Tuesday, January 31, 2017

The topology of the human form

What is topology?

"Basically, topology is the modern version of geometry, the study of all different sorts of spaces. The thing that distinguishes different kinds of geometry from each other (including topology here as a kind of geometry) is in the kinds of transformations that are allowed before you really consider something changed. (This point of view was first suggested by Felix Klein, a famous German mathematician of the late 1800 and early 1900's.)
In ordinary Euclidean geometry, you can move things around and flip them over, but you can't stretch or bend them. This is called "congruence" in geometry class. Two things are congruent if you can lay one on top of the other in such a way that they exactly match.
In projective geometry, invented during the Renaissance to understand perspective drawing, two things are considered the same if they are both views of the same object. For example, look at a plate on a table from directly above the table, and the plate looks round, like a circle. But walk away a few feet and look at it, and it looks much wider than long, like an ellipse, because of the angle you're at. The ellipse and circle are projectively equivalent.
This is one reason it is hard to learn to draw. The eye and the mind work projectively. They look at this elliptical plate on the table and think it's a circle because they know what happens when you look at things at an angle like that. To learn to draw, you have to learn to draw an ellipse even though your mind is saying `circle' so you can draw what you really see, instead of `what you know it is'.
In topology, any continuous change which can be continuously undone is allowed. So a circle is the same as a triangle or a square because you just `pull on' parts of the circle to make corners and then straighten the sides, to change a circle into a square. Then you just `smooth it out' to turn it back into a circle. These two processes are continuous in the sense that during each of them, nearby points at the start are still nearby at the end." Robert Brunner

More examples of the topology of the human form as mapped with various CAD solutions, i.e, the types of meshes that wrap the geometry. Each kind of mesh has certain parameters that it follows. This is determined by the mathematical algorithm that defines where various coordinates are in space,.For example, a mesh may be perpendicular to the ground plane, or it can be at a designated angle to the ground plane. The mesh is often thought of as being a regular matrix of polygons, projected on or dividing up a volume. The mesh by extension creates a way for the brain to visualize the displacement or occupancy of that volume in space.

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