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Magic Paper Acknowledgements Video15 COOL PAPER TRICKS
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While these goals are sometimes congruent, they are also often in tension. As an example of this tension, explanatory media are often highly polished, presenting a finished product from which distracting elements have been eliminated.
By contrast, exploration involves distractions, wrong paths, and dead ends. One sees this in myriad small ways in the video: the frequent readjustment of the position and size of the glyphs; the various unnecessary weight and bias parameters that are sometimes shown; and so on.
In a polished presentation these would have been eliminated, showing just what the viewer needs at any given moment. But such superfluous elements are inevitable in exploration.
Another example of the tension between education and exploration comes from the programming language Logo.
Logo is often touted as an example of discovery-based learning, and in many respects it's a good exploratory medium for mathematics.
However, while Logo has had considerable success in helping children understand ideas about differential geometry, differential geometers do not, as far as I'm aware, use it as a medium for research.
An interesting project would be to create a single environment which can be used for both education and research. I am optimistic this is possible; after all, beginning chess players use the same board and the same moves as the world champion.
The second question motivating Magic Paper was whether it is possible to reify deep ideas about mathematics inside the interface. That sounds cryptic, but has a straightforward meaning.
To explain that meaning, let's briefly switch topic, and recall an idea from elementary physics, the principle of conservation of energy:.
The basic point is that there's some relationship between the mass of a particle, m , its velocity v , its position x , and its potential energy U x.
The way physics is ordinarily done, we manipulate this expression using algebraic operations. But a different approach is possible.
I develop a prototype interface which reifies the principle of conservation of energy. In particular, the prototype vividly shows the user the relationship between mass, position, velocity and potential energy.
This makes it possible to directly see and adjust the potential energy, see how the trajectories respond, and so on. This gives a unique and rich experience, one enabling the user to internalize a powerful set of cognitive technologies for thinking about motion.
Chalktalk -like media seem promising for reifying such principles. Unfortunately, as far as I'm aware there are no general results about neural networks as powerful as conservation of energy.
However, as a step in this direction, directly seeing the relationship between the parameters of a network and the output of various neurons does help us understand the behaviour of the network.
It'd also be interesting to develop such an interface in a context where more deep principles are available — perhaps using Magic Paper to explore say chaotic dynamical systems, using ideas such as conservation of energy, conservation of phase space volume, Lyapunov exponents, and the KAM theorem.
Building such ideas into the interface would make them a routine, concrete part of the way the user works and thinks.
Wouldn't such keystrokes be faster and easier? This question has bothered me ever since first seeing Perlin demonstrate Chalktalk.
Even as an audience member, I found it much more satisfying to see him draw a glyph than to just hit a key and have it appear.
And this effect was even stronger as a user of Magic Paper. I don't have a satisfactory understanding of why it's satisfying to do this. Perhaps it's because the act of drawing activates associations in my brain that wouldn't have been activated as strongly by issuing a key command.
In this sense, the act of drawing is far more meaningful — and meaning-inducing — than hitting a key. It's a little like the reasons we don't name a variable meant to represent someone's name as jnwhg ; it's more meaningful to use name.
Or the reasons it's better to use a map of a coastline to navigate, rather than a log of latitudes and longitudes for points on the coast.
Unfortunately, while such explanations are suggestive they're vague and incomplete. And yet my lived experience is that this is a strong effect.