CHAPTER XII

DEVELOPABLE SURFACES

DEVELOPABLE

EVELOPABLE Surfaces. I have said that in the analysis of these complex problems it is necessary to consider parts of the subject separately, to isolate them, and study the problems in their simplest aspects. I am indebted to my son, Mr. M. Treleaven Reade, for pointing out that certain geometrical forms can be developed out of plane surfaces without extension, compression, or stress or strain. These are called developable surfaces.1

The application of this principle to the study of mountain structure is of interest in enabling us to elucidate some of the complexities of the subject. Mr. M. Treleaven Reade has been good enough to make the model shown on Plate IX., figs. 1 and 2.

1 See 'Some Properties of Flexible Surfaces and Flexible Solids (Trans. of Liverpool Engineering Society, vol. xx. Session 1898–99). A superficially rigid surface which can be unrolled so as to lie wholly in one plane is known mathematically as a developable,' and it is only this class of surfaces and their combination in various stages of development that the author treats of in this paper. The 'developable ' surfaces consist of the cylinder, cone and torse. They are a distinct geometrical conception from the truly mathematical surface, in that they are assumed to possess the material qualities of unstretchability and unshrinkability, or, as defined before in other words, they are 'superficially rigid.' For all practical purposes we may regard 'developables' as sheets of material thin enough to bend easily, but thick enough to be superficially rigid' (p. 191).

Fig. 1 shows a rectangular surface, formed out of drawing-paper, measuring 6 inches by 6 inches. The curved lines drawn upon it represent scorings made by a knife alternately on opposite sides of the paper, the firm lines representing the ridges, the dotted lines the valleys, as developed in fig. 2.

Fig. 2 shows this surface folded into curved symmetrical forms: a symmetrical ridge-and-furrow system of mountains. The surface area of the folded system is exactly that of the unfolded parallelogram, or 36 square inches. The basal area of the folded system measures 24.12 inches, or 11.88 square inches less than when flattened out as shown by the encircling white line. It will be observed that the basal plan of the folded system assumes an irregular form, and that in the development of the ridge-and-furrow folds the horn of the fold at a has moved in a spiral.

Examined analytically, it will be found that, excepting what is due to the thickness of the paper, there is, theoretically, neither stress nor strain involved in the folding; and this is true of all the forms of developable surfaces. Translated into the concrete, it will also be observed that to fold up a system of mountains of this form and character compressile movements would have to be applied in variable degrees to the exterior of the parallelogram.

These areas have all been measured with the planimeter.