Dynamic rectangle

A dynamic rectangle is a right-angled, four-sided figure (a rectangle) with dynamic symmetry which, in this case, means that aspect ratio (width divided by height) is a distinguished value in dynamic symmetry, a proportioning system and natural design methodology described in Jay Hambidge's books. These dynamic rectangles begin with a square, which is extended (using a series of arcs and cross points) to form the desired figure, which can be the golden rectangle (1 : 1.618...), the 2:3 rectangle, the double square (1:2), or a root rectangle (1: , 1: , 1: , 1: , etc.).

A dynamic rectangle is a right-angled, four-sided figure (a rectangle) with dynamic symmetry which, in this case, means that aspect ratio (width divided by height) is a distinguished value in dynamic symmetry, a proportioning system and natural design methodology described in Jay Hambidge's books. These dynamic rectangles begin with a square, which is extended (using a series of arcs and cross points) to form the desired figure, which can be the golden rectangle (1 : 1.618...), the 2:3 rectangle, the double square (1:2), or a root rectangle (1: , 1: , 1: , 1: , etc.). ==Root rectangles== A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as , , etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two longer sides of a root-2 rectangle to the length of the root-2 rectangle's diagonal. Each successive root rectangle is produced by extending a root rectangle's longer sides to equal the length of that rectangle's diagonal. ===Properties=== When a root-N rectangle is divided into N congruent rectangles by dividing the longer edge into N segments, the resulting figures keep the root-N proportion (as illustrated above). The root-3 rectangle is also called sixton, and its short and longer sides are proportionally equivalent to the side and short diameter (height) of a hexagon. Since 2 is the square root of 4, the root-4 rectangle has a proportion 1:2, which means that it is equivalent to two squares side-by-side. The root-5 rectangle is related to the golden ratio (φ). The longer side is equal to one plus two times 1/φ (0.618...). ==Root-φ rectangle== s (right triangles with edge lengths in geometric progression).]] The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles. ==Jay Hambidge== Jay Hambidge, as part of his theory of dynamic symmetry, includes the root rectangles in what he calls dynamic rectangles, which have irrational and geometric fractions as ratios, such as the golden ratio or square roots. Hambidge distinguishes these from rectangles with rational proportions, which he terms static rectangles. According to him, root-2, 3, 4 and 5 rectangles are often found in Gothic and Classical Greek and Roman art, objects and architecture, while rectangles with aspect ratios greater than root-5 are seldom found in human designs. According to Matila Ghyka, Hambidge's dynamic rectangles Two of the most complicated of these figures are; the penton, with proportions 1: is related to the section of the golden pyramid, the bipentons longer side is equal to the shorter side multiplied by 2 , or 2\sqrt{\phi^2 + 1}/\phi^2 ; the longer side of the biauron is - 1, or 2 / φ, times the shorter. The quadriagon is related to the diagon in the sense that its longer side is produced by projecting the diagonal of a quarter of a square. The trion has the height of an equilateral triangle and the width of the side. The hemidiagons longer side is half that of the root-5 rectangle and is produced by projecting the diagonal of half a square until it is perpendicular with the origin. Besides the square and the double square, the only other static rectangle included in the list is the hemiolion, which is produced by projecting 90° or 180° half the side of a square. ==References== ==Further reading== Hemenway, Priya; Divine Proportion, Phi in Art, Nature and Science; 2005, Sterling Publishing Co., Inc, NY, NY.

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