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#### The Elements of Dynamic Symmetry

**By Jay Hambidge**

**Dover Publications, Inc.**

**Copyright © 1953 Mrs. Jay Hambidge**

All rights reserved.

ISBN: 978-0-486-14025-4

All rights reserved.

ISBN: 978-0-486-14025-4

CHAPTER 1

*The Square (1 or Unity).*

THESE lessons will deal entirely with the fundamental principles of symmetry as they are found in nature and in Greek art; no attempt will be made to show their application to specific examples of nature or of art.

The square and its diagonal furnish the series of root rectangles. The square and the diagonal to its half furnish the series of remarkable shapes which constitute the architectural plan of the plant and the human figure.

The most distinctive shape which we derive from the architecture of the plant and the human figure, is a rectangle which has been given the name "root-five." It is so called because the relationship between the end and side is as one to the square root of five, 1.:2.2360 plus.

As a length unit the end cannot be divided into the side of a root-five rectangle, because the square root of five is a never-ending fraction. We naturally think of such a relationship as irrational. The Greeks, however, said that such lines were not irrational, because they were commensurable or measurable in square. This is really the great secret of Greek design. In understanding this measurableness of area instead of line the Greek artists had command of an infinity of beautiful shapes which modern artists are unable to use. The relationship between the end and side of a root-five rectangle is a relationship of area and not line, because as lengths one cannot be divided into the other, but the square constructed on the end of a root-five rectangle is exactly one-fifth the area of the square constructed on the side. The areas of rectangles which have this measurable relationship between end and side possess a natural property that enables us to divide them into many smaller shapes which are also measurable parts of the whole.

A simple method for constructing all the root rectangles is shown in **Fig. 1**.

In **Fig. 1**, AB is a square whose side is assumed to be unity, or 1. Since the diagonal of a square whose side is unity equals the square root of two, the diagonal AB equals the square root of two. With A as center and AB as radius describe the arc of a circle, BC. The line AC equals AB, or the square root of two. AD is therefore a root-two rectangle. Numerically the line KA equals unity or 1. and the line AC equals the square root of two, or 1.4142.

The diagonal of the root-two rectangle, AD, equals the square root of three. By the same process the line AE is made equal to AD, or the square root of three. AF is therefore a root-three rectangle. Numerically its height, KA, equals unity or 1.; its length, AE, equals the square root of three, or 1.732.

The diagonal of the root-three rectangle, AF, equals the square root of four. By the same process the line AG is made equal to AF or the square root of four. AH is therefore a root-four rectangle. Numerically its height, KA, equals unity or 1.; its length, AG, equals the square root of four, or 2. The root-four rectangle is thus seen to be composed of two squares, since its length equals twice its height.

The diagonal of the root-four rectangle, AH, equals the square root of five. By the same process the line AI is made equal to AH, or the square root of five. AJ is therefore a root-five rectangle. Numerically its height, KA, equals unity or 1.; its length, AI, equals the square root of five, or 2.236.

This process can be carried on to infinity. For practical purposes no rectangles beyond the root-five rectangle need be considered. In Greek art a rectangle higher than root-five is seldom found. When one does appear it is almost invariably a compound area composed of two smaller rectangles added together.

In any of the root rectangles a square on the longer side is an even multiple of a square on the shorter side. Thus a square constructed on the line AC has twice the area of the square on KA; the square on AE has three times the area of the square on KA; the square on AG has four times the area of the square on KA; the square on AI has five times the area of the square on KA. This is demonstrated graphically in the following diagrams, Fig. 2.

The linear proportions 1, √2, √3, √4, √5, etc., are based on the proportions of square areas derived by diagonals from a generating square.

The linear proportions 1, √2, √3, √4, √5, are geometrically established as in Fig. 2.

In **Fig. 2a** this unit square is indicated by shading. Its diagonal divides it into two right-angled triangles, one of which is marked by heavy lines. The 47th proposition of the first book of Euclid proves that the square on the hypotenuse of a right-angled triangle equals the sum of the squares on the other two sides. Since the area of each of these two squares is 1, the area of the square on the hypotenuse is 2; and its side (or the diagonal of the unit square) equals √2. The side of the unit square is √1, which is equal to 1. Thus the linear proportion 1: √2 has been established.

In **Fig. 2b** the diagonal of the unit square is used as the base of a rectangle with sides of the lengths 1 and √2. This is called a root-two rectangle. Its diagonal divides it into two right-angled triangles, one of which is marked by heavy lines. The squares on the two shorter sides of this triangle are equal to 1 and 2 respectively. Therefore the square on its hypotenuse equals 1 + 2, or 3; and the side of this square (or the diagonal of the root-two rectangle) equals √3. Thus the linear proportion 1: V 3 has been established.

In **Fig. 2c** the diagonal of the root-two rectangle is used to construct a rectangle with sides of the lengths 1 and V 3. This is called a root-three rectangle. Its diagonal makes, with the two adjacent sides of the rectangle a right-angled triangle, which is marked by heavy lines. The squares on the two shorter sides of this triangle are equal to 1 and 3 respectively. The square on its hypotenuse equals 1 + 3, or 4; and the side of this square (or the diagonal of the root-three rectangle) equals √4. Thus the linear proportion 1: √4 has been established.

In **Fig. 2d** the diagonal of the root-three rectangle is used as the base of a rectangle with sides of the lengths 1 and √4. This is called a root-four rectangle. Its diagonal makes with the two adjacent sides of the rectangle a triangle, which is marked by heavy lines. The squares on the two shorter sides of this triangle are equal to 1 and 4 respectively. The square on its hypotenuse equals 1 + 4, or 5, and the side of this square (or the diagonal of the root-four rectangle) equals √5. Thus the linear proportion 1: √5 has been established.

In **Fig. 2e** the diagonal of a root-four rectangle is used to construct a rectangle with sides of the lengths 1 and √5. This is called a root-five rectangle. Its diagonal makes, with the two adjacent sides of the rectangle, a triangle. The squares on the two shorter sides of this triangle are equal to 1 and 5 respectively. The square on its hypotenuse equals 1 + 5, or 6, and the side of this square (or the diagonal of the root-five rectangle) equals √6. Thus the linear proportion 1: √6 has been established. This process can be carried on to infinity.

The root rectangles are constructed within a square by the following simple method:

[ILLUSTRATION OMITTED]

In **Fig. 3**, AB is a square and CED is a quadrant arc of a circle, the radius of which is AD. The diagonal of the square AB cuts the quadrant arc at E, and FG is a line through the point E drawn parallel to AD. AG is a root-two rectangle within the square AB.

The diagonal of the root-two rectangle AG cuts the quadrant arc at H. A line through the point H similar to the line through E establishes the root-three rectangle AI.

The diagonal of the root-three rectangle AI cuts the quadrant arc at J. The area AK is a root-four rectangle.

The diagonal of the root-four rectangle AK cuts the quadrant arc at L. The area AM is a root-five rectangle, and so on.

There are many other methods for the construction of root rectangles, both outside and inside a square, but, in the writer's opinion the ones described are the simplest.

CHAPTER 2*The Rectangle of the Whirling Squares (1.618) and the Root-Five Rectangle (2.236).*

IN the preceding lesson we learned the evolution of the root rectangles from the diagonal of a square. We shall now consider the important rectangle derived from the diagonal to half a square called the "rectangle of the whirling squares," numerically expressed as 1.618.

Because of the close relationship of the 1.618 to the root-five rectangle we shall have to consider the two together before taking the root-five in the logical sequence of its development from the square.

The rectangle of the whirling squares:

[ILLUSTRATION OMITTED]

Draw a square as CB in **Fig. 4**. Bisect one side as at A. Draw the line AB and make AE equal to AB. Complete the rectangle by drawing the lines BF, FE. This rectangle, DE, is the rectangle of the whirling squares. It is composed of the square CB and the rectangle BE.

The construction of a root-five rectangle is simple. Draw a square as CB in the diagram **Fig. 5**, and bisect one side as at A. Draw a line from A to B and use this line as a radius to describe the semicircle DBE. AE and AD are equal to AB, or the line DE is twice the length of the line AB. Complete the rectangle by drawing the lines DF, FG and GE. It will be noticed that the rectangle FE is composed of the square CB and two rectangles FC and BE. This is the root-five rectangle and its end to side relationship is as one to the square root of five, 1:2.236; the number 2.236 being the square root of five. Multiplied by itself this number equals 5.

The root-five rectangle produces a great number of other shapes which are measurable in area with themselves and with the parent shape. The principal one of these is the 1.618 or that which is made by cutting a line in, what Plato called "the section." The relationship of this rectangle to the root-five rectangle is shown by its construction. It is apparent that this shape is equal to the root-five rectangle minus one of the small rectangles FC or BE of **Fig. 5**.

The construction of the shapes of vegetable and animal architecture is simple and if we wished to begin using them in design little more would be necessary. Unfortunately we cannot afford to wait to discover all the wonderful properties possessed by these simple shapes by practically employing them in our design problems. Time is too short and our needs too great. But, fortunately, we have the use of a tool which the Greek artists did not possess. That implement is arithmetic. By the use of a little adding, multiplying, dividing and subtracting, we may expedite our progress enormously. This will be apparent if we consider the square of the root-five rectangle as representing one or unity; this may be 1. or 10. or 100. or 1000, but it will always be a square as an area of sides of equal length. Regarded thus it will be apparent that the area of any rectangle may be composed of one or more squares plus some fractional part of a square. The square root of five is 2.236; if one is subtracted from this number the result will be 1.236. In this case 1.236 represents the two small rectangles on either side of the square, .618 plus .618 equals 1.236. If this number is divided by two, the result is .618. This number represents each one of the two small rectangles. We now see that the area of the root-five shape may be considered as 1. plus .618 plus .618. Also it is now apparent that the area of the rectangle of the whirling squares may be considered numerically as 1. plus .618.

It was stated in the first lesson that the relationship between the areas of the squares described on the end and side of a root-five rectangle was as one to five. The square described on the side of a whirling square rectangle is equal in area to the square described on the end, plus the area of the rectangle itself.

CHAPTER 3*The Application of Areas.*

FOR the purpose of dividing up the areas of rectangles so that the divisions would be recognizable, the Greeks had recourse to a simple but ingenious method which is called the "application of areas." This idea was used by them both in science and in art. For a description of the process as used in science, see any standard work on the history of Greek geometry.

Classic design furnishes abundant examples of its use in art. The process in design may be illustrated by either of the rectangles described in the preceding lesson.

If, to a rectangle of the whirling squares, we apply a square on the end of that shape, the operation is equivalent to subtracting 1. from 1.618. A square on the end applied to the area of a whirling square rectangle leaves as a remainder a .618 area. If a square on the end of a rectangle is applied to its side, however, the operation is not so simple unless we use the Greek method. This process is shown in **Fig. 6**.

To the rectangle AB the square AC is applied on the end.

To apply this same square to a side as GB, we must first draw a diagonal to the whole shape as GH. This diagonal cuts the side of the square AC at D. Through the point D we draw the line EF. The area EB is equal to the area of the square AC. (For proof, see similarity of figure, Lesson 7, **Fig. 35b**.)

This process applies to any rectangular area whatever which may be applied either to the end or the side of a rectangle. It will be noticed that the square on the end of a rectangle when applied to a side changes its shape, *i.e.,* it is no longer a square, though equal in area to the area of a specific square. It is clear that now it is composed of a square plus some other area which may be composed of either a square or squares or some fractional part of a square.

*The Reciprocal.*

A PROCESS connected with the science of plan-making which was thoroughly understood by the Greeks, was that of determining the reciprocal of a rectangle. This conception of a reciprocal of a shape is most important. Briefly, the reciprocal of a rectangle is a figure similar in shape to the major rectangle but smaller in size. The end of the major rectangle becomes the side of the reciprocal. Greek art shows us several simple methods for determining reciprocals, but they all depend upon the fact that the diagonal of a reciprocal cuts the diagonal of the major shape at right angles. **Fig. 7** is a rectangle of the whirling squares.

AB is a reciprocal to the shape CD and the diagonals CD and AB cut each other at right angles at E. The rectangle AB is a similar shape to the whole, *i.e.,* the area AB is exactly like the area CD, the difference being one of size only. The line BD, which is the end of the larger rectangle, is the side of the smaller one. The name, the rectangle of the whirling squares, may now be explained.

Because of the fact of similarity, the reciprocal of this shape, **Fig. 7**, is also a whirling square rectangle and the area CA, *i.e.,* the area in excess of the area AB, is a square and if we define the continued reciprocals of this specific shape, the result would appear as **Fig. 8**.

A diagonal of the whole and a diagonal of a reciprocal, enable us to draw readily the continued reciprocals. The area KB is a reciprocal of the area KL and the area KC is a reciprocal of the area KB, etc. These areas are all similar shapes. The numbered areas, 1, 2, 3, 4, 5, etc., are all squares. A rectangle of the whirling squares is so called because its continued reciprocals cut off squares and these squares arrange themselves in the form of a spiral whirling to infinity around a pole or eye. The appearance in nature of this rectangle is explained in the lesson on the phenomenon of leaf arrangement.

The arithmetical statement of the reciprocal may now be considered. If unity or 1. represents the area of a square, then, in a whirling square rectangle, the fraction .618 must represent the reciprocal and the area .618 is also a whirling square rectangle. We obtain the reciprocal of any number by dividing that number into unity, which is equivalent to dividing the side into the end. 1.618 divided into 1. equals .618. The square root of five divided into unity, 2.236 into 1., equals .4472. The area represented by .618 is a whirling square rectangle, and that by .4472 is a root-five rectangle. As far as shape is concerned a root-five rectangle, numerically, may be 2.236 or .4472; this also applies to a rectangle of the whirling squares which may be 1.618 or .618. Any rectangle may be arithmetically expressed in the same manner.

It will be noticed that a root-five rectangle may be considered as composed of a square plus two whirling square rectangles, or as a square plus two reciprocals of that shape. It may also be considered as two whirling square rectangles overlapping each other to the extent of a square, **Fig. 9**.

*(Continues...)*

Excerpted fromThe Elements of Dynamic SymmetrybyJay Hambidge. Copyright © 1953 Mrs. Jay Hambidge. Excerpted by permission of Dover Publications, Inc..

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