DR. JOACHIM LANGHEIN

On the Path to a Proportion Science in Architecture and the

Golden Section

Proposal for an Integrated Research Strategy on Building Culture and Conditions of Aesthetic Perception

[Heidelberg, 31.08.2010]

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Symmetry: Art and Science Gmuend, Austria, 2010

ON THE PATH TO A PROPORTION SCIENCE IN ARCHITECTURE & THE GOLDEN SECTION

JOACHIM LANGHEIN

Name: Dr. Joachim Langhein, Economist, Ecologist, Geographer, Japanologist, private Researcher on Proportion in Architecture since 1980 & global human ecology (since 1969) (b. Potsdam, Germany, 1942). Address: Retired as economist, (during 1970s) assistant professor at Heidelberg University, specialist in R&D and I&D; preparation of policy decision making, system theoretician, specialist food industry and development, specialist in information economics, renewable energy management, feasibility studies in food industry, Sitzbuchweg 83, 69118 Heidelberg, Germany (& Sonnenalm 157, 8983 Bad Mitterndorf, Austria) E-mail ; DrLanghein@proportions.de Home-page: www.proportions.de Fields of interest: Objective &complementarily subjective Aesthetics & Proportions in Architecture within the frame of Anthropoecology; Architecture & Urbanism in general, Systems Theory applied to all fields of Ecology (human ecology, long-term resource ecology & economy, visual phenological ecology, Resource eco-logy and renewable energies), multistability theory in Global Sustainability; Geography and subfields, includ-ing Geology, Anthropogeography, House Science & Geography (geography of traditional architecture); theory and general application of physical and ecological Entropy Theory; Economics, global economy and tropical agriculture. social sciences (first main study); Linguistics, study of all major & some minor languages of Europe, and Far & partly Near East; Japanology, Sinology and other East Asian languages; Cultural history of vernacular architecture in Europe, the Americas, Japan and categorization by proportion systems; History of Art & Architecture; History of Geometry, commented Bibliographic Database on proportion and treatises on architecture, with >56.000 records (work started in 1980, www.proportions.de); Humanistic, Gestalt & Phenomenological Psychology; Physiology of visual perception; Critical History & Geography of Religions. Awards: “Diplom-Volkswirt” (Economist) 1969, Diplom-Geographer at Heidelberg Univ. 1971, Dr. of nat. Sc. in Ecology, Heidelberg Univ. (Germany) 1986, INTBAU essay 10, 2005 (The Prince’s Foundation. UK)

Publications: see literature list, >120 publications on different fields. Abstract: Despite many publications, a synoptical and instrumental knowledge (or theory) of

architectural aesthetics, proportions and other objective and systemic parameters of aesthetics is still

in its infancy. The knowledge on the Golden Section is best developed, while other aspects and their

integration into an organized theory and system of categories are still widely missing. This paper

introduces results of independent research over 30 years. The study focusses on proportions of

domestic architecture and a multidisciplinary approach to proportion & perception.

1. INTRODUCTION TO RESEARCH PROJECT

(1) To know more on proportion means to know more on important objective reasons of beauty (good gestalt, gestalt pragnanz). Proportion is for beauty a necessary, though not sufficient precondition; since a number of mostly objectively assesseable “gestalt parameters” must be respected in addition, as was commonplace in architecture of the past, i.e, in every kind of traditional architecture. If these “gestalt parameters” – like those

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described by RASMUSSEN (1962ff.) - become violated, as occurred more and more in post-1830 construction, architecture is (subconsciously) perceived as misfigured by present and will be still more by future generations. “Gestalt parameters” are visual algorithmic patterns (like proportion, symmetry, material texture, colour & fractal har-mony, light & shade variable patterns, closedness, rhythm & motion, façade/window arrangement & shape, ornamentation, style & shape grammar unity, symbolic (“figural unities by likeness & nearness”, grouping and varied repetition, invisible features, windows, etc.) The zeitgeist can dazzle man’s perception for sustained periods in regard to judgements on architectural beauty or ugliness, but not forever (as deduceable from the Phenomenology of HUSSERL & MERLEAU-PONTY). Today, we have clear evidence, that timeless and objective criteria of beauty do exist. Beauty in architecture complies with visual patterns, wit, and workmanship, tolerates some contrasts, tensions and vagueness, but less obtrusiveness. WOTTON (1651, 1670) translated the term “venustas” of VITRUVIUS with “delight”. Beauty should be delightful, friendly in all life conditions and settings, relating to sense & senses, inspiring both the cognitive and emotional potentials in man. (2) The relationship between the terms of proportion and the Golden Section is not yet well defined, despite an astounding amount of literature and websites on the last. The Golden Section is and will ever remain a subcategory of the term of proportion, having a number of special properties. It evokes perceptual impressions of specific harmony and grace. But the term proportion relates to a much broader, general category, a more comprehensive term that has crucial significance for gestalt pragnanz (= beauty) in architecture. Like the term “rose” is not identical with the term flower, while all roses are flowers with specific qualities of beauty, nobility and symbolic meanings, only. Proportion in general is a multilevel concept. Proportions have properties of systems or patterns, being connected with the extensional-visual gestalt qualities of “objects”, thus supporting human perception (see 2.1). Stated in terms of a metatheory, an important formal pre-requisite of aesthetics lies in the balance of figural unity (order) & diversity (complexity). Proportion intermediates decisively between these contrasts, thus consti-tuting clarity and orientation through aesthetic patterns. It sets up a kind of “skeleton of sign order”. Without sign order, man’s perception confronts strong obstacles in its perceptual organization since the necessary information reduction of sign diversity to sign order, - a survival relevant capacity of perception – cannot take place smoothly. Proportion and other shape-generative parameters (= gestalt parameters) contribute to that balance, facilitating reduction and subsequent enrichment of visual information. Without such facilitation, no object will be perceived as aesthetical. Generally, beauty is no luxury, since it sustains man’s “fully functioning” and promotes productive thinking and acting (SCHILLER 1802, WERTHEIMER, 1920, 1945, 1971, MASLOW, 1969, 1970). (3) As other (geometrical) proportion systems, the Golden Section fulfils such functions. In architecture, it is both embodied in “macro-ratios” of approx. 5:8 or 8:5 among the parts as in grid meshes. The aesthetic effects of the “Golden Mean” were recently been objectively proven by neuroaesthetic MRT-based experiments (C. Di DIO et al., 2007). Aesthetical delight arouses from “macro-ratios” in bodies, buildings or ensembles. In contrast to other proportion systems, the Golden Section consists of increasing (or decreasing) ratios and may evoke impressions of dynamism, growth, perhaps joy of development. In architecture, Le CORBUSIER’s “Modulor” sustained the problematic belief that the Golden Section is a kind of exclusive, all-purpose tool or panacea for proportioning (or beautifying) of all design in architecture. This belief is still shared by a major part of present-day’s architectural communities. In regard to scientific progress, the equation of proportion = Golden Section often resulted in obstacles for scientific

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advancement of efforts to establish a consistent body of organized knowledge of propor-tion within efforts to establish concepts of architectural aesthetics. By the way, objectivity never excludes subjectivity, since both are complementary. (There exists a larger phenomenogical literature considering building culture and landscape beauty.) (4) All in all, proportion science as a constituting part of aesthetics in architecture is still in its infancy, despite the efforts of the last 200 years and perhaps over half a million of publications. Since 1830, the decay of beauty (“good gestalt”, “gestalt pragnanz”) of visual environments is on a free fall, due to this research gap and lack of practical knowledge. A dramatic deterioration of visual aesthetics overgrows the surface of the earth, eliminating the ubiquitous harmony of man-made environments of the past. Ugliness is on the march, before the eyes of all. Beauty remains in protected oases, only. In architecture, proportions can be realized either by “(grid) meshes of regulating lines” or as by ratios of whole building parts (“macro-ratios”) in relation to others and the whole – horizontally or vertically. The Golden Section offers additional possibilities. It appeared in architecture of ancient Egypt as the Golden Triangle (= isocele triangle within M:m rectangle; see 4.5). The fraction sequences 3:5, 5:8, 8:13 etc. are the first approximative steps of the Fibonacci Sequence. The ratios m:M or M:m determined by geometry were in many style periods used as thumb rules for arranging the macro-ratios in whole configurations of architecture (see my essay in www.intbau.org/essay10.htm, Fig. 18). GHYKA, MOESSEL and others show many examples in their illustrations. (5) “Patterns of regulating lines organize whole elevations” (HALE 1994, 52). Several methods for applying proportion in design exist. My preferred approach (based on my “Proportion Codes”, in the net) results in most cases in clear and appreciated representa-tions, like those shown in the figures. It consists of grid meshes of regulating lines, while using angles derived from the three most elementary polygons (identified in “Proportion Codes”, see net). This can be seen in all figures of this paper. The concept of grid mesh proportion analyses successfully results in clear analyses in c. 98% of cases. The more speculative identification of macro-rational proportions can start here. (6) In European traditional architecture, two types for analyzing proportions can be distinguished. [1] In Mediterranean Traditional Architecture (= MTA, see fig. 2 & 4; DOYON/HUBRECHT 1942, 1996, 37-49; MASSOT, 1979, 1996/2003, 2004ff.), the “starting positions” for mesh-based analyses lie in the vertical axes of windows /openings and/or columns (bays). (The MTA proportions may be called “Proportioning dominated by Window Axes” [PDWA].) Dating back to Antiquity and Middle Ages, PDWA patterns became the most important model for pan-European domestic traditional architecture (like TUA, TSHA, THA), mainly after 15c, in religious architecture much earlier. PDWA patterns “migrated” primarily in Modern Times to Central, North and East Europe, including European colonies, and later beyond. In many historic urban centres of Europe, both patterns still coexist, even e.g. in France and Spain. [2] In North European traditional architecture (better: Central, East & North European TA = NTA, see fig. 3 & 5), starting positions for mesh proportion analyses lie in diago-nals that are primarily determined by (or parallel to) roof pitches (HALE, 48); the NTA proportions may be called “Proportioning dominated by Roof Slope” [PDRS]). In NTA, timber is often the building material most used. The renewable resource wood offers a wider range of static options in building, contrary to the use of stone (rocks, bricks). Timber-framed and block architecture are typical PDRS buildings (see fig. 3, 4, 5). (7) The grid mesh systems of regulating lines and circles allow relatively simple design methods to determe basic features, structures and the arrangements of 2D surfaces

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(façades) and 3D solids (building bodies), as fig. 2-5 (and all fig. in ”essay10”) demon-strate. This is not the only possible method of proportioning. The final and sophisticated proportioning of the whole building bodies required design with macro-ratios, where the Golden Section could successfully be applied, together with grid mesh patterns. The Golden Section is suited to “bring in” or superpose macro-ratios to the whole design (e.g., as in fig. 18 in “essay10”). In the first half of 20th-century, HAMBIDGE, GHYKA et al., demonstrated beyond doubt the beauty-generating potential inhered in Golden Mean proportions /ratios. Beyond architecture, the Golden Section manifests itself often in nature and art, and has peculiar properties in mathematics and geometry. (8) The angles of the grid meshes of regulating lines are derived from three most simple regular polygons as generators of harmonious form. the equilateral triangle, square, pentagon, including their “duplications” hexagon, octogon and decagon. These polygons are at the same time basic “generative figures” of regular linear & circular geometry, most perfect symmetric figures (as they comply with all 17 planar symmetry groups), perfect topological nets and knots, elements of tesselations in the case of polygons with 3, 5, 6 and 8 corners. Also the Golden Triangle and “musical proportions” constitute tesselations. Thus the poygons are perfect gestalt generators, inhering several layers of the most elementary mathematical regularities. (9) The geometry of the regular “five-corner-polygon” (Quinture) constitutes the Golden Section; in regard to angles, builders preferred the Golden Triangle since earliest periods in all categories of architecture. The geometry of the square (Quadrature) is still everywhere present in the TVA & TUA of styles and areas. It suits somehow with balanced impressions (but can problems in statics, as in Romanesque style). The geometry of the equilateral triangle (Triangulature) imparts effects of high-rising, steep building, and was never so dominating as in Gothic style. In the macro-ratios of whole architectural bodies (of THA), two or even three proportion systems are often reprensented (in Gothic cathedrals, Baroque/Georgian THA). The Golden Section, called by PACIOLI “divina proportio”, in Latin sectio aurea, in French Nombre d’Or, denoted by the Greek letter phi (φ), requires a redefinition within an integrated framework knowledge. Some harmonious effects on perception will ever remain enigmatic. The Golden Mean is described as the division of a line into two seg-ments so that the ratio of the whole line to the larger segment equals the ratio of larger segment to the smaller one, as a recursive algorithm, being repetitive up and down. (10) Geometrical proportions were used since prehistoric and historic times and their application is both relatively simple and easily learnable for using as tools in design, work preparation & performance, and firmness (statics). Prehistoric TVA has often ground plans complying with simple polygon geometry (e.g. F. SCHUBERT, 1994, 2002). At times of a sophisticaed zeitgeist in building art, “doubled proportioning” may have been used in traditional high architecture [THA] or at times in traditional urban architecture [TUA, TSHA]), particularly in Gothic and Baroque style. Evidence of such intentions is manifest in fig. 24 of my INTBAU “essay10” (architectural design of J. LOIS, 1662). (11) Beauty of art, architecture and artifacts is rooted deep in prehistory and history of man, perhaps of all pre-industrial civilizations. Nearly always and everywhere, we en-counter balanced diversity in order (unity), while today degeneration of diversity to non-fractal chaos, and of order to monotony became the rule.- One pecularity of architecture as art is its orthogonality; in contrast to others arts, proportions in architecture comply predominantly with elementary regular geometry, particularly with regular polygons as shape generators and work controllers. (The angles of the three classes Triangulature,

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Quadrature and Quinture are indicated in my “Proportion Codes”). These figures could exactly be drawn by geometrical constructions with measuring cords only; also with compass-only were possible. Thus intuition could be professionally combined with rule in a playful manner, since past builders “saw different” (HALE 1994). In history of Western art and architecture, the year 1830 was, according to HALE, a historic turning point: After 1830, pattern balance went step-by-step out of European and the American design. HALE (1994, 16) called this overall balance in past design a “pattern dominated design”, supported by the “old way of seeing”, and today’s design as pattern-devoid. (12) In the last decades, valuable progress in mathematics contributed to increase the number of tools that can establish objective evaluation and operation in architectural aesthetics. Such tools are the mathematical symmetry, fractal mathematics & geometry (e.g. re harmonies of surfaces, colours, light & shade patterns etc.), topology, tesselation, rhythm research, etc. The aesthetic effect of the Golden Section has been objectively supported by means of recent neuroaesthetic experiments and approaches (C. Di DIO et al., 2007). Analogously, the progress of colour research improves the possibilities of

objective gestalt science the provability of most gestalt parameters. (13) Apparently, the geometry of the three basic proportion systems (Triangulature, Quadrature, Quinture) evokes different states of mood in man’s aesthetic perception and mind. Quinture was discussed above: it may evoke impressions of dynamics, growth, self-consciousness, etc. The Quadrature seems to evoke impressions of sincerity, calm-ness, balancedness, and maturity, while the Triangulature may evoke a mood of contem-plation, striving upwards, of search of transcendence, of “overcoming the world”. (14) In regard to improved possibilities of objective assessment of beauty, it seems to be a lucky coincidence that mathematics can simply be defined as “the science of patterns” (K. DEVLIN, 1994ff.) since the 1990s. One promoting factor for the decline and loss of patterns after 1830 was the accentuation of “use – including the creation of effect -. Pat-tern is an end in itself. The creation of effect … is a motive, to influence the observer” (HALE). Man lost insight how his own perception works. The motto “form follows function” has no relation to the principles of visual perception, with seeing.

2. ARCHITECTURAL AESTHETICS AND PROPORTION

(1) A basic formal key function of proportion in architectural aesthetics concerns the clarity and algorithmic unity of whole bodies of buildings and ensembles, despite complexity. Proportion relates to extentional balance between visual unity/order & diversity/ complexity in objects and their appearance in optical perception. (2) Without proportion, architecture lacks clarity and readability. The clarity promotes information reduction and subsequent information enrichment. This is an important condition of the formal functions of perception. Proportion helps man’s perception to reduce sign diversity to sign order probably best. As stated, traditional art and architecture are charcterized by algorithm-like patterns. Our perception system is able to reduce 2 mln bits to 200 bits within fractions of a second, if visual objects are supportive. This reduction brings the information amount beyond the processing capacity of man’s short-term memory. If information reduction can quickly proceed stepwise (as fig. 1 in “essay10” (www.intbau.org/essay10.htm) demonstrates) patterns like proportions, the colour harmony, fractal orders in materiality and patina can be experienced as harmonious fascilitating delightful experiences, because perception reduces complexity to the order/unity of super-signs. After super-sign patterns have been identified, perception immediately reverses to “grasp complexity”by recognizing the diversity in its context and

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parts, by detecting meaning and feeling delight. This can achieved only if the effect of unity or order is stronger than diversity. Aesthetic pleasure unfolds if the perception can freely play between the pleasing, multilevel patterns of the whole and parts. (3) Aesthetic perception and its extraordinary capacity of information processing is promoted by proportion and other gestalt parameters. In a famous statement of 1712, LEIBNIZ discussed the acoustical perception of music; this view is usseful for the research of visual perception: "Hearing music is a hidden arithmetic exercise of the mind unconscious that it is calculating, because it does many things by way of unnoticed conceptions which with clear consciousness could not do…” LEIBNIZ states that the perception processes information in a systemic way (WÄSSLE, 2004), particularly by subconscious identification of mathematical patterns, of contrasts, tensions, motions etc. Neuroaesthetics shows how this works within the eyes and the brain. (4) The principles of harmony and proportion are recognized in music beyond doubt, but not in architecture. Music without proportions is simply noise; analogously spoken, there are reasons to state architecture without proportions may to be “visual noise”.

Fig. 1a:: O.M. UNGERS’ courtyard in Berlin, 1991-1993. Well-proportioned without balance Fig. 1b:: Rome, Palazzo Sciarra. between unity & diversity/complexity. Neither information reduction nor enrichment late 16c.

(6) The figures 1a/b demonstrate that proportions alone does not suffice for aesthetic perception: Unger’s design of 1991 is well-proportioned according the Quadrature, but lacks diversity, while the Palazzo Sciarra at Rome has it.- Several hundred proportion codes (angles) can be derived by means of very simple operations found by the practical sense of the builders of the past. This simple geometry could be carried out with a single measuring cord (used simultaneously as ruler and compass). Complex curvilinear bodies appear in traditional architecture rarely, besides in vault and cupola constructions. (7) There are geometrical and mathematical methods to determining the Golden Section ratio. The most easiest method is geometrical (by cutting of the diagonals in pentagon). Other methods are arithmetical (φ = (1+√5)/2 = 0.618033988… :1:1.618033988… :2. 618033988…or by the approximation of the Fibonacci Sequence. The pentagon is the only first regular polygon where the diagonals do not cut eachother in one central point. 3. PROPORTION RESEARCH STILL IN ITS INFANCY & THE

LOSS OF HARMONIOUS LIFEWORLD ENVIRONMENTS Despite its long history, despite ancient matured knowledge in arts and crafts, the current times’ insight into architectural aesthetics and proportion lacks synopsis and productive thinking in essential fields. This very research gap (LANGHEIN, 2005) may still persist for long times. Mainstream thought continues to “seclude” itself (SALIN-GAROS, 2002). Even specialists and architects show “uncertainty and helplessness” (H. RAUTERBERG, 2009, personal email). The loss both of “the old way of seeing”, of patternedness of man-made environs after 1830 and of creative, intuitive design habits that respect proportions and form harmony are some of the reasons that the “desecration of beauty on earth” (H.-G. GADAMER, 1960) continues to gain momentum.

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4. TRADITIONAL ARCHITECTURE, PROPORTION & TYPES (1) The vast majority of traditional architecture in Europe, Asia and America comply (and complied) with well-matched, matured aesthetic patterns and proportions. They have reached often an almost perfect “gestalt pragnanz” (“good gestalt”); particularly the low-respected farmhouses and townhouses are masterpieces of the ingenuity of the commoners. They can provoke a superb positive emotional response. Nonetheless, people are dazzled by the modernist zeitgeist, remain blinded for the timeless beauty of such creations, and continue to destroy them still today. This is partly different e.g. countries like Switzerland, France, Austria, etc., and in monument protected areas. (2) The three geometrical “proportion systems” – Triangulature, Quadrature, Quinture (Golden Section) – are parts of the gestalt generators and are represented in traditional architecture (TA) in specific distribution patterns in time (THA) or space. (3) The traditional architecture should be categorized into (1) Traditional Verna-cular Architecture (TVA, particular rural architecture, any kind of farm buildings), Traditional Urban Architecture (TUA, townhouses), Traditional Semi-High Archi-tecture (TSHA, e.g., town halls, rural tax collecting building, functional monastery buil-dings etc.), and Traditional High Architecture (THA, religious architecture, high-class domestic architcture, like palaces, villas, châteaux, countryhouses etc.). Generally it can be stated, that TVA often clings to regional proportion systems for millenia, while THA change the, with style. In short, THA changes more with time, TVA more with space. As mentioned above, the basic pattern of Traditional Mediterranean Architecture (MTA), is primarily characterized by vertical axes of openings (windows, sometimes with columns, determining the proportions = PDWA) since (early?) Antiquity, while the proportions in Northern traditional architecture (NTA) are determined primarily by the roof pitches (PDRS). In the north, of course, wood was available in large quantities; timber suits better to climatic conditions, prefab-like methods were standard. Compared to stone as building material, the timber has quite different properties regarding statics and required functions, favouring other methods of geometrical constructiion and proportioning than in MTA. Already with the Roman conquest of parts of NW Europe after 58 BC, basic MTA patterns began to spread over the northwestern parts of the continent (Gallia, Britannia, Germania, Raetia etc.), first in religious and functional architecture. During and after the völkerwanderung and the medieval cultural consoli-dation, MTA-PDWA patterns spread into non-religious THA and TUA/TSHA. During the Renaissance and Baroque, they became dominant in most parts of Europe and the European colonies around the earth. Treatises on architecture, particularly that of Sebastiano SERLIO (1475-1554) and his followers exerted a strong influence on the divulgation of MTA-PDVA patterns in worldwide architecture since 16c, including the Anglo-Saxonian, Ibero-American, French, Dutch, Germanic, and Scandinavic world. The intensifive emigration of Italian and Ticino muratori to East Europe (Austria-Bohemia, Hungary, Germany, Poland-Lithuania, Russia) were part of this development. (3) On base of c. 2.500 proportion analyses prepared, it can be stated that typical distribution patterns for TVA, TUA/TSHA, and THA exists in Europe, Triangulature occurs exclusively in Gothic THA and TUA/TSHA, but probably never in TVA. (4) Quadrature is most often represented in all categories of traditional architecture, in THA intermingled with the Golden Section (classical Greece, e.g. Parthenon). Part of Quadrature proportions are e.g. those of the Knauth’s Triangle. Normally, the the square and Knauth Triangle can be found together in the same region/area or the same village,

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Fig. 2: Treatises of Sebastiano SERLIO (16c) and Pierre Le MUET (1623) showing TUA House Types that spead over Europe and America since 16th Century

Serlio Book 7, 183 Serlio, Book 7, 103 Book 6, Avery Ms Piere Le Muet (1591- 1669) Pierre Le Muet, 1 . (Cornell), LI, 2nd L’Art de bien Bastir, 1623, 39

side-by-side. In TVA, wooden architecture is in many cases dominated by Quadrature proportions, because the statics of timber in buildings – in contrast to almost all rock and bricks as buildng materials - endures tension. In THA, Quadrature was frequent in Mediterreanean Antique civilizations, also in India, the Far East, Islamic architecture, in Romanesque and Baroque styles. In TVA, Quadrature was either dominant or frequent in West Slavic TVA (from the Baltic Sea east of Kiel Förde down to Serbia /Balkan), in Alemannic TVA (of German-speaking Switzerland, most parts of Alemannic SW Germany (with exception of the upper Rhine rift valley), and Alemannic Austria (Vorarlberg, Außerfern in Tyrol), in Frisian Architecture (NE from Amsterdam through Netherlands and NW Germany along the North Sea coast up to Denmark), partly in Saxonian and Westfalian TVA in Germany, in Scottish and probably in other North and SW English, Irish and Breton TVA. (5) The rural architecture of the US East Coast show all the proportion systems; this splendid architecture seems to comply both with the main features of TVA and TUA. The “pure” Quinture can seldom be found in grid meshes of regulating lines in any traditional architecture, as regards to angles discussed during the Renaissance (54°, 36°, etc.), and if so, only in buildings built after the 15c. However, the Golden Section appears frequently in the proportions of the “Golden Triangle” (with angles like 51°49’38”, 38°10’22” and others – often with high precision) in all categories of architecture, including the oldest pyramids of Egypt (like Cheops pyramid). Like Quadrature, theGolden Triangle proportions occur in many parts of Europe in all categories of traditional architecture (TVA, TUA, THA), including the Northern Mediterranean area, e.g. in Carolingian Franconia (today’s France & Germany, e.g. the Alemannic Alsace), in Saxonian & Westfalian TVA. in Italian palace and villa architecture, including those of Michelangelo & Palladio. In late Baroque and the English Georgian style, “musical proportions”, occur sometimes. Their identification is in a similar way possible as the “polygon-derived” geometrical proportions. (6) There is not doubt, that the Golden Section often imparts impressions of grace, elegance, sophistication to buildings. This is partly best researched and demonstrated in the publications of GHYKA, BORISSAVLIETICH, MOESSEL, DOCZI, FOURNIER DE CORATS, HAMBIDGE, HUNTLEY. MAILLARD and many others. 5. RELIABILITY AND PROBABILITY RATIOS

(1) Critics sometimes conclude by pointing out to contradicting proportion analyses of same monuments (e.g. THA like cathedrals, villas, palaces) that all such efforts were useless. Such conclusions are not as conclusive as they may seem. Even defective analyses state with some probability that buildings analysed do have form order and perhaps gestalt pragnanz that exclude randomness. Any evidence of form order is a significant in itself.

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(2) One of the most reliable probability laws of natural science is the entropy law, which states that no order is caused at random. Proportion analyses identify order of form and figural unity. The vast majority of ancient builders intended order in form. The fact that TVA exists mostly as types allows the deliberate repetition of analyses in order to identify figural “invariants” established by proportion and other means (gestalt parameters). (3) The probability of correctness of proportion analyses can thus be strengthened. In addi-tion, the laws of statistical probability increase the reliability. Such repitition can be compared with supplementary experiments in experimental sciences. This ia best possible e.g. in Switzerland because of the country’s excellent documentation of elevations of traditional buildings. The differences between TVA and TSHA/TUA can sometimes be fluid or unclear. The “Zehnthaus Geislingen” (early 16c) looks like timber-framed TVA (fig. 5), but has TSHA/TIA proportions, since it was an official building in late Middle Ages. Its proportions of the 60° angle (Triangulature) can almost never encounted in TVA.

Fig. 3: TVA in Central Europe (Germanic Switzerland, Austria, North Germany)

Richterswil (ZH, 1749), Lauperswil (BE, Emmental), Maschwanden (ZH, 18c) Farmhouse in Appenzell (CH) Haus zum Bären farmhouse near Berne (CH)

Dornbirn-Oberdorf, Vorarl- Zell-Oberwinkel (Kärnten, Seeboden (Kärnten, Millstätter See), farmhouse (Gmündi.K. area) Berg, W Austria (late 18c) Karawank alpine range)

Trebsing near Gmünd (Kärnten) Granitzen (Judenburg, Central Styria) St. Wolfgang (Granitzen, Köflach, Kemetberg Austria Central Styria Austria) Central Styria) (South StyriaI

TVA Badbergen (Niedersachsen, TVA Storage TUA at Stade (18c) TVA: Typical farmhouse of Artland TUA storage at 1748) in Badbergen still some 600 still extant, here Nienburg/Weser ‚Wehlburg’, Open Air Museum Cloppenburg (Niedersachsen)

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Fig. 4: Examples of traditional Mediterranean Architecture (MTA, MTVA, MTUA)

House, Galicia (E) Sineu (Mallorca, Spain) Taganana. Tenerife Los Llanos, La Palma, Canarian Isl. (E) Conflent, Rousillon (1903), France

Rancate (TI), casa Caroni House in Liguria, Italy Oppède-le-Vieux, mas de la Cavada Outero (Galicia, Spain) (17c), Ticino, Switzerland Crémade (Provence, France)

Estreito da. Calheta (Island, São Luís (Maranhão, Brazil) Patzcuaro (Michoacán, Mexico), Ubeda (Jaén), Andalucia of Madeira, Portugal) Plaza Vasco de Quiroga Spain

Fig. 5: TSHA/TUA in Germany & Basel of early 16c (mostly Townhalls)

Town hall of Alsfeld (1512, Hessen, Tax collecting building Town halls of Sinsheim, Basel (Switzerland), Münster/Westf. (1510) Germany) (Zehnthaus) at Geis- (Baden-Württemberg) ) lingen, east of Stutt- Gart, Württemberg)

List of Abbreviations MTA Mediterranean Traditional Architecture, PDWA patterns of proportioning NTA North, Central and East European Traditional Architecture, PDRS patterns of proportioning PDRS Proportioning, dominated by Roof Slope (or Roof Pitch), as in NTA PDWA Proportioning, dominated by Window Axes (i.e. vertical axes of openings), Bays vertical

limits like columns, lisenes), as in MTA TA Traditional Architecture THA Traditional High Architecture TIA Traditional Institutional Architecture, belongs to TSHA TSHA Traditional Semi-High Architecture, as urban and monastic architecture TUA Traditional Urban Architecture, belongs to TSHA TVA Traditional Vernacular Architecture

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References (Short Literature List) (A literature list extended to ca. 120-150 pp. will be available after October 2010 on

www.proportions.de) BORASSLIEVITCH, Miloutine (1954), Traité d’esthétique scientifique de l’architecture, Paris (Author) DEVLIN, Keith (1994, 200). Mathematics. The Science of Patterns, New York (Scientific American Library) Di DIO, Cinzia, Emiliano MACALUSO & Giacomo RIZZOLATTI (2007), The Golden Beauty: Brain response to

Classical and Renaissance Sculpture, in: PLoS One 2 (11), e1201 DOCZI, György (1981, 1994, 2005), The Power of Limits: Proportional Harmonies in Nature, Art and Architecture,

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Ch. Massin et Cie, 1942), Paris (Dominique Vincent, 1996) FRANKL, Paul (1960), The Gothic. Literary Sources and Interpretations through Eight Centuries, Princeton, N.J.

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