Dynamic Symmetry is the law of proportion discovered by Jay Hambidge (1920) in plant and shell growth.It is the symmetry of organic structure.The basic pattern plan of leaf arrangement is only one phase of an architectural scheme connected with the phenomena of growth in general, but the dynamic scheme has a general base in Phyllotaxis.The basic pattern plan of Phyllotaxis was pointed out by Max Hirmer (1931) and Dynamic Symmetry must be revised by the geometrical analysis of his theory.Now I shold like to introduce a new theory of “Revised Dynamic Symmetry”.
Max Hirmer's study on Phyllotaxis is that the foundamental schemes of leaf arrangement are not based on various divergences given by Schimper-Braun's series, but on the limit divergences of that series.The limit divergences, viz., 137°30'28. “936, 99°30'6”, 77°57'19 “, 64°4' 43”, have the ratio of 0.618, 0.382, 0.2764, 0.2165, to their conjugate angle.These ratios are able to explain dynamically as in Fig.1.The fraction 0.618 is equal to a whirling square rectangle. The ratio 0.382 is equal to a square plus a whirling Square rectangle. The fraction 0.2764 which is the reciprocal of 3, 618 is equal to two squares plus a whirling square rectangle.The ratio 0.2165 is composed of three squares plus a rectangle of whirling squares.
For this reason it must be that the base of Dynamic Symmetry is the square and the whirling square rectangle.All compound rectangles must be also constituted by the square and the whirling square rectangles.A root-five rectangle is equal to a whirling square rectangle plus its reciprocal or it may be regarded as a square plus two whirling square rectangles.The root-five rectangle must be considered as a compound rectangle.The dynamic ratios which are most frequently found not only in nature but in Greek design can be explained by the compound rectangles composed of the square and the whirling square rectangle. Root rectangles, excepting the root-five, as persumed by J.Hambidge, may be considered as a type of symmetry intermediate between static and dynamic or as a minor phase of the dynamic type.
Nanzen-ji Garden is a famous garden of the Edo period in Japan.The composition of this garden can be explained geometrically by the compound rectangle of whirling square rectangles as shown in Fig.4.a and b.Revised Dynamic Symmetry cannot be used unconsiously, but it is able to say that the design of this garden closely approximates to Revised Dynamic Symmetry.Revised Dynamic Symmetry is of great value also in garden design.

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