Shadows and Structure

Jeremy Kahn

Cubism and Plato's Allegory of the Cave

To my understanding, cubism is an attempt to show all the sides of an object simultaneously. This kind of vision would be to our vision the way our vision of the world is to those people in Plato's Cave: suddenly, external structure that could only be deduced or discovered by rotation is, instead, quickly and immediately obvious.

One way in which Plato's Allegory is not entirely dimensional is that the information that the freed prisoner gains is not merely the insight into dimensionality and 3-space, but also into secondary characteristics of the actual objects: their color, heat and smell, for example, which are all unavailable to those observing only the shadow of the object.

Simplexes and Pascal's expansion

We have explored Pascal's Triangle in some detail in the first few weeks of class--or at least an application thereof. But what does Pascal's Triangle represent? Vertically, we have n, of the function (a+b)^n, and as we descend n values, the powers of a and b separate from each other.

What happens when we need to describe (a+b+c)^n, though? Pascal's Triangle can't accomodate a fourth variable.

A simplex has a fairly simple definition: (n+1) points that are as far apart from all of their companions as possible in n dimensions. In two dimensions, we can separate three points by distributing them evenly around a circle, forming the vertices of a triangle. In three, we must distribute four points around a sphere, forming the vertices of a tetrahedron. All the points in both shapes are as far as possible from each other on the surface of their respective n-spheres and each is symmetric with respect to the others: That is, a, a vertex of a 2-simplex (a triangle) relates to b and c in the same way that b relates to a and c, and c relates to a and b.

This symmetry is useful for the problem posed above: what shape can we use to treat the variables a, b and c in a symmetric manner in a single layer of n? A 2-simplex: a triangle. When you stack the 2-simplex layers (combining the diagrams for values of n), it becomes easy to see that the Pascal's object for the expansion of (a+b+c)^n is a 3-simplex: a tetrahedron, with one vertex n, and the other three vertices a, b, and c.


We use slicing in linguistics, as well--in fact, I wouldn't be surprised if another student chooses to bring this topic up. An isogloss (New Greek, iso-, same, + gloss tongue, language) is a line linguists draw on a map connecting points that share some linguistic feature. (In some contexts, it can be an attempt to divide a geographical population into two or more distinct groups. The lines that appear are very similar to those drawn by ecobiologists; after all, the difference is only in the kind of population. The data, on the other hand, can be of the same family and can be represented in similar graphs.

Jeremy Kahn x6753

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