Page 11

Tower Enlargement Measures

All measures for the revised tower design are defined in terms of the concept design with quantum increments f, k and m that come from the footing corner triangle (page 110).

The dimensions of the enlarged tower octagon are thus defined mathematically together with all related geometric forms: the footing octagon and the tower octagon open area.

The calculation of the various measures for the enlarged tower are outlined in Table 1. The parallel calculations for the same variables in the concept design are outlined in Table 2 for reference. The variable names are distinguished in the tower enlargement by the prime (') designation.

All variables are ultimately defined in terms of the concept design tower octagon side s for comparability. This gives a geometric comparison of the variable dimensions without having to know the unit of measurement.

Some of the measurements are shown in folio 111:01.

Table 1. Tower Enlargement Measures

(Dimensions as function of the concept design tower octagon side s)

s’ = s + m = s + (s / 8) ∙ tan (22.5º) = s ∙ 1.05178

u’ = s’ ∙ (1 + √2) = s ∙ 1.05178 ∙ (1 + √2) = s ∙ 2.5398

t’ = s’ / cos (67.5º) = s ∙ 1.05178 / cos (67.5º) = s ∙ 2.8022

y’ = u’ - 2 a = s ∙ 2.5398 - 2 ∙ s ∙ 0.8321 = s ∙ 0.8750

y’ = y + f = s ∙ 0.7500 + s / 8 = s ∙ 0.8750

so‘ = y’ ∙ tan (22.5º) = s ∙ 0.8750 ∙ tan (22.5º) = s ∙ 0.3624

ss’ = s’ + 2 m = s ∙ 1.05178 + 2 ∙ s ∙ 0.05178 = s ∙ 1.1553

j = s ∙ 0.8750 = y’

Enlarged tower octagon area = (s’)^{2} ∙ 4.8284 = (s ∙ 1.05178)^{2} ∙ 4.8284 = s^{2} ∙ 5.3414

Enlarged open space octagon area = (s_{o}’)^{2} ∙ 4.8284 = (s ∙ 0.3624)^{2} ∙ 4.8284 = s^{2} ∙ 0.6341

Open space dimension increment = y’ / y = s ∙ 0.8750 / s ∙ 0.7500 = 1.1667

Open space area increment = (s_{o}’)^{2} ∙ 4.8284 / s_{o}^{2} ∙ 4.8284 = s^{2} ∙ 0.6341 / s^{2} ∙ 0.4660 = 1.3607

Enlarged tower wall masonry fraction = {(s’)^{2} ∙ 4.8284 - (s_{o}’)^{2} ∙ 4.8284} / (s’)^{2} ∙ 4.8284 = 0.8813

Masonry volume change = {(s’)^{2} ∙ 4.8284 - (s_{o}’)^{2} ∙ 4.8284} / {(s^{2} ∙ 4.8284 - s_{o}^{2} ∙ 4.8284} = 1.0790

Table 2. Concept Design Measures

(Dimensions as function of the concept design tower octagon side s)

s

f = s / 8 = s ∙ 0.1250

u = s ∙ (1 + √2) = s ∙ 2.4142

t = s / cos (67.5º) = s ∙ 2.6131

k = f / cos (22.5º) = (s / 8) ∙ (1 / cos (22.5º) ) = s ∙ 0.1353

m = f ∙ tan (22.5º) = (s / 8) ∙ tan (22.5º) = s ∙ 0.05178

a = s / √2 + f = s / √2 + s / 8 = s ∙ 0.8321

y = u - 2 a = s (1 + √2) - 2 (s / √2 + s / 8 ) = s ∙ 0.7500

so = y ∙ tan (22.5º) = s ∙ 0.3107

ss = s + 2 m = s + 2 ∙ (s / 8) ∙ tan (22.5º) = s ∙ 1.1036

j = y + f = s ∙ 0.7500 + s ∙ 0.1250 = s ∙ 0.8750

Concept design tower octagon area = s^{2} ∙ 4.8284

Concept design open space octagon area = s_{o}^{2} ∙ 4.8284 = s^{2} ∙ 0.4660

Concept design wall masonry fraction = (s^{2} ∙ 4.8284 - s_{o}^{2} ∙ 4.8284) / s^{2} ∙ 4.8284 = 0.9035

The tower diagonal t' has increased by 7.2% with the tower enlargement.

The dimension of the open area inside the tower y' has increased by 16.7%; the open area surface has increased by 35.1%.

The open space in the center of the enlarged tower appears significantly larger compared to the open space in the concept design in the illustrations, because of the oversized dimension of the footing width. The footing width is in actuality smaller and the open space enlargement is modest, folio 111:02.

The tower masonry in the tower has increased in volume by 7.9%. Nevertheless, the masonry percentage of the tower volume has decreased slightly by 2.3%. The relative percentage of the masonry volume in the tower is still an overwhelming 88.1%.

Interestingly, the measure of the open space y' is exactly the same as the measure j, which is the distance of the minor diagonal intersection from the indoor face of the façade wall in the concept design.

They are both measures of open spaces and may likely have had a prominent symbolic value for the medieval architect, possibly a reason that guided this peculiar form of tower enlargement.

Putting this puzzling though aside, the remaining question is what happens to the minor diagonal intersections to finalize the design modifications.