Modeling the Mirascope Using Dynamic Technology
Figure 4c: Two light rays that are parallel to the \(y\)-axis reach an upward parabola.
Figure 4d: Hide the intermediate steps to reduce the construction complexity.
Figure 6a: A light ray from point \(P\) reaches the upper parabola and is reflected to the lower one.
Figure 6b: A light ray from point \(P\) is reflected out of the mirascope.
Figure 6d: Defining a new tool to reflect light from point \(P\) out of the mirascope. In GeoGebra, click "Tools" in the menu, select "Create New Tool," and follow the onscreen directions to click the objects involved. In our case, inputs include point \(P\), point \(G\), number \(a\), and number \(c\). Outputs may include all the objects related to the light reflection or, if so desired, the light ray q only.
Figure 6e: Using the newly defined tool to reflect a second light ray from point \(P\) out of the mirascope. In GeoGebra, place point \(H\) anywhere on the downward parabola. Point \(H\) is the location where a second ray from point \(P\) touches on that parabola. Next, click the button for the new tool at the upper-right corner. For inputs, click point \(P\) and point \(H\). When prompted for numbers, type "a" and "c". The color of the second ray is changed to distinguish it from the first one.
Figure 7c: Modeling more light rays from point \(P\) is not necessary to locate the image of \(P\).
Figure 7d: The image is well above the mirascope if the two parabolas are relatively close to each other.
Figure 7e: Finding the appropriate opening by tracing the outgoing light rays.
Figure 9: When the eye, the object, and the focus of a parabolic mirror are aligned in certain ways, one may be able to see a clear image with changes in its size and orientation.
Bu, Lingguo, "Modeling the Mirascope Using Dynamic Technology," Loci (November 2010), DOI: 10.4169/loci003595
Will other shapes?
Since the publication of article, I have received several emails about the mirascope and its GeoGebra-based explorations. Among them are (1) can we use a sphere? (2)how could you use it in the dark (place a LED inside)?, (3) what is the point of having students make their own? I welcome your ideas and suggestions. Thanks for your interest.
The role of models for transferring understanding
I have a mirascope on my desk in my office. Based on the interactions I see between office visitors and the mirascope, it is the single most intriguing object in my office (and yes, this does check my ego). I put it on my desk because of a conversation I overheard one day while sitting in a coffee shop. There were two male university students next to me, and they were highly intrigued by a question that one of them had posed. He asked "Why does our brain confulse left and right when we see a reflection in a mirror, but we don't get confused with the up and down direction in the reflections?" These two people generated several possible explanations (I suppose I might call them hypotheses), and all of their reasons were based on incoherent bits of knowledge about psychology and neuroscience rather than notions about optics or even a passing reference to a physical model. They eventually encountered their reflections in the spoons on their table, dared to ask each other how their reflection could be upside down, and thus they were completely baffled (ending the conversation with a shrug and apparent feeling of helplessness). I had a laptop with GeoGebra on my table. I was dying to introduce them to a model that might help them tie these incoherent ideas together, but I chose to finish my work so that I could return home to play with my one year old daughter. If I had known of this article, I would have shown it to the two university students and let them disentangle their ideas. Lingguo Bu has very elegantly taken these ideas to the next level with a beautiful GeoGebra-based model of the basic optics involved in the mirascope, showing a complex integration of basic ideas that result in a fascinating phenomenon. Does the author or my fellow readers know of other GeoGebra models of basic ideas and related ideas that may be even more complex? I would love to put them together into a coherent unit related to optics and geometry.