Lewis and Owen’ sought a mathematic solution to find the best retentive configuration for complete crowns. They used the term “admissible” for a form that consists of three straight line segments AB, BC, and CD as in Fig. 1, where the sum of the angles a and b is not less than 180 degrees. This describes mathematically a preparation that is not undercut. The term “form” was used interchangeably for the exterior shape of the prepared tooth and the interior shape of the finished crown, which they assumed to be geometrically identical. A force is completely determined by specifying direction, magnitude, and point of application. All possible directions of this applied force (but not the magnitude and point of application) were considered, with respect to their effect on the crown’s retention. Lewis and Owen defined the “retentivity” of an acceptable preparation to be the range of directions of the applied force for which the crown does not move in relation to the tooth. For example, if for a given preparation the crown does not move under the influence of forces of directions, say between 75 and 100 degrees, but does move for forces having directions outside this range, that form shall be said to have a retentivity of 25 degrees. With the above definitions the problem can be reformulated for admissible forms. How can the angles a and b (Fig. 1) and the lengths AB and CD be selected so that the retentivity will be maximum? The authors divided the analysis into two parts, geometric and mechanical.
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