portrait - What's the succinct reason for face-warping in wide angle lenses?

Is there an accurate way of explaining face-warping when using wide angle lenses?

I realised I can't actually explain it without diagrams, so a succinct definition (no more than a few lines) would be greatly appreciated.

Answer

To fill the frame with your subject with a wide angle lens, you have to be really close. Perspective — the way lines map from 3D reality to a 2D surface like a photograph — is naturally strongest with objects that are close and flattens as things are far away. That's all there is to it.

This is really nicely explained with pictures at What does it really mean that telephoto lenses "flatten" scenes?, and How to avoid unflattering facial distortion when using a slightly wide angle lens? is likely also interesting.