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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: \[g_2(x) = -\sin(x) \]

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