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.).[1][2][3]
Hambidge's 1920 illustration of the construction of root rectangles. The lengths of the horizontal sides of the original square and the four root rectangles derived from it, are respectively .
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.[2]
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.[4]
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).[5]
The root-3 rectangle is also called sixton,[6] and its short and longer sides are proportionally equivalent to the side and diameter of a hexagon.[7]
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.[7]
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...).[7]
Root-φ rectangle
A root-phi rectangle divides into a pair of Kepler triangles (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 congruentKepler 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.[3] 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.[4]
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 the drawing any network of parallels and perpendiculars to sides and diagonals. This produces automatically 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.[3]
Caskey's 1922 illustration of the property that a root-N rectangle divides into N reciprocal rectangles of the same proportions.
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"[9] and γονια, gonia, "angle"; "a right angled figure", which, as a consequence, is rectangular and tetragonal[10]) has been used historically by artists, architects and calligraphers to guide the placement and interaction of elements in a design.[3][11] These orthogons are:[12]
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."[13]
All 12 orthogons, when formed together, create an entire unit: a square that is developed into a double square.[14]
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:√φ is related to the section of the golden pyramid, the bipenton's longer side is equal to the shorter multiplied by two thirds of the square root of three, longer side of the biauron is √5 - 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√5) 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.[15]
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.[16]
Wersin's book has very detailed explanations for creating individual Orthogons.[17] 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.
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 Pollio in Book Three of "De Architectura" (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 is an illustration of the concept of parts relating to the work as a whole.[18]
Related Research Articles
The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangle is oriented as a "landscape".
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 and with ,
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 ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes.
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, , which is , where is approximately 1.618.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.
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.
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is where is the golden ratio, and the progression can be written: , or approximately . Squares on the edges of this triangle have areas in another geometric progression, :\varphi ^{2}} . Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles.
Wolfgang von Wersin was a Czech-born designer, painter, architect and author who developed his career in Germany.
Rabatment of the rectangle is a compositional technique used as an aid for the placement of objects or the division of space within a rectangular frame, or as an aid for the study of art.
In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from the diagonals and symmetry lines of a regular pentagon, omitting one of the symmetry lines. In turn, it can be used to construct higher-dimensional convex polytopes that cannot be given rational coordinates, having the fewest vertices of any known example. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations.
In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a parallelohedron.
The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as:
The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in surd form as:
References
↑ SKINNER, Stephen, Sacred Geometry Deciphering the Code, New York City: Sterling Publishing Company, 2006, pp. 53
↑ WERSIN, Wolfgang Von, Das Buch vom Rechteck Gesetz und Gestik des Raumlichen die Othogone-scheibe. Die Orthogone-scheibe (The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described. The Orthogons Described), Ravensburg: Otto Maier Verlag Publishers, 1956
Hemenway, Priya; Divine Proportion, Phi in Art, Nature and Science; 2005, Sterling Publishing Co., Inc, NY, NY.
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