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Mathematical Model

Castel del Monte

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        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. 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 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 evident in the mathematical model as well. The spreadsheet model, folio 103:01, starts with one variable, the measure of the base square on line 15.

The spreadsheet model gives all plant form dimensions after the side of the base square is assigned a dimension (cell E15) and the unit of measurement is defined (cell F11).

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.

A study can be performed to find the correct combination of these two variables that satisfy both the mathematical models and the actual plant form dimensions. An iterative study can be done using the spreadsheet to systematically vary the combination of these two variables until the predicated dimensions of the mathematical model match the actual dimensions. The search is not an infinite series of permutations, because the approximate dimension of the base square is known.

Ancient mysticism provides further assistance in narrowing down this search. The focus is on the geometric figures of the square and the octagon, the primal forms in the design algorithms. These forms had a mystical value in the religious Middle Ages. The base octagon size is defined by a single quantity, the measure of its major diagonal, which is also the diagonal of the base square. This diagonal, or the equivalent measure of the side of the square, is the primal measure in the mathematical model as well.

We draw from medieval culture and historical notes to rationalize about what could have been conceptually the primal measure of the plant octagon.

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 that has two squares inscribed and is itself inscribed inside the circle is the transitional figure between the square and the circle, folio 105:01.

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, folio 105:02.

The adding of two diagonals could be because the square has two diagonals or because there are two squares inside the octagon, folio 105:01.

The year was commonly taken to have 360 days plus a remainder of days in the Middle Ages. Accordingly the side of the square would be 127.279, folio 105:03.

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, folio 105:04.

In ancient times certain mathematical relationships and constants were know as approximations, 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 plant octagon. The number 360, the number of days in a year, had a particular and even mystical significance. By making 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 were two diagonals in a square that would add up to 360 when each was 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, folio 105:05.

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 architect as he decided the measure of the base square.

The base square with a side of 128 is the measure that the medieval architect chose for the dimension of the castle plant; it is the primal measure in the mathematical model.



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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.


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