A dynamic rectangle is a right-angled, four-sided figure (a

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

) with

dynamic symmetry Jay Hambidge (1867–1924) was a Canadian-born American artist who formulated the theory of "dynamic symmetry", a system defining compositional rules, which was adopted by several notable American and Canadian artists in the early 20th century. ...

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 Jay Hambidge (1867–1924) was a Canadian-born American artist who formulated the theory of "dynamic symmetry", a system defining compositional rules, which was adopted by several notable American and Canadian artists in the early 20th century. ...

's books. These dynamic rectangles begin with a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

, which is extended (using a series of arcs and cross points) to form the desired figure, which can be the

golden rectangle In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618. Golden rectangles exhibit a special form of self-similarity ...

(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

in which the ratio of the longer side to the shorter is the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...

of an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

, such as , , etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a

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.Jay Hambidge. (1926, 1948, 1967)
The Elements of Dynamic Symmetry
'. Courier Dover Publications. pp. 9–10.

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 diameter of a

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

. *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 In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

(φ). The longer side is equal to one plus two times 1/φ (0.618...).

Root-φ rectangle

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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

Kepler triangle A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...

Jay Hambidge

, as part of his theory of dynamic symmetry, includes the root rectangles in what he calls ''dynamic rectangles'', which have

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

and geometric

fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

as ratios, such as the

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 Prince Matila Costiescu Ghyka (; born ''Matila Costiescu''; 13 September 1881 – 14 July 1965), was a Romanian naval officer, novelist, mathematician, historian, philosopher, academic and diplomat. He did not return to Romania after World ...

, Hambidge's dynamic rectangles Root rectangles Caskey 1922

The 12 orthogons of Wersin

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 In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a par ...

and

tetragon In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

al) has been used historically by artists, architects and calligraphers to guide the placement and interaction of elements in a design. These orthogons are: * Square (1:1 or 1:) * Diagon (1:) * Hecton or sixton (1:) * Doppelquadrat (1:2 or 1:) * Hemiolion (2:3) * Auron (the

, 1: φ) * Hemidiagon (1:½) * Penton (1:) * Trion (1:⅔) * Quadriagon (1:(1+)/2) * Biauron (1:2φ) * Bipenton (1:2) Wolfgang von Wersin's book includes an extraordinary copy of text from the year 1558 (

Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...

), with diagrams of seven of the 12 orthogons and an invitation from the passage to pay careful attention as the "ancient" architects believed "nothing excels these proportions" as "a thing of the purest abstraction." All 12 orthogons, when formed together, create an entire unit: a square that is developed into a double square. Perhaps the most popular among the ortogons is the ''auron'' or

, which is produced by projecting the diagonal that goes from the middle point of a side of a square to one of the opposite vertexes, until it is aligned with the middle point. Four of these orthogons are harmonic rectangles: the ''diagon'' or '' root-2 rectangle'' is produced by projecting the diagonal of a square; the ''sixton'', ''hecton'' or '' root-3 rectangle'' is produced by projecting the diagonal of a diagon; the double square or '' root-4 rectangle'' is produced by projecting the diagonal of an hecton; the '' root-5 rectangle'' is produced by projecting the diagonal of a double square (or by projecting 180° both diagonals that go from the middle point of a side of a square to the opposite vertexes). Two of the most complicated of these figures are; the ''penton'', with proportions 1: is related to the section of the

golden pyramid The Golden Pyramid Award is the highest prize for best film in the international competition of the Cairo International Film Festival, hosted annually in Cairo, Egypt Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a List of ...

, the ''bipentons longer side is equal to the shorter multiplied by two thirds of the square root of three, longer side of the ''biauron'' is - 1 or 2τ 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 ''hemidiagon'' (1:½) longer side is half the one 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.

Constructing an orthogon

The dimensions of orthogons relate to each other and to the Orthogon as a whole. For this reason, use of Orthogons as a template or under-structure is of interest to artists, architects and designers. Orthogons always begin with a square, any square. Once an individual Orthogon is constructed, additional related measurements are determined (small, medium, large). These measurements can then be used to guide the design (painting, architecture, pottery, furniture, calligraphy, auto, etc.). Diagrams for all twelve orthogons are available. Wersin's book has very detailed explanations for creating individual Orthogons. The measurements derived are then applied in a design. The artwork of

Giorgio Morandi Giorgio Morandi (July 20, 1890 – June 18, 1964) was an Italian painter and printmaker who specialized in still life. His paintings are noted for their tonal subtlety in depicting simple subjects, which were limited mainly to vases, bottles, b ...

exemplifies how measurements of varying sizes (derived from an Orthogon) can create visual harmony.

Orthogons and design

Use of dimensions related to an orthogon as an under-structure system (or template for a design) ensures that the various parts will relate to the design as a whole. Marcus

Vitruvius Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...

Pollio in Book Three of "

De Architectura (''On architecture'', published as ''Ten Books on Architecture'') is a treatise on architecture written by the Roman architect and military engineer Marcus Vitruvius Pollio and dedicated to his patron, the emperor Caesar Augustus, as a guide f ...

" (known currently as "The Ten Books of Architecture") explains:

"Therefore, since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme. Hence, while transmitting to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings in which merits and faults usually last forever."

Leonardo's drawing of the

Vitruvian Man The ''Vitruvian Man'' ( it, L'uomo vitruviano; ) is a drawing by the Italian Renaissance artist and scientist Leonardo da Vinci, dated to . Inspired by the writings by the ancient Roman architect Vitruvius, the drawing depicts a nude man in two s ...

is an illustration of the concept of parts relating to the work as a whole.HEMENWAY, pp. 95

References

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