Simply put, contributors to the Roanoke Valley Coral Reef will use basic crochet techniques to produce pieces that are strikingly similar to naturally growing coral. The many individual pieces are then assembled into a "reef" of diverse and lovely shapes.
The piece shown to the left starts with a single line of chain stitches. In each successive row, the number of single crochet stitches is doubled. The rapid increase in the number of stitches causes the relatively stiff yarn to curve in an interesting way. Mathematicians classify this type of curvature as negative. Negative curvature is the rule for surfaces in what is called hyperbolic geometry. More information about crocheting hyperbolic surfaces can be found at the website for Daina Taimina's book Crocheting Adventures with Hyperbolic Planes and at the Institute For Figuring website.
Hyperbolic geometry is beautifully illustrated in some of M.C. Escher's prints (link here for some details), and is a mathematically consistent set of geometric rules. In standard Euclidean geometry, you might recall that the angles of a triangle always add up to 180 degrees. One byproduct of the rules of hyperbolic geometry is that the angles in a triangle add up to less than 180 degrees. This is illustrated in the figure below.
The piece shown below illustrates another property of hyperbolic geometry. Believe it or not, the same amount of tan yarn and grey yarn was used in this hyperbolic piece.
The length of the outer edge of the surface grows very rapidly. After a few complete circuits, it takes a lot of yarn to make the next loop! If you scrunch up the surface, as in the following picture, you get a better sense of how extensive the outer edge is.