Average architecture
The middle dimensions in architecture describe a system of proportions in which different sizes, such as length, width and height of a room, are proportionally related by being harmoniously linked. Vitruvius describes the first application in architecture when it comes to determining the height of a room. Leon Battista Alberti and Andrea Palladio give more detailed descriptions ; the latter also provides examples in his four books on architecture .
Accordingly, the arithmetic , the geometric or the harmonic mean should help to find the height for a fixed floor plan for the selected length and width. The proportions of the following room can be determined from these dimensions. Even an entire building can be proportionally divided by the whole using the mean dimensions. An example:
For example, assume a room is 12 m long and 6 m wide. For the arithmetic mean (better known as the average) the height would be 9 m, the harmonic mean results in the height of 8 m, the geometric mean is 8.46 m.
Philosophical approach to the importance of mediocrity
Alberti writes in his architecture : "In this way, the architects use these mean proportions to make the most beautiful designs for the whole building as well as for the parts of the building ..." Alberti defines beauty as "a kind of correspondence and harmony of the parts into a whole." which was carried out according to a certain number, a special relationship and arrangement, as required by symmetry, that is, the most perfect and supreme law of nature. "
The aesthetics of the philosophers of the Middle Ages are very much based on ancient ancestors. Umberto Eco explains that Augustinus quotes Cicero almost literally when he asks: "What does physical beauty consist of? In the correct relationship between the parts ..." For Eco, the aesthetics of proportion is an aesthetics of quantities.
Plato and Aristotle expressed themselves in various places in their writings on proportions and mean measures. So Plato considers it impossible to beautifully bring two things together without a third, connecting one. Only an intermediary band can establish a connection between them. The mediation of two different sizes take over the middle sizes to tie the previous two over a third.
The arithmetical determination of the mean dimensions
The mean dimensions can be determined arithmetically. What is surprising is the different approaches that reveal different perspectives and meanings of the mean.
The calculation of the arithmetic mean
The arithmetic mean (aM) of two numbers (here always a and b) is usually determined using the formula: aM = (a + b) / 2 .
Boethius does this around 500 AD:
- a = 40, b = 10
- Form the difference: d = a - b = 40 - 10 = 30
- Halve the difference: 30/2 = 15
- Add half the difference to the smaller number: 15 + 10 = 25 = aM
The calculation of the geometric mean
The geometric mean (gM) can only be determined numerically by multiplying the two starting numbers and taking the square root:
- a = 40, b = 10
- √ (a * b) = √ (40 * 10) = 400
- √400 = 20 = gM
Since it was hardly possible to pull the roots up until the Renaissance, according to Alberti and Palladio, they refer to the graphic definition of the geometric mean. Both use the same numerical example (proportion 9: 4 with the gM 6).
Calculating the harmonic mean
In the literature there are different approaches to the arithmetical determination of the harmonic mean (hM) from two numbers:
Today the formula applies: hM = 2 * a * b / (a + b)
- a = 40, b = 10
- hM = 2 * a * b / (a + b) = 2 * 40 * 10 / (40 + 10) = 800/50 = 16
Boethius does the math:
- a = 40, b = 10
- Add the two numbers: s = a + b = 40 + 10 = 50
- Form the difference: d = a - b = 40 - 10 = 30
- Multiply the smaller number by the difference: m = b * d = 10 * 30 = 300
- Divide the product by the sum of the two numbers: t = m / s = 300/50 = 6
- Add this result to the smaller number: hM = t + b = 6 + 10 = 16
Alberti does the math:
- a = 40, b = 10
- Find the ratio of the two numbers to each other, here about 4: 1
- Add the two ratios: 4 + 1 = 5
- Form the difference: d = a - b = 40 - 10 = 30
- Divide the difference by the sum of the ratios: 30/5 = 6
- Add the result to the small number: hM = 6 + 10 = 16
The difficulty with this method is to find the appropriate ratio to the two measures.
The easiest way is Palladio (his way corresponds to today's formula by changing):
- a = 40, b = 10
- Find the arithmetic mean, aM = (a + b) / 2 = (40 + 10) / 2 = 25
- Multiply the two numbers m = a * b = 40 * 10 = 400
- Divide the multiplication by the arithmetic mean: hM = (a * b) / aM = 400/25 = 16
The calculation of the median measures at musical intervals
The Pythagoreans had particularly emphasized the tetraktys , the ancient Greek symbol for the number 10, as this enabled them to illustrate their worldview in a concise way. The proportions 1: 2, 2: 3 and 3: 4 and also the proportions 1: 3 and 1: 4 with a monochord can be experienced both acoustically and visually (using the string lengths). These proportions (octave, fifth and fourth; duodecime and double octave) have long been part of music theory .
The room with the dimensions 12 m long and 6 m wide forms the proportion of the octave in the floor plan, the ratio 2: 1. The harmonic mean 8 m with the width of 6 m forms the proportion of the fourth (ratio 4: 3). The remaining area of 4 m by 6 m gives the proportion of the fifth (ratio 2: 3) which is the arithmetic mean of the octave. The octave consists of the fourth and the fifth, i.e. its harmonic and arithmetic mean. Other subdivisions of the octave can also be calculated. If you choose the height over half the width (3 m) for the room of 12 m by 6 m, the proportions are 12: 3, i.e. 4: 1 for the double octave. The double octave finds its geometric mean in the octave, which is found again as the proportion for the front wall of the room (6 m: 3 m).
The geometric determination of the mean dimensions
Since it made particular difficulties to determine the geometric mean, Alberti and Palladio refer to the geometric determination of the mean. Again, there were different methods. Often these are based on Euclid.
The drawing of the arithmetic mean
The simplest method is to extend the drawn length with the width and determine the middle of the total distance with two circular strokes - the arithmetic mean.
- Extend the length by the width
- 2 circular strokes
- Plumb to the baseline to determine the center of the total line - the arithmetic mean
Francesco di Giorgio Martini uses a construction to determine the arithmetic mean in the drawing.
- Enter the width in the length
- Draw a square
- Cross out remaining area
- Plumb line from the cross to the baseline = the arithmetic mean
Drawing of the geometric mean
Euclid, Serlio and Palladio graphically determine the geometric mean as follows:
- Find the arithmetic mean by extending the long side with the width.
- Circular stroke with the center and radius of the arithmetic mean
- The perpendicular on the extension of the width to the semicircle is the geometric mean.
According to Thales, the geometric mean can also be determined as follows:
- Make a semicircle over the rectangle.
- Circle the width to the length.
- Draw a plumb line on the semicircle
- The connection between the plumb line and the corner is the geometric mean we are looking for
A special feature is the determination of the geometric mean from the quadrature . A square is quartered. Two smaller squares form a double square; the diagonal of a small square gives the geometric mean of the length and width of the double square. The decreasing wall thickness on higher floors was determined in this way.
Drawing of the harmonic mean
Palladio uses a method that corresponds to his mathematical way. It draws the same path as for the determination of the geometric mean, for this the arithmetic mean for the radius has to be determined beforehand.
- Draw the geometric mean with the radius of the arithmetic mean.
- Extend the long side by the arithmetic mean using a circular stroke.
- A straight line from the side lengthened by the arithmetic mean to the corner of the width is lengthened to the likewise lengthened straight line of the opposite width. The perpendicular corresponds to the harmonic mean.
Albrecht Dürer published a simpler method, which he actually used for the visual subdivision of a window opening, for example; he called this Verkehrer . He draws the two different lines parallel, connects them, crosses them and draws a line at the point of intersection which, limited by the connecting lines, results in the harmonic mean of the two lines of origin. Transferred to the same example rectangle:
- Enter the width on the length using a compass.
- Draw the missing connection line.
- Cross out the polygon.
- The perpendicular at the crossing point, limited by the connecting line, gives the harmonic mean.
The methods of Francesco di Giorgio and Dürer can be combined so that arithmetic and harmonic mean can be determined with just a few lines.
The application of mediocrity to examples
The Pantheon in Rome is the best preserved ancient Roman building. Inside, the hall with the distinctive dome circumscribes a hemisphere. In the section, this becomes a semicircle, which encloses a rectangle in the proportion of the octave. The arithmetic mean defines the intersection of the dome with the outer wall, the geometric mean the upper edge of the abutment, the harmonic mean the outer edge of the eaves.
The use of mediocrity can be demonstrated in many examples on Gothic cathedrals. The Chartres Cathedral measures m in overall width (including aisles) and the height of the walls depending 32.50, which corresponds to 100 altfranzösische foot (p = 1 du pied Roi = 32.47 cm). The width of the main nave is a good 50p, the arithmetic mean of 75p corresponds to the upper edge of the capital . At this point, the thrust from the vault is transferred to the side aisles and the buttress . Since today's calculation methods were not available to the master builders of the Gothic period, the mean dimensions offer the possibility of deriving the resultant from the pressure and thrust forces.
Alberti pointed out the mediocrity in his work On Architecture . He designed the facade for the Florentine church of Santa Maria di Novella. This describes a square, which can be divided into four squares. One upper square removed and the remaining one pushed into the middle, the facade can be read through the median dimensions. If you halve a lower square, the harmonic mean determines the lower edge of the entablature, the arithmetic mean the upper edge. In the upper square, the geometric mean determines the lower edge of the entablature. Since the geometric mean is always larger than the harmonic mean, the upper story appears higher than the lower.
Criticism of the mediocrity
The use of mediocrity in architecture is hardly known, only the more recent literature tries to review and evaluate the few sources. Even if Alberti and Palladio are two of the most important architects of the Renaissance with considerable aftereffects on subsequent epochs, the mediocrity in proportion research has so far only had little echo. The procedures for calculating or graphically determining the median dimensions are simple, so the assumption is that architects at the time did not need to refer to the median dimensions, or were taken for granted, or considered superfluous.
literature
- Roger Popp: The mediocrity in architecture - nature, meaning and application from antiquity to the Renaissance. Hamburg 2005, ISBN 3-8300-1973-4 .
- Leon Battista Alberti: Ten books on architecture. Darmstadt (1450, 1485) 1975, ISBN 3-534-07171-9 .
- Sebastiano Serlio: The Five Books Of Architecture. New York, ISBN 0-486-24349-4 . (Reprint of the 1611 English edition)
- Andrea Palladio: The four books on architecture. Zurich / Munich (1570) 1983, ISBN 3-7608-8116-5 .
- Rudolf Wittkower: Basics of Architecture in the Age of Humanism. dtv-Wissenschaft, Munich 1983, ISBN 3-423-04412-8 .
- Paul von Naredi-Rainer: Architecture and Harmony. Cologne 1982, ISBN 3-7701-1196-6 .
- Lionel March: Architectonics Of Humanism. Chichester (West Sussex) 1998.
Individual evidence
- ^ Vitruvius: Architecture. Basel 1995, ISBN 3-7643-5518-2 , V / 2, p. 208 and VI / 4.
- ^ Leon Battista Alberti: Ten books on architecture. Darmstadt 1975, ISBN 3-534-07171-9 , p. 503.
- ^ Alberti, Baukunst, p. 492.
- ↑ Umberto Eco: Art and Beauty in the Middle Ages. Munich 1981, ISBN 3-423-30128-7 , p. 49.
- ^ Augustine: Epistula. 3, CSEL 34/1, p. 8.
- ↑ Cicero: Tusculanae IV. 31, 31.
- ↑ Eco, p. 63.
- ^ Plato: Timaeus. c7.
- ↑ Anicius Manilus Severins Boethius, Five books on music , German Leipzig 1872, reprint Hildesheim 1973, p. 56.
- ^ Alberti, Baukunst IX / 6, p. 503; Palladio: four books. P. 87.
- ^ Alberti, Baukunst, p. 503.
- ↑ Albrecht Dürer: From the human proportion. Nuremberg 1528. (Reprint: Nördlingen 1980)
- ↑ Eberhard Schröder establishes the reference to the harmonic mean, see Rainer Gebhardt (Ed.): Arithmetic books and mathematical texts of the early modern times. Annaberg-Buchholz 1999, pp. 49-56.
- ↑ Wolfgang Trapp : Handbook of dimensions, numbers, weights and the calculation of time. Augsburg 1969, ISBN 3-86047-249-6 , p. 227.
- ↑ See bibliography, here the works of Wittkower, Naredi-Rainer and March