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.

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