## Vertical And Horizontal Volumes And Their Combinations Snowing Increased Beauty Tmrougn Variety

FIG* 33* VOLUME BASED ON TMC GOLDEN OBLONG^ WHIRLING SQUARE, OR XM AREA* RATIOS OP SIDES l.<3IQ* THIS RATIO 13 THE MOST POPULAR IN THE FIELD OF DESIGN •

note transfer of dimensions*

FIG« 34»FRONTAND SIDE VIEW OF FIG* NOTE TRUE. SHAPES*

FIG* 35* LEFT SIDE R00T4* RIGHT SI DC XM »TOP I.Z3*

FIG* 35* LEFT SIDE R00T4* RIGHT SI DC XM »TOP I.Z3*

FIG* 30* LEFTSIDE ROOT 5* RIGHT SIDE ROOT2.® TOP XM*

fig* 3ft* the root rectangles drawn within a square* "1 *** t? ffr

FIG.39. ROOT RECTANGLES FROM THt SQUARE

FIG* 40» /METHOD OF DRAWING RECTANGLES OF SIMILAR RATIOS« HORIZONTAL POSITION-

FIG» 41* VERTICAL RATIOS SIMILAR TO FIG* 40° NOTE DIAGONALS AT RIGHT ANGLES TO EACH OTHER*

Eh*«**« <f/ | |

Room/ | |

Cab/not |

FIG.39. ROOT RECTANGLES FROM THt SQUARE

volumetric casings on Plate 3, while their relations to each other are depicted in Figure 32, Plate 3. Grouped as they are in Figure 32, their wonderfully rhythmic relationships are shown to advantage.

Plate 4 was drawn to illustrate fully these rectangles in various volumetric combinations through which their beauty of rhythmic proportionship, their vigorous character, and strong dynamic qualities may be appreciated and so justify Mr. Hambidge's designation of "Dynamic Rectangles." For illustrative purposes, the tops of all volumes on Plate 3 were drawn as squares, while on Plate 4, as this is no longer necessary, much variety is shown by the various combinations.

The dynamic rectangles can be constructed readily by means of the metric scale. If 100 millimeters has been adopted as the unit of measure for the front edge of the volume, multiply 100 by the desired ratio, as 100x1.73, the Root Three ratio. The result, 173, is the desired length of the side, and 173 millimeters may be measured, as in Method One, page 31. For convenience, the metric scale is appended, while a small celluloid scale is available for a few cents. 10 millimeters equal one centimeter 10 centimeters equal one decimeter 10 decimeters equal one meter Obviously all pieces of furniture cannot be designed within the volumes based upon various combinations of Root Rectangles, as indicated in Figures 33 to 37 on Plate 4. But, at any rate, practice in drawing these rectangles will lead towards better understanding of proportionate relations.

You will observe that Figures 35, 36, and 37 on Plate 4 are vertical volumes, composed of variously proportioned Hambidge rectangles. For these vertical volumes, select a unit of measure as 50 or 100 millimeters, and multiply it by the desired ratio as follows: For Root 2— .705, Root 3— .576, Root 4— .5, Root 5— .447, and, for the Whirling Square, .618. Lay the result off on the line of measure, project to the desired edge, and the proportion has been completed. Example: It is desired to make one face of the volumetric casting a Root Three ratio in a vertical position. Select a unit of measure as 100 mm. and multiply by .576. The product, 57.6 mm., is then measured off on the line of measure and checked up by familiar procedure with the foreshortening triangle. Horizontal ratios and volumes are formed by multiplying the unit of measure by the ratios referred to in the opening paragraphs of this section.

For those individuals who prefer the geometric approach to Root Rectangle construction, reference is directed to Figure 38, Plate 4, in which the Root Rectangles are drawn within the square. The construction and geometric processes are self-evident. In Figure 39, the Root Rectangles have been drawn in order outside the square and again, in their juxtaposed positions, their rhythmic relationships are revealed.

### Similar Rectangles

This chapter cannot well be closed without reference to that useful device for repeating similar rectangles; that is, rectangles of exactly similar ratios. Thus, by duplicating similar forms, we have produced harmonious beauty through repetition. These similarly proportioned rectangles may be either larger or smaller than the parent form.

For an example of similar rectangles, note the 1.618 ratios of Figures 33, 35, and 36, Plate 4. You will observe that their diagonals are all parallel, regardless of their varying sizes. Referring to the right-hand drawing of Figure 40, Plate 4, you can readily understand the method of creating a number of rectangles of varying sizes but of similar ratios, drawing any desired line far enough to intersect the diagonal x of the rectangle y, which gives the corner of a new and similar rectangle.

The left-hand figure of 40 shows the plan of a room and the diagonal method of proportioning the table to repeat the room ratio, giving unity through similarity. To make a rectangle of exactly the same proportions as the room plan of Figure 40, construct the line b, Figure 41, at right angles to the line a, the room diagonal. By constructing a rectangle based on the diagonal, as shown, one can create furniture elevations of exactly the same ratios as found in the room plan.

Application of this scheme is seen in Figure 41, in which a cabinet has one side harmonizing in its ratio with the room plan of Figure 40 and the wall elevation as well. Naturally these are almost ideal conditions, but repetitions of similar ratios wherever possible are desirable moves in the direction of integrating ratios throughout the room into harmonious and unified proportionate beauty.

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