Section Ii: Reference; The Geometry Of A "Sphere - 4ms Company Spherical Wavetable Navigator User Manual

SECTION II: REFERENCE

The Geometry of a "Sphere"

A torus
While we use the term "Sphere" when talking about the wavetables in the SWN, the structure of each wavetable
is actually a 3-torus. What's a 3-torus? It's a three-dimensional structure existing in a four-dimensional space. To
understand what a 3-torus is, first think about a circle. A circle is a single dimensional object that wraps around to
its beginning point as it reaches its end point. Any point on the circumference of a circle can be specified by a
single number (by an angle from 0° to 360°), so a circle is one-dimensional. However, in order to draw a circle you
need a two-dimensional space such as a piece of paper. So, a circle is a one-dimensional object existing in a two-
dimensional space. Now, think about a doughnut (a torus, see picture above). You can make a doughnut if you
extrude a circle into a cylinder and then bend the cylinder around so the top and bottom faces are touching. You
can specify any point on the surface of the doughnut with just two numbers (an angle of the original circle and a
position along the extruded cylinder), so the torus surface is two-dimensional and clearly exists in a three-
dimensional space.
The next step is not as easy to visualize — imagine you took a doughnut and extruded it through a fourth
dimension, and then connected the beginning to the end. This is a 3-torus. It's a three-dimensional object that
exists in a four-dimensional space. If you happened to be on the surface of a 3-torus and you looked far enough
in any direction you'd see the back of your own head! If you walked far enough in any direction you'd end up
exactly where you started, facing the same direction. The same is true for the SWN's wavetables: if you navigate
far enough in any direction, you end up back to where you started.
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