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The illustration on the cover of this book appears with the permission of its creator, Roger Burrows.  This image is the result of a multi-step process.  The first step employs algorithms that arrange spheres of different sizes to form stable, closely packed three-dimensional structures.  Slicing these structures with flat planes exposes two dimensional cross sections of the type shown on the upper half of the next page.

The next step involves transforming these cross sections into lattices by connecting pairs of circle-to-circle contact points with straight lines. A partially completed lattice is shown  on the lower half of the next page.  For more information about close packing algorithms and lattices, see “Making Space – Finding Space” by Roger Burrows and the website www.rogerburrowsimages.com.

The final step involves artistic creativity rather than algorithmic precision.  By selectively coloring individual segments of the complete lattice and adding a few artistic flourishes, it is possible to create the cover’s whimsical undersea panorama.  Selective coloring processes of this type form the basis for Roger Burrows’ series of Ultimate Coloring Experience books: Images 1, 2, 3, 4 and 5.  See also his Hidden Images series and his other publications.

Note that the discovery of images that are hidden within patterns generated by geometric algorithms is similar in spirit to the discovery of algebraic relationships that are hidden within numerical data generated by random processes.  Hidden algebraic relationships of this type play a crucial role in observational stochastics.  Many of the results and examples presented in this book are based on such relationships.