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Mathematical Model

Castel del Monte

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        Unit of Measurement Determination

The result for the iterative spreadsheet analysis for the unit of measurement is shown graphically in folio 107:01

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 folio 107:02.

The overall deviation is considered for:

  1. all 20 variables -- Group A in folio 107:01
  2. 19 variables, excluding variances above 2%, -- Group B in Folio 107:01
  3. 17 variables, excluding variances above 1% -- Group C in folio 107:01

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 c26" 28.453 28.45 0.000 0.00% 0.00% 0.00% 0.00%
9 h26" 26.287 26.29 -0.003 -0.01% 0.01% 0.01% 0.01%
10 c18" 23.830 23.81 0.025 0.11% 0.11% 0.11% 0.11%
11 h14" 20.247 20.26 -0.011 -0.05% 0.05% 0.05% 0.05%
12 h12" 17.745 17.70 0.049 0.28% 0.28% 0.28% 0.28%
13 wg 6.411 6.40 0.010 0.16% 0.16% 0.16% 0.16%
14 h6g" 11.202 11.29 -0.085 -0.75% 0.75% 0.75% 0.75%
15 h2g" 8.928 8.93 -0.002 -0.02% 0.02% 0.02% 0.02%
16 bg" 2.340 2.37 -0.030 -1.27% 1.27% 1.27%
17 ddt18" 10.462 10.40 0.066 0.63% 0.63% 0.63% 0.63%
18 ddf18" 9.696 9.63 0.071 0.74% 0.74% 0.74% 0.74%
19 pg" 1.038 1.05 -0.012 -1.14% 1.14% 1.14%
20 d2" 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, bg" and pg, with percent variances between 1% and 2%, because of the some unique circumstances. The variance for bg" 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 pg is the thickness 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 folio 107:01 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 plant measures with a variance of less than 0.55% with 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 the proof that the geometric algorithms resulting from this study is correct.

This is not a chance result, because of the complexities of the algorithms and the large number of variables involved.

This is the design that the medieval architect followed. These geometric algorithms correctly decode the work of these ancient and unknown architects.

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 blue print cast in stone.

This also proves the choice for the base square dimension discussed previously, when it was theorized that the dimension assigned by the architect to the primal geometric form was a square with a side of 128 units.

It is desirable to have a physical proof at the castle for this unit of measurement. The exploration may become more fruitful now that we know what to search for.

One such candidate is the capping stone for the footing (f), folio 107:03.

The cross section of the stone blocks that cap the footing octagonal ring have an intricate (decorative) profile on the visible side, and a square angle at the opposite, buried side. 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.


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