By Walker R. J.

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**Example text**

September 7, 2011 10:37 World Scientific Book - 9in x 6in 24 Carter˙Red˙to˙Blue An Excursion in Diagrammatic Algebra Fig. 3 A branch point: this does not occur during the eversion A surface when projected to a plane, like the orange that we look upon, may have a segment upon which the tangent plane is perpendicular to the plane of projection. Such a segment is called a fold line; an example is illustrated in Fig. 4. The important fact to understand is that a drawn surface, or a surface that is seen, has a fold.

The figure demonstrates a red fold, a red cusp, a double point arc, and a triple point by enclosing these in green squares. The triple point indicated lies below a red sheet. When a point of interest lies below a sheet of the sphere, it is said to be veiled. There may be several veils between you, the viewer, and the interesting point. There are three other triple points in the figure; try and find them. The cusp at the bottom of the figure is blue, and the red sheet veils it. On the right of the illustration a blue fold emerges from behind a red fold and this blue sheet intersects a red sheet towards the bottom of the illustration.

A fold appears that has an up-left cusp and a downleft cusp as its end points. The birth (or death) of a pair of cusps that are connected by a pair of fold lines is called the lips change because when the folds are drawn sideways, their introduction looked like a pair of lips. I think that the terminology is due to French mathematician, Ren´e Thom. To change a round red sphere into a round blue sphere, an arc of blue folds has to be introduced. One way of doing so is to introduce folds via the introduction of lips.