Ambiguous objects are in the display case!

Do Over: Double Integrals over Regions

Introduction Over the semester we've looked at many topics and created 3D models. For this we are going to revisit an old topic, double integrals over a region. In this we found the volume of a surface in the xyz-plane bounded by two curves. From the many topics I chose to revisit this topic. I have a couple reason to why I chose to redo this. First, the model did not print correctly. The print added spaces between the rectangular prisms. Another reason was that I think the surface and curves did not represent the topic entirely. The surface I chose just increased between the curves. Improvements When making the model on Onshape there were no spaces between the rectangles, which can be seen on the right. However, when printing this spaces were being added. The second issue was with the surface I chose which was \(f(x,y)=xy+x\). This function only increased over the two curves I chose \begin{align*} f(x) &= \sqrt{x} ...

The Septoil Knot

Knots are a very interesting topic and a field that has not quite been fully discovered, so mathematicians are still discovering new ideas and invariances about knots even today. While it may seem like knots are a simple skill you learn at camp, they actually have a lot of mathematical properties and in this blog post we are going to look at just a few. By mathematical definition, a knot is a closed curve in three dimensional space that does not intersect itself. Since we are working with three dimensional space and you are reading this on a two dimensional screen, we need a way to look at knots in two dimensions and that is where knot projections come in. A knot projection is simply a picture of a knot in two dimensions and where a knot crosses itself in the projection is simply a crossing of that projection. The number of crossings of a knot is the smallest number of crossings among all projections of a knot. Since a knot is not necessarily solid, one...

Knot 10-84

Introduction In mathematics, a knot is simply a closed loop. The simplest version of this is the unknot, which is a just a closed circle (imagine a ponytail holder). Knots, however, quickly become more complicated than this more basic example. This post will examine a particular knot (knot 10-84) and a few of its knot invariants. Crossing Number Knots are often defined by their crossing number, which is the number of times the knot’s strands cross each other. As indicated in its name, knot 10-84 is a 10 crossing knot. In order to visualize the knot, we can look at its knot projection, in which the knot is represented by a line segment broken only at its undercrossings: Tricolorability Now that we’ve looked at knot crossings, we will examine a potential property of knots: tricolorability. In order to understand tricolorability, it is first important to know that one strand of a knot is defined as an unbroken line segment in the knot p...

MA 391 Composition and Communication

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