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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:√φ, 1:√2, 1:√3, 1:√5, etc.).
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 √2, √3, 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.
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, 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
can produce the most varied and satisfactory harmonic (consonant, related by symmetry) subdivisions and combinations, and this by the very simple process [...] of drawing inside the chosen rectangle a diagonal and the perpendicular to it from one of the two remaining vertices (thus dividing the surface into a reciprocal rectangle and its gnomon), and drawing any network of parallels and perpendiculars to sides and diagonals. This automatically produces surfaces correlated by the characteristic proportion of the initial rectangle, and also avoids (automatically again) the mixing of antagonistic themes like √2 and √3 or √5. √5 and Φ, on the contrary, are not antagonistic but consonant, also with √Φ, Φ2, et cetera.
According to Wolfgang von Wersin's The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described (1956), a set of 12 special orthogons (from the Gr. ορθος, orthos, "straight" and γονια, gonia, "angle"; "a right angled figure", which, as a consequence, is rectangular and tetragonal) has been used historically by artists, architects and calligraphers to guide the placement and interaction of elements in a design. These orthogons are:
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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:√φ, 1:√2, 1:√3, 1:√5, etc.).
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 √2, √3, 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.
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, 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
can produce the most varied and satisfactory harmonic (consonant, related by symmetry) subdivisions and combinations, and this by the very simple process [...] of drawing inside the chosen rectangle a diagonal and the perpendicular to it from one of the two remaining vertices (thus dividing the surface into a reciprocal rectangle and its gnomon), and drawing any network of parallels and perpendiculars to sides and diagonals. This automatically produces surfaces correlated by the characteristic proportion of the initial rectangle, and also avoids (automatically again) the mixing of antagonistic themes like √2 and √3 or √5. √5 and Φ, on the contrary, are not antagonistic but consonant, also with √Φ, Φ2, et cetera.
According to Wolfgang von Wersin's The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described (1956), a set of 12 special orthogons (from the Gr. ορθος, orthos, "straight" and γονια, gonia, "angle"; "a right angled figure", which, as a consequence, is rectangular and tetragonal) has been used historically by artists, architects and calligraphers to guide the placement and interaction of elements in a design. These orthogons are: