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Castel del Monte

Mathematical Model of the Plant Design
(File 144-120)

Page 5

        Medieval Mysticism and the Search for the Unit of Measurement

A key to understand architectural design is the unit of measurement. There is no information on the builders and the construction work of this castle all the way back to the Middle Ages. Nothing is known about the unit of measurement used in the design and construction of the castle, nor there seems to be evidence found of this measure anywhere in the castle.

Scholars have speculated about the cubit and the palm (Cardini 2000, 57), common measures from ancient times. H. Götze theorizes that the unit of measurement may have been the Roman foot, approximated to about 0.3 m (Götze 1998, 169), which he seems to reconcile to the measurement of some major forms in his plant model hypothesis.

The mathematical model is the tool used to discover the actual unit of measurement used in the design and construction of Castel del Monte.

All plant forms spring from a single geometric figure in the design algorithm, the base square. This is also the case in the mathematical model; the dimensions of all plant forms are derived in the spreadsheet following the assignment of a measurement to the base square dimension L.

The base square dimension and the unit of measurement are therefore the only independent variables (unknowns) in this mathematical model. Knowing one allows the determination of the other.

The study objective is then to find the numerical combination of these two variables that gives a match between the plant form dimensions as calculated in the mathematical model and the actual plant form dimensions.

The spreadsheet is adept at doing an iterative analysis to find the correct combination of values for L and the foot measurement. Such a brute force analysis is handicapped by the lack of some data, the lack of precision for some of the actual measurements, and the possible errors/approximations during construction 800 years ago.

However, the search is facilitated by a number of considerations. Medieval mysticism provides clues in this search of a dimension for the base diagonal; this is the diagonal that is common to both the base octagon and the inscribed base square.

The primal forms in the design algorithms, the base square and the base octagon, had a mystical significance in the religious Middle Ages. We draw from medieval culture and historical notes to rationalize about what could have been conceptually the primal measure of the base diagonal.

It is known that in the symbolism and religiosity of medieval culture the square was associated astrologically with the earth and religiously with a state of sin and imperfection. The circle, on the other hand, was associated with the universe and infinity, and religiously to a state of perfection, God and paradise.

The octagon, which has a square inscribed and is itself inscribed inside a circle, is the transitional figure between the square and the circle, Fig. 7.



Figure 7. Octagon, a transitional geometric form between square and circle. .

Fig-7. Square, octagon and circle.

The octagon is the human connection between earth and the universe, man’s ascension from sin to a state of perfection. This subject is addressed more exhaustively by other scholars (Götze 1998, 117; Cardini 2000, 57).

The base octagon is drawn around the base square, which is taken to be the primal geometric form. Scholars report that the square was the starting geometric form in antiquity to build an octagon geometrically (Götze 1998, 115-117 and 164-165).

A logical measure for the base square, which is associated with the earth, would be an earthly measure. One such measure in astrological sense is the number of days in a year on earth. It is theorized therefore that the ideation of the concept for Castel del Monte may have started from a base square whose two diagonals added up are equal in linear units to the number of days on earth, Fig. 8.



Figure 8. Base square dimension inspiration.

Fig-8. Base square dimension inspiration.

The adding of two diagonals could be because the square has two diagonals or because there are two squares inside the octagon, Fig. 7 and 8.

The year was commonly taken in the Middle Ages to have 360 days plus a remainder of days. The diagonal would be therfore 180 units (360 days divided by two). Accordingly the side of the square would be 127.279 units, Fig. 9



Figure 9. Side of base square from Phytagorean relationship.

Fig-9. Side of base square from Phytagorean relationship.

Choosing a side of 128 units for the side of the base square results in a diagonal dimension that when doubled up, for two diagonals or two squares, yields 362.04 units, Fig. 10.



Figure 10. Theorized base square dimension, L=180.

Fig-10. Theorized base square dimension, L=180.

Certain mathematical relationships and constants were know as approximations in ancient times, much more so than today; approximations facilitated common usage in everyday computations. The square root of 2 (√2) was commonly approximated as 1.4; nowadays this quantity is relegated to a function key on a calculator.

Ancient mathematicians were aware that there was a remainder, the difference between the true value of √2 (1.414136...) and 1.4. The perception was that the remainder was an imperfection that could be ignored, although in geometric construction this approximation to 1.4 was clearly noticeable and ignoring it would lead to errors.

The same is true for the number of days in a year. The year was approximated to 360 days. The other 5 days were considered imperfections, actually bonus days at the end of the year.

It is no different in modern times when the year is thought as being 365 days with a bonus extra day every fourth leap year, and we clearly forget about the other “imperfections,” the minutes and seconds that are ignored but make up the exact solar year (the solar tropical year is 365.24219878… days).

We have in these approximations the genesis for a solution to the dimensioning of the base square. The number 360, the number of days in a year, had a particular and even mystical significance. By making the base diagonal, T, equal to 360/2 units (180), L is 128 units using 1.4 for √2 and dropping the tiny remainder here too, an imperfection. The corresponding value for l is 64 units. Making T equal to 360 would have resulted in a castle too big; after all there are two diagonals in a square that would add up to 360 when each is 180.

The measure of 128 units for L is a propitious one, because the quarter square (after dividing the base square into four equal square portions) has a diameter that is the radius of the circle circumscribing the base square and base octagon, fig. 11.



Figure 11. Base square and quarter square dimensions.

Fig-11. Base square and quarter square dimensions.

The side of this quarter square is 64 units (half of the side of the base square). This number is the result of the product of two eights, 8 x 8 = 64. Eight is the mystical number at this castle. It is also opportune to note that the castle has eight towers, each with eight sides, and their product is 64. Furthermore, there are two floors, each with eight rooms; two eights in the product that yields 64.

The coincidence of these numbers, 360, 180, 128, and 64, and the mystical connections of 360 being the number of days in a year and 64 being the number of tower octagon sides is inescapable nowadays and must have been known in the superstitious Middle Ages.

While these may seem good subjects for numerologists, they may have very well been in the mind of the medieval architects as they decided the measure of the base square.

The dimension of 128 units is the measure adopted in the mathematical model for the base square; it is the best hypothesis at this point, but one that is proved correct in the end.



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Ref.

Cardini, F. 2000. Castel del Monte. Bologna: Il Mulino.

Götze, H. ed. 1998. Castel del Monte, Geometric Marvel of the Middle Ages. New York: Prestel-Verlag.