Users of current solid modelors can be forgiven for hesitating to ask the basic question: what is a manifold geometry? You’re more likely to ask what a non-manifold geometry is, particularly if the program you’re using has issued a stern warning regarding the geometry you’re trying to construct.

The mathematical answers can be found in several places:  from the rigorous and comprehensive but often intimidating World of Wolfram to the more egalitarian but still abstruse Wikipedia.  One of the better answers is at the intriguingly named Applet Magic website.

The answer in this article is an alternative, intended for the mathematically challenged engineer.

The quick answer for the world of 3D modelers is that a manifold geometry is one that satisfies the Euler Poincare equation: E + 2 = V + F, where E is the number of edges, V is the number of vertices and F is the number of faces. Only such an object can be found in the “real” world.  A  non-manifold geometry, of course, is one that violates the equation. It is a mathematical entity only, one that has no physical or real equivalent. In other words, if you try to manufacture it you’ll be out of luck unless M.C.Escher’s on your team. Or your customers favor Timothy Leary.

Now the maths. No, not the equations, but the interesting stuff. This is derived from the remarkable book by E.T.Bell. Any mathematically challenged engineer should read this book, but that’s another story. Back to the topic at hand.

A two-dimensional manifold is a class, or set, of objects that can be precisely described by exactly 2 numbers. For instance, if you specify a sound using its pitch and its loudness, you have created a 2-dimensional manifold: any sound in this set can be prescribed by exactly 2 numbers.

A surface is a 2-dimensional manifold. Why? Since it consists of points, and therefore represents a set of points. And you need exactly 2 parameters (commonly called u and v or t and s by many CAD programmes)  to specify uniquely and completely the position of any point on the surface.

Such definitions can be extended to n-dimensions.  So when your computer complains, it’s saying that the geometry you’re building doesn’t belong to the class of objects it’s been programmed to build / analyse / display / whatever.

Any time you construct a solid, and then delete a face, you’re reprimanded for creating a  non-manifold geometry. It may not do you much good, but you would be within your rights to email the programmers that your geometry is certainly manifold but not a part of the same set the programmer wanted.

Programmers, by the way, are to be pitied rather than censured since they sometimes have to tie themselves in knots to apply the Euler-Poincare equation. Try applying it to a sphere and you’ll see what I mean.

And if you’re seriously interested in this topic, you must catch up the Ted Nugent of Physics - Clifford Stoll - whose lecture here at ted.com was fun, if somwehat baffling. If I’d seen this way back when I wrote the article above, perhaps I wouldn’t have been so dismissive of Leary. And perhaps I’d have written about Klein bottles instead.