three polygons. Notice that this is the case for all of the five regular polyhedra: the tetrahedron has three triangles connected at each vertex while the octahedron has four and the icosahedron has five. The cube has three squared at each vertex and the dodecagon has three pentagons at each vertex.

Remember that a vertice has 360 degrees to work with in two dimensions. Now, remember that a 3D object must have a height. In order for a polyhedra to obtain this height, its regular polygon faces must be at angles to one-another, or they must bend at their edges. When the faces bend, the vertices no longer have a 360 degree radius. In other words, the vertices can only connect a total number of polygons whose interior angles equal a total less that 360 degrees!

Notice that each for each of the five regular polyhedra, this is the case. The equilateral triangle has interior angles of 60 degrees. In a tetrahedron, each vertex needs only to hold 180 degrees. In an octagedron, each vertex needs only hold 240 degrees. In a icosahedron, each vertex needs only to hold 300 degrees of interior angles. However, notice that there exists no regular polyhedron where each vertex connects six triangles. This is because six triangles would require a vertex to hold 360 degrees, something possible in 2D but not in 3D. The same applies to the cube which has vertices connecting three squares with interior angles of 90. The sum of these three interior angles is 270 which is possible, but four squares would contain a sum of interior angles too high for a 3D vertex. The dodecagon has three pentagons at every vertex. Pentagons have interior angles of 108 degrees. Three pentagons would therefore take up 324 degrees on a vertex which is enough for the vertex to handle. Four, once again, is too much.

The next regular polygon, the hexagon, does not form a regular polyhedron. This is because the hexagon has interior angles of 120 degrees. Because at least three regular polygons are needed at each vertex in order to form a regular polyhedron, a vertex would need to be able to connect three hexagons. However, 120 multiplied by three is 360 which just barely exceeds the maximum angular capabilities of a vertex. Three are able to share a vertex in 2D but not 3D as shown in the figures below.

photo 1    photo 2

How do we know that no regular polygons besides triangles, squares, and pentagons can form regular polyhedra, through? Notice that, as sides are added to regular polygons, their interior angles are always increasing. Therefore, the interior angles of the hexagon are smaller that that of every other regular polygon with a larger number of sides. Therefore, their angles will never be small enough to form a regular polyhedra.

Carrie and Randomized Response Surveys

A. I think that the fact that this was invented almost 50 years ago is interesting and I think is cause enough to study it.

B. I have always been interested in human behavior and this is one way to study it. Also the relation with Randomized Response Surveys and public policies among other controversial topics.

Carrie and Bioinformatics

“How might disease spread in populated areas in the event of a bioterrorism incident, and how would it be contained?”

It is questions like the one above that motivated me to become an applied mathematics major. The general definition of Bioinformatics is an interdisciplinary field that develops and improves methods for storing, retrieving, organizing and analyzing biological data. The part that I am interested in is the study of disease. One part of bioinformatics involves taking the data collected from studies to form a complete picture for interpretation and analysis.

In the Journal for Cancer Research, “Serum Protein Fingerprinting Coupled with a Pattern-matching Algorithm Distinguishes Prostate Cancer from Benign Prostate Hyperplasia and Healthy Men” (http://cancerres.aacrjournals.org/content/62/13/3609.full#sec-2) is an article that uses a decision tree algorithm. This is a powerful tool for classification and prediction. It is a classifier in a tree form having a

  • Decision node: specifies a test on a single attribute
  • Leaf node: indicates the value of the target attribute
  • Arc/edge: split of one attribute
  • Path: a disjunction of test to make the final decision

Source: www.cse.ust.hk/~twinsen/Decision_Tree.ppt

I think that bioinformatics is a huge and very important tool in mathematics that I hope to be apart of one day.