Page 6
Unit of Measurement Examination
Having established a dimension for the primal measure, a base square with a side of 128 units, the mathematical model is left with only one unknown, which is the dimension for the unit of measurement used by the medieval architect at Castel del Monte.
Any proposed unit of measurement can be tested with the mathematical model by plugging its metric value in the spreadsheet, cell F11 of the spreadsheet on page 3, and letting the spreadsheet calculate the many variable dimensions. The unit that provides the best match between the predicated dimensions in the spreadsheet and the reported measurements is logically the unit of measurement that the architects used in the design and construction of Castel del Monte.
Because there are dozens of variables and over one hundred formulas in the mathematical model, establishing a match between predicated and actual measurements requires some iterative examination. The search is complicated by a number of circumstances.
Not all variable dimensions addressed in this model are available from published literature. Because scholars focus on different aspects of the castle and have diverse theories, the dimensions that they focus on and for which they report dimensions in the literature are not necessarily the variables in this mathematical model.
This study relies primarily on measurements reported in the literature. The most extensive and reliable documentation source on various plant dimensions is provided by Götze, Fig. 12, and Shirmer, Fig. 13.
Figure 12. Plant dimensions – Götze.
Figure 13. Plant dimensions – Shirmer.
Another complication is that actual measurements include uncertainty due to a number of sources. Foremost are the measurement errors for the reported measurements, which are not entirely avoidable.
Some dimensions are known with great certainty because they are measured with greater precision and repeatedly for the many instances of this dimension. One such example is the measure for the 32 equal sides of the towers provided by Götze. The many instance measurements of this variable provide a means to refine further the nominal tower octagon side measure using the statistics of 32 measurements.
The footing dimension, which is considered an ornamental feature by scholars and is physically more difficult to measure precisely is mentioned deferentially as a single measure by Götze, and is known with less precision.
There are additional sources of unknowns contributing to expectable deviations in comparing the mathematical model predicates to the actual measures. Some deviations come from the precision capabilities of medieval builders and how exactly they were able to reproduce the dimensions in the architectural plans.
It is remarkable the level of precision achieved by medieval builders, both in design and construction, which is verified by the predicated dimension that deviate from the measurement by less than four parts in ten thousands (0.4%), as shown later.
A final consideration is the extreme range of dimensions that the mathematical model predicates, which are being matched to actual measurements.
The search for the unit of measurement involves matching variables that are very large and very small at the same time and in the presence of measurement uncertainties.
An example of such extremely different dimensions is the plant octagon major diagonal (T), which is approximately 56 m, and the finishing wall layer (v), witch is about 4.5 cm; one being more than a thousand times bigger than the other.
The iterative search for the unit of measurement focus on nearly two dozen key variables for which measurements are known with reasonable certainty, Fig. 12 and Fig. 13.
Table 1 shows the key variables for which measurements are available and the source for these measurements.
Table 1. Mathematical model variables with measures
Line | Variable | Meas. | Source |
---|---|---|---|
1 | f | 0.380 m | Lanera |
2 | m | 0.154 m | Lanera |
3 | u' | 7.768 m | Götze |
4 | s' | 3.229 m | Götze |
5 | ss' | 3.543 m | Lanera |
6 | tt' | 9.283 m | Schirmer |
7 | a' | 2.560 m | Götze |
8 | c_{26}" | 28.450 m | Shirmer |
9 | h_{26}" | 26.290 m | Shirmer |
10 | c_{18}" | 23.810 m | Shirmer |
11 | h_{14}" | 20.258 m | Shirmer |
12 | h_{12}" | 17.696 m | Shirmer |
13 | w_{g} | 6.401 m | Shirmer |
14 | h_{6g}" | 11.287 m | Shirmer |
15 | h_{2g}" | 8.930 m | Götze |
16 | b_{g}" | 2.370 m | Götze |
17 | ddt_{18}" | 10.396 m | Shirmer |
18 | ddf_{18}" | 9.625 m | Shirmer |
19 | p_{g}" | 1.050 m | Shirmer |
20 | d_{2}" | 7.380 m | Shirmer |
The search consists in assigning a metric value to the unit of measurement (cell F11), letting the spreadsheet calculate the predicated dimensions for this set of variables, and measuring the overall deviation from the reported measurements.
The overall deviation is calculated as the sum of the individual variable deviations (the absolute deviation value, ignoring the deviation sign). The unit dimension that provides the lowest overall deviation is considered to be the one that provides the best match and likely the true unit of measurement in the design of Castel del Monte.
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Ref.
Götze, H. ed. 1998. Castel del Monte, Geometric Marvel of the Middle Ages. New York: Prestel-Verlag.
Shirmer, W. ed. 2000. Castel del Monte. Research results of the years 1990 to 1996. Mainz. ISBN 3-8053-2657-2