Ambiguous Hearts

 

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 Illusion. He created his objects using linear combinations of parameterized curves. 

Through the method of parameterization, a viewer can see a function \(f(x)\) from a certain angle, and a different function \(g(x)\) from a different angle. Sugihara was able to create his famous "Impossible Cylinder" with this method. 

In the article DO THE MATH!: Sugihara’s Impossible Cylinder, written by David Richenson, and published by Taylor & Francis in 2016, Richenson recreated the "Impossible Cylinder" using paper, and is shown below. 

By viewing the cylinder from different angles, it takes on a square or an oval shape. Richenson showed that the cylinder can be expressed from the two curves \(f(x) = 1 - |x|\) and \(g(x) = \sqrt{1 - x^2}\)

However, alone these curves will not produce an ambiguous image, but linear combinations of the two will. In order to create such a combination, a new parametized function \(r(t) = (t, \frac{1}{2} h(t), \frac{1}{2}k(t))\) where \(h(t) = f(t) + g(t)\) and \(k(t) = f(t) - g(t)\)

Though, the first coordinate mapped by \(r(t)\) does not need to be \(t\) itself, but can be a function such as \(s(t)\) in which case, we have \(r(t) = (s(t), \frac{1}{2} h(t), \frac{1}{2}k(t))\) and  both \(h(t)\) and \(k(t)\) remain the same as before. We can do this because our two functions will still have their first coordinate change in the same matter. 

For my ambiguous heart, my \(f(t)\) is the y-coordinate for the parameterization of a heart, and my \(g(t)\) is \(f(t)\) offset by a phase of -25 degrees. I chose -25 degrees as the phase with which to offset \(k(t)\) by because it distorts its parent curve to a large degree, but not enough to have difficulty finding the angle with which to observe the solid that will give the appearance of being created by \(k(t)\). Larger phases resulted in rather odd, and awkward angles with which to view the solid. 

\[f(t) = (1.3 \cos(t) - 0.5 \cos(2t) - 0.2 \cos(3t) - 0.1 \cos(4t) \] 

\[g(t) = (1.3 \cos(t-25) - 0.5 \cos(2(t-25)) - 0.2 \cos(3(t-25)) - 0.1 \cos(4(t-25)) \]

My curve for  \(s(t)\) is the x-coordinate for the parameterization of a single heart. 

 \[s(t) =  \sin(t) ^3 \]

The curve with which to represent my solid is thus:

\[r(t) = (t, \frac{1}{2} h(t), \frac{1}{2}k(t))\] where  \[h(t) = f(t) + g(t)\] and \[k(t) = f(t) - g(t)\]

Below I give the images of my curves, followed by how they each appear on the solid and viewed at the optimal angles, and finally how the solid will appear without the curves and viewed at an arbitrary angle. 

The lopsided shape originates from \(g(t)\) and the classic heart shape comes from \(f(t)\)

To view the solid to see the image of \(f(t)\), requires a steep angle, and a view slightly off to the side at the back. The rectangle closes to the front of the image signifies where is best to view the solid to see \(f(t)\).

To see \(g(t)\) requires a less steep angle than to view \(f(t)\), but must be viewed from the front. The wide rectangle (rendered prior to adding the rectangle for \(f(t)\)) shows which angle necessary to face the object to see \(g(t)\)

Finally, the blue solid shows the function \(r(t)\) and both rectangles with which to view the solid such that either \(f(t)\) or \(g(t)\) can be seen. From this angle, neither function alone can be seen, and appears to take on visual characteristics of both curves. 

Due to the length of the rectangles, the final print will be four inches long, and span roughly three inches in width. The print will be roughly an inch tall. These parameters are given subject to change due to possible constraints imposed by print time, and cost of the print. 

As mentioned in the thesis of this blog post, I ridiculously propose that this ambiguous object represents disappointment of dashed idealization of an arbitrary person. When viewed from the front, a perfect heart shape appears, which can signify how lack of context and background knowledge of a new person creates an infallible version of that person to the viewer. However, as time goes on, the infallibility of the person is revealed and perhaps even negative characteristics are as well, and can be signified by the view of the lopsided heart which is seen from the back of the solid.

I reached out to the Guggenheim museum about this boundary pushing 3D print, but I have not yet heard back. Though I fully expect in short time, after its appearance as a flagship piece at the Guggenheim, it will be auctioned off at a Sotheby's not at all near you.