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Shell Space: The 'Snugness' Condition

David Raup (among others, including Stephen Wolfram) proposes that all naturally occurring shells conform to the same basic design, varying only in a limited number of quantifiable ways. In Raup's simple model, you can specify a shell by setting the flare, (the rate of growth of the spiral), the verm, (the "tightness" of the shell cavity), and the spire, (the rate of creep parallel to the spiral axis). If we insist that shells are snug (that is, the shell cavity is just loose enough so that each wind touches the previous and the next exactly), then the space of shells can be parametrized by flare and spire alone.

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This simplified model of "shell space" was later adopted by Richard Dawkins, who coined the terms "flare," "verm," and "spire" for the parameters , and , respectively, as an illustration of the notion of "design space" in the book Climbing Mount Improbable.

Here are precise definitions of the parameters.

1. If is the distance between a point on the generating spiral and that spiral's axis, and the distance from the corresponding point one turn later, then , which is strictly greater than 1.

2. If is the distance between the spiral axis and the inner edge of the shell cavity, and the distance between the spiral axis and the outer edge on the same turn, then , which is greater than or equal to 0 and strictly less than 1.

3. If the acute angle between the line and the horizontal is , then .

The "snugness" condition states that

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