MA 391 Composition and Communication

How I chose which project to repeat and what needs fixed Over the course of the semester, I've learned a lot about not only explaining math but also the logistics of printing a helpful 3D model. In many of my calculus courses, professors have stood at the chalkboard and tried to draw a 3D object, only for the students to be more confused than before the visual aid was added. When I thought about which project to do again, I tried thinking of a project where my print was difficult to visualize and only made the topic more confusing. My integration over regions in the plane project came to mind. Originally, I picked the function \(f(x, y) = \frac{1}{3}(x^2)(2y)\) on the plane bounded by \(y = e^{x-5}\) and \(y = ln(x)+5\) from [0, 7] x [0, 7]. This function grows very quickly, which was seen on my model because I picked such a large rectangular region. The print was very tall and super skinny. It was difficult to see the rectangles of the approximation because...

The first time we took on stereographic projection, I discussed how it reminded me of the little kid night lights that projected starry nights onto the ceiling. When attempting to replicate the starry night, I failed and turned it into a cloudy night projection. Because I was annoyed I couldn't make it work and because right after I finished my project last time, I came up with new ideas, I decided to repeat the stereographic projection example and make it better. Last time, I used an array of clouds out of frustration because I could not get stars to work. My clouds were each made up of four circles and then I just multiplied them across an array. Well, that was boring. And while I still haven't been able to create the perfect star, I have decided to create a recognizable constellation rather than an array. Since I still couldn't figure out how to use the hull function properly with stars, I decided to create my constellation using lines....

Introduction In my previous blog post on solids of revolution, we looked at the object formed by rotating the area between \(f(x) = -\frac{1}{9}x^2+\frac{3}{4} \) and \( g(x)=\frac{1}{2}-\frac{1}{2}e^{-x} \) around the \( x \) axis and bounded by \( x = 0 \) and \( x = 1.5 \). When this solid is approximated using 10 washers, the resulting object looks like this: When I was looking back over the 3D prints I’d created for this course, I noticed that the print for this example was the least interesting of the bunch. Looking at the print now, I feel like the shape is rather uninteresting. The curve I chose has such a gradual slope that each of the washers are fairly similar in size and causes the overall shape to just look like a cylinder. Since calculating the changes in the radiuses of the washers is a big part of the washer method, I don’t think this slowly decreasing curve was the best choice to illustrate the concept. The reason I had done this o...

What is an ambiguous object? An ambiguous object is simply an object that has two different appearances. One appearance can be seen when viewed "head on" and the other appearance can be seen in conjunction by viewing its reflection in the mirror. This is depicted below. As you can see, the roof over the car appears differently when viewed head on compared to when viewing it in its reflection in the mirror behind the car. (https://thekidshouldseethis.com/post/ambiguous-cylinder-illusion-by-kokichi-sugihara) How does an ambiguous object work? An ambiguous object works from the joining of two shapes or objects with mean curves that combine the curves and thus the shape of the objects and ultimately create a new shape entirely. For example, let us look at a series of squares that are reflected as circles (and vice versa) shown below. (https://www.ndtv.com/offbeat/the-ambiguous-cylinder-illusion-is-blowing-peoples-mind-1840738) The squares ...

How the ambiguous object works In arts and entertainment, optical illusions have been used to create the impression of impossibility. In 2016, Professor Kokichi Sugihara in Japan found ambiguous cylinders that appear drastically different when viewed from two specific viewpoints. Suppose we have two viewers looking down on a curve floating above the \(xy-\)plane. The first viewer is a person looking at the curve, and the second viewer is in the mirror. We would like the two viewers to look down on the curve at 45-degree angles on opposite sides of the \(xy-\)plane. From the viewers’ perspectives, it looks as if they are viewing the curves and respectively, as shown in the following picture. (https://m-repo.lib.meiji.ac.jp/dspace/bitstream/10291/19877/1/jma_12_1_2.pdf) The key to the ambiguous cylinder is that the top of the cylinder is not a planar curve. This can be explained by the natural assumption of human visual perception. Natural assumption regard i...

  I will preface this by saying I have no creativity whatsoever, and that my most artistic ability in OpenScad peaked with my ruled surface of a heart. Though, these curves also show in a simple manner the concept of an ambiguous object. I struggle with seeing illusions, and the curves below allowed me to see the ambiguity without too much difficulty, which arose when I picked more exotic curves.  And so, I drag out this example again, to demonstrate the ability to make a heart ambiguous, which I shall ridiculously propose to represent the disappointment from the dashed idealization of an arbitrary person.  An ambiguous object appears as multiple shapes when viewed from different angles. While such objects are hard to conceptualize, they can easily be created mathematically using parameterized curves. Kokichi Sugihara achieved both a second place prize Neural Correlate Society's competition for best illusion, as well as virality for his video Ambiguous Optical...

 Ambiguous: open to more than one interpretation; having a double meaning (according to the Oxford Dictionary). You typically use this adjective in an English or art class rather than a math class because literature and artwork can sometimes have multiple meanings or viewpoints. But not math. Two plus two is always four. The square root of nine is always three. And a function always results in that specific graph. I think that’s why I’ve always had a certain liking to math because it was always direct and to the point with not a lot of room for interpretation. However, by this week’s project, my mind has been blown by the ability to use math functions to create an ambiguous object. Before we get into the math for the object, I’d also like to point out a real-world example of ambiguous objects, optical illusions. They use the same method that we are going to use by find a shared visual point or line that forces the audience to view one thing or the other. Have you ever be...

A math major is an excellent pursuit for any individuals time in college. Learning about the nearly uncountable concepts and ideas that are available to learn and master is staggering. This alone makes it a worthwhile investment. An added bonus of becoming a math major: becoming a magician. From card tricks to optical illusions, becoming a math major/magician can leave your friends and family speechless. Ambiguous Objects For this adventure, we will be looking at the optical illusion created by ambiguous objects. An ambiguous object is an object that looks dramatically different in shape, based on the angle at which the object is viewed. From one side of an object, a person may see a star, and from the other side, a circle. The following image shows just that: Credit to Dr. Kokichi Sugihara to the creation of both the object and the image. This contrast in shape seems impossible to create from just a change in perspective; it almost seems like magic (or ph...