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Unit of Measurement Determination
The result for the iterative spreadsheet analysis for the unit of measurement is shown graphically in Fig. 14.
Figure 14. Unit of measurement optimization for Castel del Monte.
The overall variance is considered for slightly different sets of the 20 targeted variables. The reason is the range of these variances and the circumstances associated with the variables with the most deviations.
Using a unit of 0.3020 m as a tentative best match for the lowest overall variance, the individual variable variances are shown graphically in Fig. 15.
Figure 15. Variance per variable for unit of measurement of 0.3020 meter.
The overall deviation is considered for:
- all 20 variables -- Group A in Fig. 14
- 19 variables, excluding variances above 2%, -- Group B in Fig. 14
- 17 variables, excluding variances above 1% -- Group C in Fig. 14
The actual variances are shown in Table 2.
Table 2. Variance analysis for a unit of measurement of 0.302 m.
Line | Variable |
Predicated
(meter) |
Measure
(meter) |
Variance
(meter) |
Variance
(%) |
A-Group
Abs |
B-Group
Abs |
C-Group
Abs |
---|---|---|---|---|---|---|---|---|
1 | f | 0.383 | 0.38 | 0.004 | 0.92% | 0.92% | 0.92% | 0.92% |
2 | m | 0.159 | 0.154 | 0.005 | 3.25% | 3.25% | ||
3 | u' | 7.776 | 7.80 | -0.022 | -0.28% | 0.28% | 0.28% | 0.28% |
4 | s' | 3.221 | 3.23 | -0.008 | -0.25% | 0.25% | 0.25% | 0.25% |
5 | ss' | 3.539 | 3.54 | -0.004 | -0.11% | 0.11% | 0.11% | 0.11% |
6 | tt' | 9.247 | 9.28 | -0.036 | -0.39% | 0.39% | 0.39% | 0.39% |
7 | a' | 2.549 | 2.56 | -0.011 | -0.43% | 0.43% | 0.43% | 0.43% |
8 | c_{26}" | 28.453 | 28.45 | 0.000 | 0.00% | 0.00% | 0.00% | 0.00% |
9 | h_{26}" | 26.287 | 26.29 | -0.003 | -0.01% | 0.01% | 0.01% | 0.01% |
10 | c_{18}" | 23.830 | 23.81 | 0.025 | 0.11% | 0.11% | 0.11% | 0.11% |
11 | h_{14}" | 20.247 | 20.26 | -0.011 | -0.05% | 0.05% | 0.05% | 0.05% |
12 | h_{12}" | 17.745 | 17.70 | 0.049 | 0.28% | 0.28% | 0.28% | 0.28% |
13 | w_{g} | 6.411 | 6.40 | 0.010 | 0.16% | 0.16% | 0.16% | 0.16% |
14 | h_{6g}" | 11.202 | 11.29 | -0.085 | -0.75% | 0.75% | 0.75% | 0.75% |
15 | h_{2g}" | 8.928 | 8.93 | -0.002 | -0.02% | 0.02% | 0.02% | 0.02% |
16 | b_{g}" | 2.340 | 2.37 | -0.030 | -1.27% | 1.27% | 1.27% | |
17 | ddt_{18}" | 10.462 | 10.40 | 0.066 | 0.63% | 0.63% | 0.63% | 0.63% |
18 | ddf_{18}" | 9.696 | 9.63 | 0.071 | 0.74% | 0.74% | 0.74% | 0.74% |
19 | p_{g}" | 1.038 | 1.05 | -0.012 | -1.14% | 1.14% | 1.14% | |
20 | d_{2}" | 7.396 | 7.38 | 0.016 | 0.22% | 0.22% | 0.22% | 0.22% |
N | 20 | 19 | 17 | |||||
Avg | 0.55% | 0.41% | 0.31% |
Group B excludes the variable m because it has a small dimension that
magnifies the small variance. The dimension of m is 15.4 cm;
the variance of the predicated measure is only five millimeters,
yet the variance percentage is 3.25%.
Group C excludes two more variable, b_{g}" and p_{g}, with percent variances between 1% and 2%, because of some unique circumstances. The variance for b_{g}" is 3 cm for a dimension of 2.37 m; it is a very small shortage that is associated with the finishing layer part of the algorithm, which is the subject of a separate discussion. The variable p_{g} is the width of the separation wall; the variance is a mere 12 mm. This variable is not part of the geometric algorithm, it was defined heuristically.
A visual inspection of fig-14 indicates that the unit of measurement is in the range of 0.3018-0.3022 m, with a leaning toward the 0.3018 measure. The variance is a range of half a millimeter. The unit of measurement is therefore approximated at 0.3020 meter, with a likely precision of ±1 millimeter.
This unit of measurement provides an average variance of 0.31% for 17 key variables (Table 2).
The data also indicates that this mathematical model, with a unit of measurement of 0.302 m, predicates the overall plant measures with a variance of less than 0.55% and 95% statistical certainty.
This is an amazing result that proves the geometric design algorithm. The predication of all geometric forms for the plant design and a replication of their dimension with an overall deviation of less than 0.55% is proof that the geometric algorithms resulting from this study is correct.
This is the design process that the medieval architect followed. It is not a chance result, because of the complexities of the algorithms and the large number of variables involved.
This is also a testimonial to the great achievement of medieval architects and builders. It is worth repeating that it is because the medieval architect followed a strict algorithm and the builders realized the construction with extreme precision that we have had the opportunity to decipher the design from the blueprint cast in stone.
The finding on the unit of measurement also supports the educated assignment made for the base square dimension.
A base square with a side dimension of 128 feet and a measure of 0.302 meters for the foot provide the best match between the plant dimensions predicated by the geometric algorithms and the actual measurements as reported by Götze and Shirmer.
It is desirable to have a physical proof at the castle for this unit of measurement. The search is made easier having the likely dimension for the “foot” dimension at Castel del Monte.
One such candidate is the capping stone for the footing (f), Fig. 16.
Figure 16. Capstone fot the tower footing space.
The cross section of the stone blocks that caps the footing octagonal ring has a complex decorative outline on the outside, and a square angle at the opposite side buried in the tower masonry. The cross section has a linear width slightly larger than the footing dimension f to allow for a portion of this stone block to be embedded into the tower masonry.
It is hypothesized that the cross sectional height of the stone blocks could be the unit of measurement of Castel del Monte. Not having any other rational basis to determine the height of this stone element, it is thought that the architect may have chosen to make the height of this most visible and important stone piece exactly one unit, 0.302 m. Lke the other dimension of this stone block, f, is "fundamental" in the geometric deasign of the plant, so is "fundamental" the measure of the foot in the construction of the castle.