Skip to main content

Mathematical Illusions: Ambiguous Objects

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 photoshop). This magic depicted is the math behind ambiguous objects in action.

The Math Behind the Magic

The first concept that is relevant to the trick is perspective. As cool as it would be, an object cannot morph into two completely different shapes. The object is one shape that roughly looks like some amalgam of the two shapes. This combination of the two shapes is created by placing two curves flat on a plane, and then angling them in such a way that takes advantage of perspective. For simplicity, it is assumed that the two angles that the object is being viewed is 45 degrees on one side, and 45 on the other. It is also assumed that the onlooker is far enough away that the view of all points of the object is 45 degrees. The goal is to have one viewer see one curve while the other viewer sees the other curve. For this example consider the following two curves:
The curves currently lie flat on the xy-plane, but a parameterization can be written such that one viewer sees a point on the curve denoted as (x, f(x), 0) and the other viewer, at the same point, sees (t, f(t),0). The line that is parallel to the 45 degree angle vector (0,1,1) and goes through the first curve can be written as \[h(s,t) = (t, s + f(t), s)\]. The other curve that is at the other 45 degree angle can be written as \[k(s,t) = (t, -s+g(t),s) \] where f(t) and g(t) are the curves flat on the plane. This definition leads to the parameterization of the ambiguous object: \[ c(t) = (t, .5(f(t) + g(t)), .5(g(t) - f(t))) \] This curve gives the necessary curve to create the object that looks like two different shapes and concludes the magical explanation.

Explaining the Example

To keep with the trend of magic, the example consist of a curve of a diamond and the curve of a spade. These two curves are different enough to show the illusion in action, and it came out quite nicely. The following two images show the opposite perspectives of the object in OpenSCAD:

The Curves

These curves were both created in a piecewise fashion. There are 2 parts to each curve for a total of four parts. The diamond is formed by the curves: \[ f_1(x) = \sin(x)^4 and f_2(x) = -\sin(x)^4\] Mapped over domains [0,180] and (180,360] respectively. The top part of the spade was formed by the curve: \[g_1(x) = .5|\sin(x)|^.5 \] With domain [0,180] and the bottom was formed over the remaining domain by the curve: \[g_2(x) = -\sin(x) \]

Comments

Post a Comment

Popular posts from this blog

Finding an object's center of mass using integration

Thinking about center of mass As a kid, I would sit in class and balance my pencil on my finger. What I didn't know then was that I was finding the pencil's center of mass. An object's center of mass is a point where the average weight of the object lies. In other words, the weighted position vectors of the object at this point sum to zero (source: https://www.dictionary.com/browse/center-of-mass ). The exact location of this point can be found using calculus. First, it is important to understand the moment . The moment for a two-dimensional object is given with respect to both the x- and y-axes. It is a physical characteristic of the object a certain distance from these axes. It's easiest to think about the axes as reference points for where the characteristic acts; in this case, where the mass is distributed. When computing these moments, the hardest concept to grasp is that the moment with respect to y, for example, is given by the x-coordinat...
Post a Comment
Read more

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Finding the Center of Mass of a Toy Boat

Consider two people who visit the gym a substantial amount. One is a girl who loves to lift weights and bench press as much as she possibly can. The other is a guy who focuses much more on his legs, trying to break the world record for squat weight. It just so happens that these two are the same height and have the exact same weight, but the center of their weight is not in the same part of their body. This is because the girl has much more weight in the top half of her body and the boy has more weight in the bottom half. This difference in center of mass is a direct result of the different distributions of mass throughout both of their bodies. Moments and Mass There are two main components to finding the center of mass of an object. The first, unsurprisingly, is the mass of the whole object. In this case of the boat example, the mass will be uniform throughout the entire object. This is ideal a majority of the time as it drastically reduces the difficulty...
Post a Comment
Read more