Polyhedra; The Five Shapes

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.

What are p-adic numbers?  They are a different set of numbers first introduced by Kurt Hensel in 1897.  The motivation at that time was an attempt to bring the ideas and methods of power series methods into number theory.  They have been used in proving Fermat’s Last Theorem and have other applications in number theory.  See http://mathworld.wolfram.com/p-adicNumber.html for more information.

A little terminology needs to be introduced.  The p in p-adic represents any prime number.  For each prime, there is a new and different set of p-adic numbers.  Q2 identifies the 2-adics, Q5 represents the 5-adics, Q17 represents the 17-adics.  To keep the same notation, Q will represent the real numbers.

Another term to consider is “close.”  The basic idea is that given a number n, it is close to 0 if it is highly divisible by a prime p.  Consider the numbers 25 and 625.  Relatively speaking, 625 has more factors of 5 than does 25, or in other words, 625 has a higher divisibility by 5.  Therefore 625 is closer to 0 than 25.  This idea will be made a little bit clearer in future postings.

Numerical Integration Methods Part 2

Within our lab, we plan to explore, technically 6, different methods of numerical integration, though a couple of the methods are very similar or even just different variations of the same method.  While Luke’s post will tell you all about why numerical integration is important, and why you would want to have numerical ways to compute integrals, I will provide some of the important definitions to help you remember the specifics about the different types of numerical methods of integration and the different advantages and disadvantages each method has. I’ll be covering the left-hand sum, right-hand sum, and Simpson’s rule.

The left and right-hand sums are actually variations of the Riemann sum method. All methods approximate the curve by finding the area of rectangles that cover a similar area as the curve.  As Kenny will elaborate on in his definition of the Riemann sum, these methods are calculated by dividing the interval over which a given function is to be integrated, into subintervals which will serve as the base of the rectangles which will be formed. The height of the rectangles is determined by picking a point on the function. Then the areas of each rectangle is found and added together. The sum of the areas is the approximation of the integral.

Left-Hand Sum

The distinguishing factor for a specific type of sum is the point used from which the height is determined. In the left-hand sum, the left-hand end-point of the sub interval is used as to determine the height. This means the length of the rectangle is found by extending the point a the left-hand side of the subinterval to the function. A picture I found on Wikipedia helps illustrate this concept:

As can be seen in the illustration, for monotonically increasing functions, the left-hand sum approximation is an underestimate of the integral. For monotonically decreasing functions, this method provides and over-estimation.

The formula for this method is as follows:

$L_n = \sum_{i=0}^{n-1}f(x)d$

Where f(x) is the function we’re integrating and d is the width of the subinterval.

Right-Hand Sum

Conversely, the right-hand sum method, uses the right-hand point of the subintervals. And can be pictured as:

With a formula of: $R_n = \sum_{i=1}^{n}f(x)d$

Where f(x) is the function we’re integrating and d is the width of the subinterval.

Wikipedia actaully had a nice summary if you’d like more explanation: http://en.wikipedia.org/wiki/Riemann_sum#Left_sum

Simpson’s rule

SImpson’s rule allows us to compute an integral using quadratic polynomials. This method, like the other methods, separates the area that is to be integrated into subintervals, but differs in the sense that it finds the area of parabolas that encompass the subinterval, by using quadratic polynomials which approximate the function.

Simpson’s rule is formally defined as: $\frac{(b-a)}{6}[f(a)+4f(\frac{a+b}{2})+f(b)]$ and it is very accurate when calculating integrals of polynomials to a cubic degree.

Wolrfram MathWorld has a very succinct description of Simpsons rule which can be found at: http://mathworld.wolfram.com/SimpsonsRule.html.

Also, an interesting fact from Wikipedia about Simpson’s rule, this rule is widely used by naval architects to numerically integrate hull offsets and cross-sectional areas to determine volumes and centroids of ships or lifeboats.

http://en.wikipedia.org/wiki/Simpson%27s_Rule

Prime Number Lab: Post 1

Prime numbers are important to study because they are largely the basis for what we consider to be modern number theory. Understanding prime numbers and their distribution can help us solve problems in fields such as cryptology. Prime numbers have also been looked at since mathematics as we understand it came into being. So, by studying prime numbers, we are studying a vast history of research and theorems that mathematicians have discovered over the years.

We relate this motivation to process with an introduction into the process of finding primes, and move onto proving why we can find these primes by proving that the list of primes is infinite.
Our first observation begins with the Sieve method, which is the most basic method to find primes, but also the most computationally expensive as the numbers we work with get larger.
Euclid’s proof shows that the Sieve method can be used infinitely to find any number of primes we desire.

Euclid’s Proof of Infinite Primes:
Theorem: There are an infinite number of primes.

For the sake of argument, let’s let there be a finite number of primes, so n primes exist.

Let’s let M be an integer greater than 1. Like all integers greater than 1, it is either prime, or composite. Let’s let M be composite, so M is the product of primes. So, let M be the product of all the primes in our list. This means that M=p1*p2*p3*…*pn. What do we know about M+1? Well, it’s an integer greater than 1, so it’s either prime or composite. Let’s let M be prime. Then, either M+1 is a prime not on our list. Uh oh :(.

Now, let’s let M+1 be composite. This means that p|(M+1), where p is a prime from our list. But, that same p would also divide M. This is a contradiction because consecutive numbers are co-prime. This means that they cannot share a divisor greater than 1. So there must be a prime not on our list which divides M+1. So, there is at least one more prime that is not included in our list, so our list cannot be complete. Thus, the set of all primes is an infinite set, and there are an infinite number of primes.

To Next Post: Expanded Motivations and Definitions

Numerical Integration Methods

As a follow up on our motivation, I will be introducing a few of the methods that we will be testing. Namely, this post will introduce Riemann sums, Trapezoidal sums, and the Midpoint method. I wanted to put these three together because they are very similar in computation (we do not know how similar they are in accuracy though).

Riemann Sums

The Riemann sums method is one of the simplest methods to compute definite integrals. All this method does is sum the evaluation of the function at some point, $x_i$, multiplied by some small value, $d$. This gives us the following equation:

$$\sum_{i=1}^{n}{f(x_i)d}$$ where $n$ is the number of elements in the range of our interval.

An example of what this may look like using Python/Sage code:

 import numpy f = lambda x: x**3 # Some function f a, b = (0, 3) # Some interval d = 0.001 # some small delta value numpy.sum((f(x)*d) for x in numpy.arange(a, b, d))) 

Riemann Sums Method. Src: Wikipedia

Trapezoidal Sums

The Trapezoidal Sums method is similar to the Riemann sums method as in it computes the sums of the function evaluated at some point, $x_i$. However, each term that is summed is the average between two points. That is, our sum looks like the following:

$$\sum_{i=1}^{n-1}{(f(x_i)+f(x_{i+1})d/2}$$ where $n$ again is the number of elements in the range of our interval.

An example of what this may look like in Python/ Sage code:

 f = lambda x: x**3 # Some function f a, b = (0, 3) # Some interval d = 0.001 # Some small delta value x = np.arange(a, b, d) np.sum((f(x[i]) + f(x[i+1]))*d/2 for i in range(0, len(x)-1)) 

Midpoint Method

The midpoint method, like the Trapezoidal method, is very similar to the Riemann sums method, except, while using the midpoint method, we are computing the sums of the “middle” of the rectangle. That is, our summation looks as follows:

$$\sum_{i=1}^{n-1}{((f(x_i)+f(x_{i+1}))/2)d}$$ where $n$ is the number of elements in the our interval.

An example of what this method may look like in Python/Sage is:

 f = lambda x: x**3 # some function f a, b = (0, 3) # some interval d = 0.001 # some small delta value x = np.arange(a, b, d) np.sum((f((x[i] + x[i+1])/2)*d) for i in range(0, len(x)-1)) 

Numerical Integration Methods-Motivation

I’m sure most of us have at least some experience with integrals being tricky and hard to compute. Whether it be trig substitution or some other method integrals can be very difficult. There are also functions, real functions that do not have an antiderivative, functions like:

$e^{-x^2}$ or $\frac{sin x}{x}$

However, there are ways to evaluate these integrals, and these are contained in the numerical methods. It is worth studying these methods one, to find the area under the cover for these certain functions, and two, to test the accuracy of these methods. There are certain methods in this chapter which my teammates will tell you more about.

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.