Author Archives: Marc Garland

Marc on Randomized Response

My name is Marc and I have an interest in randomized response which is a technique developed by Stanley Warner in 1965 that is used to obtain accurate results on surveys that ask delicate questions. For example, if you want to ask people in a classroom if they are sexually active, you can create a decoy question involving the flipping of a coin. The more delicate question can then be asked in tangent with this decoy question and nobody really knows if the participants are answering yes to the decoy question or the real question. This is done in order to make people feel more comfortable about their privacy. A program is used in this lab that simulates a randomized response survey.

One of the questions involves finding a function for a particular survey. By using a proportion concerning the number of yes answers versus the number of total answers, it is asked to find an equation that can help in estimating what the true number of yeses should be to the real question. Further questions build on determining how to uncover accurate results using the probabilities in possible answers from the real question and the decoy question.

I think this is a very useful lab for statistical analysis because you want people to be honest when gathering data but you simply cannot rely on honesty. Finding an accurate way to collect data while helping others to feel comfortable in being honest helps in gathering accurate data.

 

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.

 

Polyhedra

A polyhedron is a geometrical 3D shape whose faces are all polygons. A regular polygon is a 2D shape that has equal length sides as well as equal interior angles. A regular polyhedron, therefore, is a polyhedron with sides that are all equal regular polygons that meet at their vertices. Each vertex on a regular polyhedron connects an equal number of regular polygons. These figures have been studied since ancient times by the Greeks, and remnants of architecture has been uncovered in Scotland as well as Egypt.

There are five currently known regular polyhedra. Three have been formed with regular triangles: the tetrahedron with 4 faces, the octahedron with 8 faces, and the icosahedron with 20 faces. The commonly used cube is also a polyhedron with equilateral squares for faces, and the dodecahedron  is represented as a 12-sided shape with pentagonal faces. The five known regular polyhedra can be seen here.

The Greeks believed these to be the only regular polyhedra possible. In this lab, we will attempt to prove (or disprove) this theory of the ancient Greeks. We also plan to analyze the truncation of these regular polyhedra. Truncating is when the vertices of a 3D shape are cut off, creating a shape in their place.  Truncating a regular polyhedron at all vertices can create entirely new shapes and this interesting phenomenon will be explored further. As we delve further into the lab, more questions might arise that we will analyze and perhaps attempt to answer.

Marc and Prime Numbers

A lab that I found interesting in the textbook was the Prime Numbers lab. Prime numbers are numbers that cannot be divided by any number other than one and itself. Because any number can be written as a product of prime numbers, prime numbers are basically the building blocks of arithmetic. Even though one cannot be divided by any number other than itself, it is not considered a prime number due to mere convenience because every number would have one as one of its prime multipliers which would be tedious. Yay, simplicity!

One question in this lab involves finding a proof that there are infinitely many prime numbers. This has always been a complicated question in the realm of mathematics!

Because prime numbers are such a fundamental and important group of numbers, I find it interesting to learn more about them for it always leads to learning more about the natural numbers in general. I can’t wait to see what labs others are interested in on Wednesday!

Marc on why Calculus is the Bomb

My name is Marc and I am a math student at Boise State University in Boise, Idaho which is best known for its potatoes as well as recreational opportunities. One area of mathematics that I enjoy learning and studying about is calculus. Calculus is the study of limits, derivatives, and integrals which are useful in studying continuous functions. Limits play a massive role in a variety of fields ranging from economics to engineering, and limits help to define derivatives. Derivatives are functions that describe the slope of a continuous function. The opposite of a derivative, an integral, can be used to find areas under curves in two dimensions as well as volumes in three dimensions. All of these tools can be used for optimization purposes which are significant in practically any field.

For more information about calculus, you can visit the Wikipedia page here.

Because the derivative represents the slope of a function,  the derivative of a function is thus calculated with this idea of change in distance in mind. The equation associated with the derivative is:

$m=\lim_{h\to 0 }\frac{f(a+h)-f(a)}{h}$

Notice that this equation represents a change in distance on a graph. Slopes are meant to represent changes of functions and functions represent a variety of trends! For example, the derivative of a position function in physics would produce a velocity graph which represents speed, or how the distance is changing over time. The derivative of a velocity function would give an acceleration function which reveals how the speed is changing over time. Pretty cool, isn’t it!?