Dynamic rectangle
Encyclopedia
A dynamic rectangle is a right-angled, four-sided figure (a rectangle
) with dynamic symmetry, which in this case, means that aspect ratio
(height divided by width) 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 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.
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.
, as part of his theory of dynamic symmetry, includes the root rectangles in what he calls dynamic rectangles, which have irrational
and geometric fraction
s 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
'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:
Wolfgang Von Wersin's book includes an extraordinary copy of text from the year 1558 (Renaissance
), 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 golden rectangle, 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 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.
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
exemplifies how measurements of varying sizes (derived from an Orthogon) can create visual harmony.
Pollio in Book Three of "De Architectura
" (known currently as "The Ten Books of Architecture") explains:
Leonardo's drawing of the Vitruvian Man
is an illustration of the concept of parts relating to the work as a whole.
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
) with dynamic symmetry, which in this case, means that aspect ratio
Aspect ratio
The aspect ratio of a shape is the ratio of its longer dimension to its shorter dimension. It may be applied to two characteristic dimensions of a three-dimensional shape, such as the ratio of the longest and shortest axis, or for symmetrical objects that are described by just two measurements,...
(height divided by width) is a distinguished value in dynamic symmetry, a proportioning system and natural design methodology described in Jay Hambidge
Jay Hambidge
Jay Hambidge was an American artist, born in Canada. He was a pupil at the Art Students' League in New York and of William Chase, and a thorough student of classical art. He conceived the idea that the study of arithmetic with the aid of geometrical designs was the foundation of the proportion and...
's books. These dynamic rectangles begin with a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
, 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
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
in which the ratio of the longer side to the shorter is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
, such as , , etc.
The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
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 diameter 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 ratioGolden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
(φ). The longer side is equal to one plus two times 1/φ (0.618...).
Root-φ rectangle
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
Kepler triangles.
Jay Hambidge
Jay HambidgeJay Hambidge
Jay Hambidge was an American artist, born in Canada. He was a pupil at the Art Students' League in New York and of William Chase, and a thorough student of classical art. He conceived the idea that the study of arithmetic with the aid of geometrical designs was the foundation of the proportion and...
, as part of his theory of dynamic symmetry, includes the root rectangles in what he calls dynamic rectangles, which have irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
and geometric fraction
Fraction
In common usage a fraction is any part of a unit.Fraction may also mean:*Fraction , one of more equal parts of something, eg...
s as ratios, such as the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
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
The 12 orthogons of Wersin
According to Wolfgang von WersinWolfgang von Wersin
Wolfgang von Wersin is a Czech-born designer, painter, architect and author that developed his career in Germany....
'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
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
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:
- Square (1:1 or 1:)
- Diagon (1:)
- Hecton or sixton (1:)
- Doppelquadrat (1:2 or 1:)
- Hemiolion (2:3)
- Auron (the golden rectangle, 1:φGolden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
) - Hemidiagon (1:½)
- Penton (1:)
- Trion (1:⅔)
- Quadriagon
- Biauron (1:2φ)
- Bipenton
Wolfgang Von Wersin's book includes an extraordinary copy of text from the year 1558 (Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
), 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 golden rectangle, 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
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
rectangle is produced by projecting the diagonal of a square; the sixton, hecton or root-3
Square root of 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property...
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 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
Giorgio Morandi was an Italian painter and printmaker who specialized in still life. His paintings are noted for their tonal subtlety in depicting apparently simple subjects, which were limited mainly to vases, bottles, bowls, flowers, and landscapes.-Biography:Giorgio Morandi was born in Bologna...
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 VitruviusVitruvius
Marcus Vitruvius Pollio was a Roman writer, architect and engineer, active in the 1st century BC. He is best known as the author of the multi-volume work De Architectura ....
Pollio in Book Three of "De Architectura
De architectura
' is a treatise on architecture written by the Roman architect Vitruvius and dedicated to his patron, the emperor Caesar Augustus, as a guide for building projects...
" (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
Vitruvian Man
The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci circa 1487. It is accompanied by notes based on the work of the famed architect, Vitruvius. The drawing, which is in pen and ink on paper, depicts a male figure in two superimposed positions with his arms and legs apart and...
is an illustration of the concept of parts relating to the work as a whole.