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the mathematics of art - model 3

MODEL 3 - MULTIPLE SIGMOID

This is nothing but a generalization of model 2. Instead of an artist starting off slowly, making some rapid progress, and then seeing his development taper off, one can maintain that what seems as a steady state is but a temporary plateau before a second wave of progress kicks in and his art is moved to a higher level.

Mathematically, its equation would be y = c + M1 / [1 + exp(-a1(x-t1))] + M1 / [1 + exp(-a1(x-t1))] + ... = c + sum{M1 / [1 + exp(-a1(x-t1))]}. Note that 1, 2, ..., n do not index different artists, but different stages of development for the same artist. Visually, the graph looks like a rising staircase. The starting point is c, the first plateau is at c+M1 centered at t1, the second plateau is at c+M1+M2 centered at t2 etc. All factors and all determinants have the same properties as in the simple sigmoid model.

I personally like this model the best for three reasons.

First, I like the implied process for an artist's career. I keep thinking of it as analogous to the phase change diagram in thermodynamics. In the heat curve (say, of water), as we constantly add heat to the ice, the ice begins to warm up and its temperature will increase until it reaches the melting point. At that point, the temperature will stop rising, but the ice will start to melt gradually until all the ice has been turned into water. Only when the phase change has been completed does the temperature of the water start rising again. The same mechanism is repeated as water turns into steam: the temperature of the water will keep rising until the first molecule that has enough kinetic energy to break from the liquid mass evaporates as steam. Then the temperature will stop rising while the water boils (i.e. the liquid turns into steam). When all the water has turned into steam, the temperature of the water (in its steam phase now) will start rising again with the addition of heat. It is very similar with an artist. At the beginning, not much is happening. As he starts his artistic training (the equivalent of adding heat), his artistic production improves (the equivalent of rising temperature). After he has reached a phase-change point, he doesn't seem externally to make any more progress (no temperature rise), but in fact he is still absorbing knowledge (heat). The analogy with thermodynamics works beautifully at this stage because the terminology works well for both cases: there is no increase in kinetic energy, but there is an increase in potential energy (the sum of the two equaling heat content, whereas temperature measures kinetic energy only). There is something boiling inside the artist (pun intended) at this seemingly quiet stage. He is trasforming, he is changing into a new phase. When that phase change is complete, then his accomplishment starts increasing again, albeit in a different form.

Second, it combines the linear model's optimism with the the simple sigmoid model's accuracy. While it maintains the simple sigmoid model's realistic description, it allows for the sky to be the limit. The artist is still bound by his talent, but this only translates in shorter steps in his staircase if he is less talented taller steps if he is more talented. There is no absolute, global maximum. It is, therefore, entirely up to the artist to maximize his potential (that is, to reach each plateau and move toward the next one) by packing his t's closer together and working harder and harder to increase his a's. As in the linear model, it is a race against time and death. Comment on this: although this is a neat theoretical model, historically, it is only rarely that you see an artist move beyond the first plateau before time catches up with them. Picasso is clearly one such case. Beethoven is another. Maybe in this model we can define genius as someone with a much higher i (i being the number of steps in the staircase or, formally speaking, the number of simple sigmoid functions comprising the multiple sigmoid curve).

Third, it models better real-world competition among artists. The less talented artist does not only get one window of opportunity to surpass the more talented artist, as was the case in the simple sigmoid model. Instead, there are endless possibilities of a race between two artists with endless turning points (formally, intersection points). In fact, in the extreme underdog supporting scenario, if the less talented artist keeps adding heat, meaning moving fast and working hard, he can do as well as quickly overcome the talented artist's initial head start (this is inevitable since the latter's c will always be higher) and stay ahead of the game for the rest of his life. In this respect, this model allows another similarity with the linear model in that, once the less talented artist catches up with the more talented one, he then consistently exceeds him.

posted by h ampelofilosofos @ 8:04 PM