PROPORTION (ARCHITECTURE)
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'Proportion' is the relation between elements and a whole.
In architecture the whole is not just a building but the set and setting of the site. The things that make a building and its site "well shaped" include the orientation of the site and the buildings on it to the features of the grounds on which it is situated. Light, shade, wind, elevation, choice of materials, all should relate to a standard and say what is it that makes it what it is, and what is it that makes it not something else.
Vitruvius thought of proportion in terms of unit fractions[1] such as those used in the Greek Orders of Architecture [2].
Scribes had been using unit fractions for their calculations at least since the time of the Egyptian Mathematical Leather RollRhind Mathematical Papyrus [3] in Egypt and the Epic of Gilgamesh [4] in Mesopotamia.
One example of symmetry might be found in the inscription grids [5] of the Egyptians which were based on parts of the body and their symmetrical relation to each other, fingers, palms, hands, feet, cubits, etc; Multiples of body proportions would be found in the arrangements of fields and in the buildings people lived in. [6]
A cubit could be divided into fingers, palms, hands and so could a foot, or a multiple of a foot. Special units related to feet as the hypotenuse of a 3/4/5 triangle with one side a foot were named remen and introduced into the proportional system very early on. Curves were also defined in a similar manner and used by architects in their design of arches and other building elements.
These proportional elements were used by the Persians, Greeks, Phoenicians and Romans, in laying out cities, stadiums, roads, processional ways, public buildings, ports, various areas for crops and grazing beasts of burden, so as to arrange the city as well as the building to be well proportioned [7], [8]
Architectural practice has often used proportional systems to generate or constrain the forms considered suitable for inclusion in a building. In almost every building tradition there is a system of mathematical relations which governs the relationships between aspects of the design. These systems of proportion are often quite simple; whole number ratios or easily constructed geometric shapes (such as the vesica piscis or the golden ratio).
Generally the goal of a proportional system is to produce a sense of coherence and harmony among the elements of a building.
Among the Cistercians, Gothic, Renaissance, Egyptian, Semitic, Babylonian, Arab, Greek and Roman traditions; the harmonic proportions, human proportions, cosmological/astronomical proportions and orientations, and various aspects of sacred geometry (the vesica piscis), pentagram, golden ratio, and small whole-number ratios) were all applied as part of the practice of architectural design.
In the design of European cathedrals the necessary engineering to keep the structures from falling down gradually began to take precedence over or at least to have an influence on esthetic proportions. Other concerns were symbolic astronomical references such as the towers of the Sun and Moon at Chartres and references to the various astrological and alchemical relationships being discovered by the natural philosophers and sages of the renaissance.
The Roman Mille passus became the Myle of medieval western Europe and , sound and silence, texture and smoothness, mass and airy lightness, as in a forest glade where the leaves move gently on the wind or a sparkle of metal catches the eye as a ripple of water on a pond.
The Classical orders(R. A. Cordingley ref. 30)
here illustrated by the Temple of Hephaestus in Athens, showing columns with Doric capitals are largely known through the writings of Vitruvius, particularly ''De Archetura'' (The Ten Books of Architecture) and studies of classical architecture by Renaissance architects and historians. Within a classical order elements from the positioning of triglyphs to the overall height and width of the building were controlled by principles of proportionality based on column diameters. Typically Ionic column bases are molded and about 1/2 the diameter of the column. They reduce in detail from the Temple of Artemia of Ephesus built c 560 BC and the Heraeum of Samos c 550 BC to elongated detail in the Temple of Athena c 535 BC, then begin to soften their lines in the Temple of Nike at Apteros c 342 BC and begin to emphaise circular rounds in the north porch of the Erechteum c 421 BC. This establishes the elements of the form which remains virtually unchanged through the Temple of Fortuna c 40 BC, the Thermae of Diocletian AD 306 and the classical Greek orders of Andreas Palladio in the 16th century.
Long before the Greeks international trade and commerce led to standardization of units and the facilitation of calculations in unit fractions throughout the civilized world. In architectural terms, the dimensions of structural elements like posts, beams, columns, arches, openings and fenestrations constructed of wood and stone were slowly standardized as regards the expected load and span so that a given dimension could support a given load without failure.
By way of contrast to the elongated Ionic order, Doric orders never became so slender as to require a base but do have entasis as the column shaft tapered upwards like a degree of the earths surface. The column shafts of the Doric order are always fluted and twenty flutes is the usual number. The column capital has an abacus square in plan and a rounded echinus which supports it. The Doric entablature has a deep plain architrave, large Triglyphs in the frieze and a series of mutles in the cornice sloping as the roof rafters of a wooden structure. The greatest change in the dentiled entablature from Ionic through Corinthian is in the addition of the frieze and scrolled modillions to the cymia in Corinthian styles.
The proportions of Entablature to Parapet remain the same at 2:2 in all styles as do the proportions of cap, die and base at 1/4:1:3/4 in the parapet. In all styles the Cornice has the proportion of 3/4 but the frieze and architrave vary from 3/4:1/2 in the Doric style to 5/8:5/8 in the Ionic and Corinthian styles. Capitals are 1/2 in all styles except Corinthian which is 3/4. The shaft width is always 5/6 at the top. Column shaft heights are Tuscan 7, Doric 8, Ionic 9 and Corinthian 10. Column bases are always 1/2. In the Pedestal, caps are always 1/4, dies are 8/6 and bases are 3/4. In the quarter of the column entasis, Tuscan styles are 9/4, Doric are 10/4, Ionic are 11/4 and Corinthian columns are 12/4.
Having established the column proportions we move on to its arcade which may be regular with a single element at a spacing of 3 3/4 D, coupled with two elements at 1 1/3 D spaced 5 D, or alternating at 3 3/4 spaced 6 1/4 D. Variations include adding a series of arches between column cap and entablature in the Renaissance style arcade , adding a Keystone in the Archvolt in the Roman style arcade, and adding more detail in the Palladian arcade. Exterior door widths W, have trim 1/5 W for exterior doors and 1/6 W for interior doors. Door heights a re 1 D less than column heights. Anciently if a door is two cubits or between 36" and 42" in width, then its trim is between a fist and a span in width.
The Greek classical orders are all proportioned rather than dimensioned or measured modules and this is because the earliest modules were not based on body parts and their spans, fingers, palms, hands, feet, remen, cubits, ells, yards, paces and fathoms which became standardized for bricks, and boards, before the time of the Greeks, but rather column diameters and the widths of arcades and fenestrations
Typically one set of column diameter modules used for casework and architectual moldings by the Egyptians, Romans and English is based on the proportions of the palm and the finger, while another less delicate module used for door and window trim, tile work, and roofing in Mesopotamia and Greece is based on the proportions of the hand and the thumb. Board modules tend to round down for planing and finishing while masonry tends to round down for mortar. Fabric, carpet and rugs tend to be manufactured
in feet, yards and ells
In Palladian or Greek revival Tuscan architecture as in Jeffersonian Classic, modern modular dimensional systems based on the Golden Section and other pleasing proportional and dimensional relationships begin to influence the design as with the modules of the volute. One interface between proportion and dimension is the Egyptian inscription grid. Grid coordinates can be used for things like unit rise and run
The architectural foot as a reference to the human body was incorporated in architectural standards in Mesopotamia, Egypt, Greece, Rome and Europe. Common multiples of a foot in buildings tend to be decimal or octal and this affects the modulars used in Building materials. Elsewhere it is a multiple of palms, hands and fingers which are the primary referents. Feet were usually divided into palms or hands, multiples of which were also remen and cubits.
The first known foot referenced as a standard was from Sumer, where a rod at the feet of a statue of Gudea of Lagash from around 2575 BC is divided into a foot and other units. Egyptian foot units have the same length as Mesopotamian foot units, but are divided into palms rather than hands converting the proportional divisions from Sexagesimal to septenary units. In both cases feet are further subdivided into digits.
In Greeks, there are several different foot standards generally refered to in the literature as short, median and long which give rise to different architectural styles known as Ionian, and Doric in diccussions of the Greek |orders of Architecture. The Roman Romans foot or pes is divided into digitus, uncia and palmus which are incorporated into the Corinthian style.
Some of the earliest records of the use of the foot come from the Persian Gulf bordered by India(Meluhha), Pakistan, Beluchistan, Oman (Makkan), Iran, Iraq, Kuwait, Bahrain (Dilmun), the UAE and Saudia Arabia where in persian architecture it is a sub division of the Great circle of the earth into 360 degrees. In Egypt, one degree was 10 Itrw or River journey's. In Greece a degree was 60 Mia chillioi or thousands and comprised 600 stadia, with one stadion divided into 600 pous or feet. In Rome a degree was 75 Mille Passus or 1000 passus. Thus the degree division was 111 km and the stadion 185 m. One Nautical mile was 10 stadia or 6000 feet. The incorporation of proportions which relate the building to the earth it stands on are called sacred geometry.
Vitruvius described as the principal source of proportion among the orders the proportion of the human figure. .
Though he was certainly aware of the work of Pythagoras, it does not appear that he took the harmonic divisions of the octave as being relevant to the disposition of form, preferring simpler whole-number ratios to describe proportions.
However, beyond the writings of Vitruvius, it seems likely that the ancient Greeks and Romans would occasionally use proportions derived from the golden ratio (most famously, in the Parthenon of Athens), and the Pythagorean divisions of the octave. These are found in the Rhynd papyrus 16. Care should be taken in reading too much into this, however, while simple geometric transformations can quite readily produce these proportions, the Egyptian were quite good at expressing arithmetic and geometric series as unit fractions. While, it is possible that the originators of the design may not have been aware of the particular proportions they were generating as they worked, its more likely that the methods of construction using diagonals and curves would have taught them something.
The Biblical proportions of Solomons temple caught the attention of both architects and scientists, who from a very early time began incorporating them into the architecture of cathedrals and other sacred geometry.
Regarding the Pythagorean divisions of the octave mentioned above, these are a set of whole number ratios (based on core ratios of 1:2 (octave), 2:3 (fifth) and 3:4 (fourth)) which form the Pythagorean tuning. These proportions were thought to have a recognisable harmonic significance, regardless of whether they were perceived visually or auditorially, reflecting the Pythagorean idea that all things were numbers.
The Renaissance tried to extract and codify the system of proportions in the orders as used by the ancients, believing that with analysis a mathematically absolute ideal of beauty would emerge. Brunelleschi in particular studied interactions of perspective with the perception of proportion (as understood by the ancients). This focus on the perception of harmony was somewhat of a break from the Pythagorean ideal of numbers controlling all things.
Leonardo da Vinci's Vitruvian Man is an example of a Renaissance codification of the Vitruvian view of the proportions of man. Divina proportione took the idea of the golden ratio and introduced it to the Renaissance architects. Both Palladio and Leone Battista Alberti produced proportional systems for classically-based architecture.
Alberti's system was based on the Pythagorean divisions of the octave. It grouped the small whole-number proportions into 3 groups, short (1:1, 2:3, 3:4), medium (1:2, 4:9, 9:16) and long (1:3, 3:8, 1:4).
Palladio's system was based on similar proportions with the addition of the square root of 2 into the mix. 1:1, 1:1.414..., 3:4, 2:3, 3:5. (Harmony and Proportion, J. Boyd-Brent).
The work of de Chambrey, Desgodet and Perrault (Tzonis and Lefaivre, 1986, p. 39) eventually demonstrated that classical buildings had reference to standards of proportion that came directly from the original sense of the word geometry, the measure of the earth and its division into degrees, miles, stadia, cords, rods, paces, yards, feet, hands, palms and fingers
Based on apparently arbitrary proportions of an "ideal man" (possibly Le Corbusier himself) combined with the golden ratio and Vitruvian Man, Le Modulor was never popularly adopted among architects, but the system's graphic of the stylised man with one upraised arm is widely recognised and powerful. The modulor is not well suited to introduce proportion and pattern into architecture (Langhein, 2005), to improve its form qualities (gestalt pragnance) and introduce shape grammar in design in building.
The plastic number is of interest primarily for its method of genesis. Its creator, Hans van der Laan, performed experiments on human subjects to attempt to discover the limits of human beings ability to perceive relationships between objects. From these discovered limits he extrapolated a system of proportions (the particular set he chose are quite close to the Pythagorean divisions of the octave). The range of scales over which the plastic number is considered functional is limited, so it is possible to construct a set of all proportional forms within it. The plastic number has not been widely adopted by practicing architects.
★ 30. R. A. Cordingley
★ 23. Vitruvius
★ 24. Claudias Ptolemy
★ 25. Herodotus War with Judah, Sennacherib, siege of 701 BC
★ 26. Michael Grant
★ 27. Lucas N. H. Bunt, Phillip S.Jones, Jack D. Bedient
★ 28. H Arthur Klein
★ 29 Francis H. Moffitt
★ 3. William H McNeil and Jean W Sedlar,
★ 4. Andrew George,
★ 5. James B. Pritchard,
★ 8. Michael Roaf
★ 10. Gerard Herm
★ 13. Gardiner
★ 14. Antonio Loprieno
★ 15. Michael Rice
★ 16. Gillings
★ 17. Somers Clarke and R. Englebach
★ 18. Marie-Loise Thomsen,
★ 19. Silvia Luraghi
★ 20. J. P. Mallory
★ 21. Anne H. Groton
★ 22. Hines
1. (Gillings ref. 16)
2. (R. A. Cordingley ref. 30)
3. (Michael Grant ref 26)
4. (Andrew George Ref 4)
5. (Gillings ref 16)
6. (Somers Clarke and R. Englebach ref.17)
7. (Herodotus ref. 24)
8. (Claudius Ptolomy ref. 25)
★ Tzonis, A. and Lefaivre L., ''Classical Architecture: The Poetics of Order'' (1986), MIT Press. ISBN 0-262-20059-7
★ Dictionary of the History of Ideas, Pythagorean Harmony
★ Padovan, R., ''Proportion: Science, Philosophy, Architecture'' (1999), Routledge. ISBN 0-419-22780-6
★ Langhein, J., ''Proportion and Traditional Architecture'' (2005), INTBAU Essay (London, The Prince's Foundation /INTBAU), [1]
A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral.
A tiling with squares whose sides are successive Fibonacci numbers in length
Architectural proportions
In architecture the whole is not just a building but the set and setting of the site. The things that make a building and its site "well shaped" include the orientation of the site and the buildings on it to the features of the grounds on which it is situated. Light, shade, wind, elevation, choice of materials, all should relate to a standard and say what is it that makes it what it is, and what is it that makes it not something else.
Vitruvius thought of proportion in terms of unit fractions[1] such as those used in the Greek Orders of Architecture [2].
Orders of Architecture
One example of symmetry might be found in the inscription grids [5] of the Egyptians which were based on parts of the body and their symmetrical relation to each other, fingers, palms, hands, feet, cubits, etc; Multiples of body proportions would be found in the arrangements of fields and in the buildings people lived in. [6]
A cubit could be divided into fingers, palms, hands and so could a foot, or a multiple of a foot. Special units related to feet as the hypotenuse of a 3/4/5 triangle with one side a foot were named remen and introduced into the proportional system very early on. Curves were also defined in a similar manner and used by architects in their design of arches and other building elements.
These proportional elements were used by the Persians, Greeks, Phoenicians and Romans, in laying out cities, stadiums, roads, processional ways, public buildings, ports, various areas for crops and grazing beasts of burden, so as to arrange the city as well as the building to be well proportioned [7], [8]
Architectural practice has often used proportional systems to generate or constrain the forms considered suitable for inclusion in a building. In almost every building tradition there is a system of mathematical relations which governs the relationships between aspects of the design. These systems of proportion are often quite simple; whole number ratios or easily constructed geometric shapes (such as the vesica piscis or the golden ratio).
Generally the goal of a proportional system is to produce a sense of coherence and harmony among the elements of a building.
Sacred proportions
Among the Cistercians, Gothic, Renaissance, Egyptian, Semitic, Babylonian, Arab, Greek and Roman traditions; the harmonic proportions, human proportions, cosmological/astronomical proportions and orientations, and various aspects of sacred geometry (the vesica piscis), pentagram, golden ratio, and small whole-number ratios) were all applied as part of the practice of architectural design.
West end of Chartres
In the design of European cathedrals the necessary engineering to keep the structures from falling down gradually began to take precedence over or at least to have an influence on esthetic proportions. Other concerns were symbolic astronomical references such as the towers of the Sun and Moon at Chartres and references to the various astrological and alchemical relationships being discovered by the natural philosophers and sages of the renaissance.
The Roman Mille passus became the Myle of medieval western Europe and , sound and silence, texture and smoothness, mass and airy lightness, as in a forest glade where the leaves move gently on the wind or a sparkle of metal catches the eye as a ripple of water on a pond.
Classical orders
The Classical orders(R. A. Cordingley ref. 30)
Long before the Greeks international trade and commerce led to standardization of units and the facilitation of calculations in unit fractions throughout the civilized world. In architectural terms, the dimensions of structural elements like posts, beams, columns, arches, openings and fenestrations constructed of wood and stone were slowly standardized as regards the expected load and span so that a given dimension could support a given load without failure.
By way of contrast to the elongated Ionic order, Doric orders never became so slender as to require a base but do have entasis as the column shaft tapered upwards like a degree of the earths surface. The column shafts of the Doric order are always fluted and twenty flutes is the usual number. The column capital has an abacus square in plan and a rounded echinus which supports it. The Doric entablature has a deep plain architrave, large Triglyphs in the frieze and a series of mutles in the cornice sloping as the roof rafters of a wooden structure. The greatest change in the dentiled entablature from Ionic through Corinthian is in the addition of the frieze and scrolled modillions to the cymia in Corinthian styles.
The proportions of Entablature to Parapet remain the same at 2:2 in all styles as do the proportions of cap, die and base at 1/4:1:3/4 in the parapet. In all styles the Cornice has the proportion of 3/4 but the frieze and architrave vary from 3/4:1/2 in the Doric style to 5/8:5/8 in the Ionic and Corinthian styles. Capitals are 1/2 in all styles except Corinthian which is 3/4. The shaft width is always 5/6 at the top. Column shaft heights are Tuscan 7, Doric 8, Ionic 9 and Corinthian 10. Column bases are always 1/2. In the Pedestal, caps are always 1/4, dies are 8/6 and bases are 3/4. In the quarter of the column entasis, Tuscan styles are 9/4, Doric are 10/4, Ionic are 11/4 and Corinthian columns are 12/4.
Having established the column proportions we move on to its arcade which may be regular with a single element at a spacing of 3 3/4 D, coupled with two elements at 1 1/3 D spaced 5 D, or alternating at 3 3/4 spaced 6 1/4 D. Variations include adding a series of arches between column cap and entablature in the Renaissance style arcade , adding a Keystone in the Archvolt in the Roman style arcade, and adding more detail in the Palladian arcade. Exterior door widths W, have trim 1/5 W for exterior doors and 1/6 W for interior doors. Door heights a re 1 D less than column heights. Anciently if a door is two cubits or between 36" and 42" in width, then its trim is between a fist and a span in width.
Proportioned vs Dimensioned modules
The Greek classical orders are all proportioned rather than dimensioned or measured modules and this is because the earliest modules were not based on body parts and their spans, fingers, palms, hands, feet, remen, cubits, ells, yards, paces and fathoms which became standardized for bricks, and boards, before the time of the Greeks, but rather column diameters and the widths of arcades and fenestrations
Typically one set of column diameter modules used for casework and architectual moldings by the Egyptians, Romans and English is based on the proportions of the palm and the finger, while another less delicate module used for door and window trim, tile work, and roofing in Mesopotamia and Greece is based on the proportions of the hand and the thumb. Board modules tend to round down for planing and finishing while masonry tends to round down for mortar. Fabric, carpet and rugs tend to be manufactured
in feet, yards and ells
In Palladian or Greek revival Tuscan architecture as in Jeffersonian Classic, modern modular dimensional systems based on the Golden Section and other pleasing proportional and dimensional relationships begin to influence the design as with the modules of the volute. One interface between proportion and dimension is the Egyptian inscription grid. Grid coordinates can be used for things like unit rise and run
The architectural foot as a reference to the human body was incorporated in architectural standards in Mesopotamia, Egypt, Greece, Rome and Europe. Common multiples of a foot in buildings tend to be decimal or octal and this affects the modulars used in Building materials. Elsewhere it is a multiple of palms, hands and fingers which are the primary referents. Feet were usually divided into palms or hands, multiples of which were also remen and cubits.
The first known foot referenced as a standard was from Sumer, where a rod at the feet of a statue of Gudea of Lagash from around 2575 BC is divided into a foot and other units. Egyptian foot units have the same length as Mesopotamian foot units, but are divided into palms rather than hands converting the proportional divisions from Sexagesimal to septenary units. In both cases feet are further subdivided into digits.
In Greeks, there are several different foot standards generally refered to in the literature as short, median and long which give rise to different architectural styles known as Ionian, and Doric in diccussions of the Greek |orders of Architecture. The Roman Romans foot or pes is divided into digitus, uncia and palmus which are incorporated into the Corinthian style.
Some of the earliest records of the use of the foot come from the Persian Gulf bordered by India(Meluhha), Pakistan, Beluchistan, Oman (Makkan), Iran, Iraq, Kuwait, Bahrain (Dilmun), the UAE and Saudia Arabia where in persian architecture it is a sub division of the Great circle of the earth into 360 degrees. In Egypt, one degree was 10 Itrw or River journey's. In Greece a degree was 60 Mia chillioi or thousands and comprised 600 stadia, with one stadion divided into 600 pous or feet. In Rome a degree was 75 Mille Passus or 1000 passus. Thus the degree division was 111 km and the stadion 185 m. One Nautical mile was 10 stadia or 6000 feet. The incorporation of proportions which relate the building to the earth it stands on are called sacred geometry.
Vitruvian proportion
Vitruvius described as the principal source of proportion among the orders the proportion of the human figure. .
Though he was certainly aware of the work of Pythagoras, it does not appear that he took the harmonic divisions of the octave as being relevant to the disposition of form, preferring simpler whole-number ratios to describe proportions.
However, beyond the writings of Vitruvius, it seems likely that the ancient Greeks and Romans would occasionally use proportions derived from the golden ratio (most famously, in the Parthenon of Athens), and the Pythagorean divisions of the octave. These are found in the Rhynd papyrus 16. Care should be taken in reading too much into this, however, while simple geometric transformations can quite readily produce these proportions, the Egyptian were quite good at expressing arithmetic and geometric series as unit fractions. While, it is possible that the originators of the design may not have been aware of the particular proportions they were generating as they worked, its more likely that the methods of construction using diagonals and curves would have taught them something.
The Biblical proportions of Solomons temple caught the attention of both architects and scientists, who from a very early time began incorporating them into the architecture of cathedrals and other sacred geometry.
Regarding the Pythagorean divisions of the octave mentioned above, these are a set of whole number ratios (based on core ratios of 1:2 (octave), 2:3 (fifth) and 3:4 (fourth)) which form the Pythagorean tuning. These proportions were thought to have a recognisable harmonic significance, regardless of whether they were perceived visually or auditorially, reflecting the Pythagorean idea that all things were numbers.
Renaissance orders
The Renaissance tried to extract and codify the system of proportions in the orders as used by the ancients, believing that with analysis a mathematically absolute ideal of beauty would emerge. Brunelleschi in particular studied interactions of perspective with the perception of proportion (as understood by the ancients). This focus on the perception of harmony was somewhat of a break from the Pythagorean ideal of numbers controlling all things.
Leonardo da Vinci's Vitruvian Man is an example of a Renaissance codification of the Vitruvian view of the proportions of man. Divina proportione took the idea of the golden ratio and introduced it to the Renaissance architects. Both Palladio and Leone Battista Alberti produced proportional systems for classically-based architecture.
Alberti's system was based on the Pythagorean divisions of the octave. It grouped the small whole-number proportions into 3 groups, short (1:1, 2:3, 3:4), medium (1:2, 4:9, 9:16) and long (1:3, 3:8, 1:4).
Palladio's system was based on similar proportions with the addition of the square root of 2 into the mix. 1:1, 1:1.414..., 3:4, 2:3, 3:5. (Harmony and Proportion, J. Boyd-Brent).
The work of de Chambrey, Desgodet and Perrault (Tzonis and Lefaivre, 1986, p. 39) eventually demonstrated that classical buildings had reference to standards of proportion that came directly from the original sense of the word geometry, the measure of the earth and its division into degrees, miles, stadia, cords, rods, paces, yards, feet, hands, palms and fingers
Le modulor
Based on apparently arbitrary proportions of an "ideal man" (possibly Le Corbusier himself) combined with the golden ratio and Vitruvian Man, Le Modulor was never popularly adopted among architects, but the system's graphic of the stylised man with one upraised arm is widely recognised and powerful. The modulor is not well suited to introduce proportion and pattern into architecture (Langhein, 2005), to improve its form qualities (gestalt pragnance) and introduce shape grammar in design in building.
The plastic number
The plastic number is of interest primarily for its method of genesis. Its creator, Hans van der Laan, performed experiments on human subjects to attempt to discover the limits of human beings ability to perceive relationships between objects. From these discovered limits he extrapolated a system of proportions (the particular set he chose are quite close to the Pythagorean divisions of the octave). The range of scales over which the plastic number is considered functional is limited, so it is possible to construct a set of all proportional forms within it. The plastic number has not been widely adopted by practicing architects.
Architectural References
★ 30. R. A. Cordingley
Classical References
★ 23. Vitruvius
★ 24. Claudias Ptolemy
★ 25. Herodotus War with Judah, Sennacherib, siege of 701 BC
Historical References
★ 26. Michael Grant
Mathematical References
★ 27. Lucas N. H. Bunt, Phillip S.Jones, Jack D. Bedient
Mensurational References
★ 28. H Arthur Klein
★ 29 Francis H. Moffitt
Near Eastern References
★ 3. William H McNeil and Jean W Sedlar,
★ 4. Andrew George,
★ 5. James B. Pritchard,
★ 8. Michael Roaf
★ 10. Gerard Herm
Egyptological References
★ 13. Gardiner
★ 14. Antonio Loprieno
★ 15. Michael Rice
★ 16. Gillings
★ 17. Somers Clarke and R. Englebach
Linguistic References
★ 18. Marie-Loise Thomsen,
★ 19. Silvia Luraghi
★ 20. J. P. Mallory
★ 21. Anne H. Groton
★ 22. Hines
See also
Footnotes
1. (Gillings ref. 16)
2. (R. A. Cordingley ref. 30)
3. (Michael Grant ref 26)
4. (Andrew George Ref 4)
5. (Gillings ref 16)
6. (Somers Clarke and R. Englebach ref.17)
7. (Herodotus ref. 24)
8. (Claudius Ptolomy ref. 25)
References
★ Tzonis, A. and Lefaivre L., ''Classical Architecture: The Poetics of Order'' (1986), MIT Press. ISBN 0-262-20059-7
★ Dictionary of the History of Ideas, Pythagorean Harmony
★ Padovan, R., ''Proportion: Science, Philosophy, Architecture'' (1999), Routledge. ISBN 0-419-22780-6
★ Langhein, J., ''Proportion and Traditional Architecture'' (2005), INTBAU Essay (London, The Prince's Foundation /INTBAU), [1]
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