# Geometry of the Icosikaidigon in Orvieto Cathedral

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## Abstract

The rose window on the main façade of Orvieto cathedral is unique because its design is based on a 22-sided polygon. It is well known that the icosikaidigon cannot be constructed using only a compass and a straightedge. Therefore, using a geometric/statistical approach, we intend to unveil which approximate construction method was used by the sculptor to draw this unique rose window.

## Keywords

Rose window Icosikaidigon Orvieto cathedral Classic Geometry## Introduction

### Geometric Foundation to the Design of a Rose Window

“Rose window” is the name given to the most important stained glass window of a Gothic cathedral. This element could have been designed based on the geometrical knowledge of medieval master masons, commonly referred to as *geometria fabrorum* (Shelby 1965). Another thesis maintains that certain learned geometries were used, in line with similar methods used to build astrolabes, such as *De mensura astrolabii* (c.1050) by Hernán de Contracto (1013–1050) (Pez 1853). Also, other methods are linked to some *Practica Geometriae* (1125–1130), such as the one by Hugo de San Víctor (c. 1100–1141) (Baron 1956); the *Artis cuiuslibet consummatio* (c.1193–c.1220), which is attributed to Gérard de Bruxelles (f.1220) (Victor 1979); or the *Practica* *Geometriae* (1346) by Dominicus de Clavasio (+c.1362) (Busard 1965).

### Gathering of Information

Our analysis relies on massive data acquisition techniques by means of terrestrial laser scanning (TLS). This procedure allows for a precise study of the architectural heritage, provides a great geometric definition and makes it possible to measure objects which are inaccessible in the building without having to make contact with them. Besides, it is a very reliable and thoroughly tested technique (Guarnieri et al. 2004). However, it has some limitations regarding the error range for the data acquisition.

As we have already mentioned above, we will focus our analysis on the outermost crown (of the five crowns which make up the rose window) because, in principle, it is the most rigid one.

*Cyclone*so as to obtain a complete point cloud of the main façade. This cloud is composed of more than 100 million points, of which more than 10 million points correspond to the rose window (Fig. 5). After assigning coordinates to all the points (Fig. 5), we select the points corresponding to the rose window area and, using the vectorial software AutoCad, we obtain all necessary parameters to make the calculations.

Coordinates of the 22 vertices and side lengths

| Coordinates \(\left( {x_{i} ,y_{i} } \right)\) of vertices | Side lengths |
---|---|---|

1 | (3.685,6.211) | 0.7037 |

2 | (4.381,6.106) | 0.7082 |

3 | (5.008,5.776) | 0.7051 |

4 | (5.566,5.346) | 0.7017 |

5 | (5.950,4.758) | 0.7417 |

6 | (6.193,4.057) | 0.7067 |

7 | (6.144,3.352) | 0.6957 |

8 | (5.951,2.684) | 0.7073 |

9 | (5.589,2.076) | 0.7059 |

10 | (5.052,1.618) | 0.7168 |

11 | (4.405,1.310) | 0.7071 |

12 | (3.701,1.239) | 0.7225 |

13 | (2.982,1.308) | 0.7007 |

14 | (2.349,1.610) | 0.7439 |

15 | (1.818,2.090) | 0.6935 |

16 | (1.438,2.670) | 0.7167 |

17 | (1.212,3.350) | 0.7092 |

18 | (1.230,4.059) | 0.6905 |

19 | (1.400,4.729) | 0.7155 |

20 | (1.787,5.331) | 0.7155 |

21 | (2.312,5.817) | 0.7079 |

22 | (2.958,6.107) | 0.7352 |

## Approximate Methods for Constructing an Icosikaidigon

In this section we compile thirteen approximate methods for constructing an icosikaidigon. We have divided them into two subsections. Firstly, we present six methods (*M* _{1–6}) which were frequently used in classic and medieval architectures; and secondly, we show seven methods (*M* _{7–13}) for which there is no evidence of use in the Middle Ages but are comprised of elemental geometric steps.

### Methods That We Are Certain Were Known in the Middle Ages

#### *M* _{1–3}: Construction Based on Approximations to Pi

*pi*which were initially put forward by Archimedes and then by Vitruvius and Ptolemy. This method was used in the design of many classical Greco-Roman buildings and consists of the following steps (Fig. 6): given a circle with centre

*O*and diameter \(\overline{AB}\), we divide the diameter into

*N*parts, each of them being the unit

*u*. Next, using a compass set with a radius which is

*n*times

*u*and its centre on point

*A*, we draw an arc which cuts the given circle at the point

*C*. We define \(\overline{AC}\) as the segment of the polygon which is inscribed in the given circle. Next, with centre on point

*C*and radius \(\overline{AC}\), we draw an arc in order to find another vertex of the polygon. Lastly, similar to the previous step, we find the rest of the vertices. Based on the ratio \(\frac{{\overline{AB} }}{N}\) we can find the approximate side length of a regular polygon.

In the particular case of the icosikaidigon, there are several historical references and examples of architectural elements which were built based on specific ratios. Here are a few examples: according to Thomas Heath (1908), ratios 14/25 and 2/7 were commonly used in classical Greek constructions; Wilson (2000) shows that the circular floor of the mausoleum Le Carceri Vecchie in Capua consists of 22 modules which were laid out based on the ratio 1/7; and finally, with respect to architectural ornamentation, in the city of Tarragona there is a 22-part circular Clypeus from the first century A.D. which includes an image of Jupiter-Ammon and was designed on the basis of the ratio 14/25 (Toldrà et al. 2014).

In this paper we consider the three most common ratios used in architecture for an approximate construction of the icosikaidigon and the hendecagon (Fig. 6). These ratios are: 14/25 (*M* _{1}), 2/7 (*M* _{2}) and 1/7 (*M* _{3}). The analytical equations of the angles of these methods are shown in “Geometric and Statistical Control’’ section.

#### *M* _{4}, *M* _{5}: Construction by Means of Set-Square Rotation

Set squares were defined in Plato’s *Timaeus*, where the triangle is considered to be the most important shape in geometry (Lisi 1997). Although classic set squares continued to be used (Recht 1989), a new kind of set square appeared in the thirteenth century. However, throughout history we find variations, such as: canonical set squares (30°–60°) by Hugues Libergier (Wu 2000); Pythagorean set squares (3u, 4u, 5u) by Niederhaslach; root-five rectangle set squares; decagonal coordinates set squares (36°–54°); and lastly, those set squares which are designed based on the golden ratio (31.72°–58.20°).

*L*in length, it could be rotated while a leg was laying on the hypotenuse. The number of movements in a full rotation depended on the set-square proportions and the polygon to be constructed.

However, this method did not offer an exact solution in all cases. In this paper we will consider two ratios which are known to have been used to construct the 11-sided polygon and the 44-sided polygon (divisor and multiple of the icosikaidigon) and have already been applied in other architectural references (Wilson 2000). These ratios are: 16/25 (*M* _{4}) and 1/7 (*M* _{5}). The analytical equations of the angles of these methods are shown in “Geometric and Statistical Control’’ section.

#### *M* _{6}: Construction Using an Astrolabe

The popularity of the astrolabe in the Middle Ages was mainly due to the possibility of measuring angles using trigonometric methods, thus enabling measurement of distances in inaccessible areas, height calculations and even deepness deduction. In the astrolabes which were typically made and used during the twelfth to fourteenth centuries, the rim of the matter is graduated with 72 marks, one approximately every 5° (Eagleton 2007). This geometric construction was obtained by bisecting the 30°-angle and then trisecting the 15°-angle with an approximate classical method (Eisner 1976). The approximate trisection of the 15°-angle is shown in the explanation of method *M* _{8}.

In order to construct an icosikaidigon, the \(16.\widehat{36}^{{^\circ }}\)-angle has to be constructed. If this is to be done using an astrolabe, the corresponding construction is doubly approximate and, therefore, generates a double error: the first error is due to approximate trisection of the 15°*-*angle using a compass and a straightedge (this error is stated in the first part of the analytical expressions listed in “Results” section for method *M* _{6}); the second error arises because each of the 21 angular differences \(\Delta = \frac{\pi }{132} {\text{rad}} = 1.\widehat{36}^{{^\circ }}\) was determined by the sculptor based on his visual assessment (error *ɛ* _{ i }, where *i* = 1 ÷ 21, i.e. *i* ranges from 1 to 21; his error is stated in the second part of the above-mentioned analytic expressions). In this method we assume that the errors made by the sculptor are all equal and correspond to a tenth of the angular subdivisions of the astrolabe, \(\varepsilon_{i} = \frac{0.1\pi }{180}\). The analytical equations of the angles of this method are shown in “Geometric and Statistical Control’’ section.

### Methods That We Are Not Certain Were Known in the Middle Ages

In this subsection we describe seven approximate methods for constructing an icosikaidigon using a compass and a straightedge (Gonzalo 2001; Schneider and Sappert 1990; Virasoro 1976; Malara 1970); five of which derive from the construction of the hendecagon (*M* _{7–11}), and the remaining two are direct methods (*M* _{12–13}).

#### Indirect Methods

Here, we show two general approximate methods (*M* _{7–8}) for constructing any regular polygon, and three specific approximate methods for constructing a hendecagon (*M* _{9–11}). Based on these five methods, the icosikaidigon can be obtained by bisecting each of the angles.

#### Direct Methods

Next, we show two specific methods (*M* _{12–13}) for constructing directly the icosikaidigon.

## Geometric and Statistical Control

### Geometric Regression

In addition to the foregoing, and in order to answer the research question, we must define the geometric basis for calculations. To that effect, and on the basis of the cloud of coordinated points which replicate the rose window, we assign 22 perimeter points *v* _{ i } (where *i* = 1 ÷ 22, i.e.: *i* ranges from 1 to 22) corresponding to the intersections of the 22 perimeter arcades (Fig. 5). Before further description of the process, it should be noted that the maximum distance between any two points of the laser scan—obtained with a Leica ScanStation P20 scanner—is 0.0002 m. Even though it makes little sense to work with units smaller than millimetres (especially if we are dealing with medieval techniques), we will use four decimal places for our calculations.

*v*

_{ i }(Fig. 5), we can see that, owing to the structural deformations and the building process itself there is no circle passing through more than three points (of these 22). For this reason, considering the positions (

*x*

_{ i },

*y*

_{ i }) of these 22 vertices

*v*

_{ i }(Table 1) and using Gauss normal equations we obtain the Eq. (1) corresponding to the quadratic regression circle \({\mathcal{C}}\):

The position of the centre of the circle \({\mathcal{C}}\) is (3.6892, 3.7133), and its radius \(R = 2.4937\,{\text{m}}\).

After applying the *F*-test by Fisher–Snedecor in this circular quadratic regression, the null hypothesis *h* _{0} of non-correlation is rejected at a significance level *α* = 1 − *G*(2827.54; 3, 18) = 0. The percentage of *y* coordinates (vertex coordinates) which are explained by the circular quadratic regression is calculated with Pearson’s adjusted coefficient of determination, and it is \(\eta_{adj}^{2} \times 100 = 100\,\%\) because \(\eta_{adj}^{2} = 1 - \left( {1 - \eta^{2} } \right)\frac{22 - 1}{22 - 3 - 1}\), where \(\eta^{2} = \frac{{\mathop \sum \nolimits_{i = 1}^{22} \left( {\hat{y}_{i} - \bar{y}} \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{22} \left( {y_{i} - \bar{y}} \right)^{2} }} = 1\) is Pearson’s coefficient of determination for \({\mathcal{C}}\), where \(\bar{y} = \mathop \sum \nolimits_{i = 1}^{22} \frac{{y_{i} }}{22}\), and \(\hat{y}_{i}\) is the *y* coordinate corresponding to the *x* _{ i } coordinate obtained as a result of the circular regression. \(G\left( {x;d_{1} ,d_{2} } \right) = I_{{\frac{{d_{1} x}}{{d_{1} x + d_{2} }}}} \left( {\frac{{d_{1} }}{2},\frac{{d_{2} }}{2}} \right)\) is the *Fisher*–*Snedecor* distribution, where *I* is the regularized incomplete beta function. In other words: \({\mathcal{C}}\) provides a circular quadratic correlation between the coordinates of the 22 vertices of the rose window. The probability that this correlation exists is 1, and 100 % of the points’ coordinates are accounted for by this correlation. Therefore, circle C is not only the best approximation but a perfect approximation, and for that reason we can consider it to be the initial circle of the rose window’s real pattern.

Table 1 shows the \(\left( {x_{i} ,y_{i} } \right)\) coordinates of the 22 vertices *v* _{ i } together with the amplitudes *φ* _{ i } of the rose window angles \(\angle \left( {\overrightarrow {Ov}_{i} ,\overrightarrow {{Ov_{i + 1} }} } \right)\), where *i* = 1 ÷ 21 and *φ* _{22} is the amplitude of angle \(\angle \left( {\overrightarrow {{Ov_{22} }} ,\overrightarrow {{Ov_{1} }} } \right)\). However, in order to compare the methods with the rose window, we will scale circle \({\mathcal{C}}\) so that its radius is \(R = 1\,{\text{m}}\) (the same radius used for all methods in our analysis) and we will move the rose window so that the centre of circle \({\mathcal{C}}\) is the origin (0, 0) of the coordinate system.

### Geometric Comparison

*v*

_{ i }and the pattern generated by each method, next we describe the geometric/statistical steps which are followed in “Method

*M*

_{2}” section in order to obtain results:

- 1.
Each of the 13 methods

*M*_{ j }(where*j*= 1 ÷ 13, i.e.:*j*ranges from 1 to 13) shown in “Introduction” section generates a particular point pattern on the plane (also called*M*_{ j }) which is made up by 22 points \(\left\{ {u_{i,j} } \right\}_{i = 1}^{22}\). More specifically, in “Geometric and Statistical Control’’ section we present the exact analytical expressions for the coordinates of these 22 vertices \(\left\{ {u_{i,j} } \right\}_{i = 1}^{22}\) for each method \(\left\{ {M_{j} } \right\}_{j = 1}^{13}\). - 2.
For each of the vertices

*v*_{ i }in Table 1, we will rotate the rose window around point (0, 0) until the coordinates of that vertex are (x, 0), i.e., until that vertex reaches the positive x-semiaxis. This rotation repositions the remaining vertices and generates a particular pattern of 22 points \(\left\{ {v_{i,k} } \right\}_{k = 1}^{22}\), where \(k = 1 \div 22\), on the plane. We will call this pattern*R*_{ i }. However, since this rotation is made for each vertex*v*_{ i }, we will obtain a total of 22 different point patterns \(\left\{ {v_{i,k} } \right\}_{i,k = 1}^{i,k = 22}\) on the plane. We will call these patterns*R*_{ i }, where*i*= 1 ÷ 22. - 3.Next, after choosing the pattern
*M*_{ j }generated by one of the 13 methods and choosing one of the 22 rose window patterns*R*_{ i }, we will compare both geometric patterns by calculation of the following parameters:- (a)The probability
*P*of linear correlation between pattern*M*_{ j }and pattern*R*_{ i }. More precisely, we will calculate the probability of rejecting the null hypothesis*h*_{0}= “There is no linear correlation between pattern*M*_{ j }and pattern*R*_{ i }”. In order to calculate this probability*P*, we apply Student’s*t*test with 22 − 2 degrees of freedom. The second column in Table 2 shows these probability values calculated for pattern*M*_{1}and pattern*R*_{ i }(where*i*ranges from 1 to 22), together with the resulting mean. So as not to extend this paper, for each of the following methods we will only present the respective means of parameter*P*on the second column of Table 3Table 2Parametric values corresponding to the comparison between pattern

*M*_{1}and the patterns*R*_{ i }*i**P**η*_{adj}^{2}× 100*d*1

0.99982

99.96267

0.0037

2

0.99978

99.95385

0.0034

3

0.99980

99.95900

0.0032

4

0.99981

99.96065

0.0030

5

0.99976

99.94970

0.0030

6

0.99990

99.98050

0.0028

7

0.99986

99.97256

0.0029

8

0.99979

99.95665

0.0032

9

0.99976

99.95035

0.0035

10

0.99971

99.93942

0.0038

11

0.99974

99.94711

0.0041

12

0.99973

99.94394

0.0044

13

0.99982

99.96360

0.0047

14

0.99975

99.94859

0.0050

15

0.99981

99.96056

0.0051

16

0.99972

99.94231

0.0053

17

0.99977

99.95346

0.0052

18

0.99977

99.95268

0.0051

19

0.99960

99.91706

0.0051

20

0.99964

99.92579

0.0048

21

0.99967

99.93170

0.0045

22

0.99961

99.92016

0.0042

Means

0.99976

99.94965

0.0041

Table 3Methods ranked in descending order according to the comparison results

Methods

*P**η*_{adj}^{2}× 100*d**M*_{12}0.99996

99.9922

0.00073

*M*_{13}0.99995

99.9913

0.00075

*M*_{11}0.99995

99.9895

0.00077

*M*_{4}0.99990

99.9808

0.00209

*M*_{3}0.99987

99.9745

0.00222

*M*_{8}0.99987

99.9735

0.00269

*M*_{6}0.99978

99.9554

0.00323

M

_{1}0.99976

99.9496

0.00413

*M*_{5}0.99973

99.9438

0.00387

*M*_{9}0.99947

99.8907

0.00658

*M*_{7}0.99911

99.8141

0.00845

*M*_{2}0.99887

99.7642

0.00964

*M*_{10}0.99664

99.2966

0.01772

- (b)
The extent to which one pattern

*M*_{ j }accounts for the other pattern*R*_{ i }. This extent is calculated by means of Pearson’s adjusted coefficient of determination and Pearson’s coefficient of determination. More precisely, by means of the above-mentioned correlation we will find out to what extent pattern*M*_{ j }accounts for pattern*R*_{ i }. This extent is calculated by means of \(\eta_{adj}^{2} \times 100\) , where \(\eta_{adj}^{2} = 1 - \left( {1 - \eta^{2} } \right)\frac{22 - 1}{22 - 2 - 1}\),*η*is Pearson’s coefficient of determination, and*η*_{ adj }is Pearson’s adjusted coefficient of determination. The third column in Table 2 shows the adjusted coefficients calculated for pattern*M*_{1}and pattern*R*_{ i }(where*i*ranges from 1 to 22), together with the resulting mean. So as not to extend this paper, for each of the following methods we will only present the respective means of the adjusted coefficient on the third column of Table 3. - (c)
The regression deviation. More precisely, we will calculate the deviation

*d*between the regression line (the straight line showing the linear correlation between pattern*M*_{ j }and pattern*R*_{ i }) and the first and third quadrants bisector. The equation of such bisector is*y*=*x*, and the equation of the regression line for patterns*M*_{ j }and*R*_{ i }is*y*=*b*+*xtanθ*. If the architect used method*M*_{ j }to design the rose window, then the maximum deviation*d*between both straight lines within the parameter range of the problem considered here (vertex coordinates for patterns*M*_{ j }) must be small. More specifically, being \(d = \hbox{max} _{{x \in \left[ { - 1,1} \right]}} \left| {b + x\tan \theta - x} \right|\), we can see that the more similar are pattern*M*_{ j }and pattern*R*_{ i }, the closer is*d*to zero. Parameters*b*and*θ*are determined using the classical means, variances and covariances calculation. The fourth column in Table 2 shows this deviation*d*for pattern*M*_{1}and pattern*R*_{ i }(where*i*ranges from 1 to 22), together with the resulting mean. So as not to extend this paper, for each of the following methods we will only present the respective means of the deviation*d*on the fourth column of Table 3.

- (a)
- 4.
These calculations result into 13 × 22 = 286 geometric comparisons with their respective parametric values

*P*, \(\eta_{adj}^{2} \times 100\) and*d*(13 methods*M*_{ j }× 22 patterns*R*_{ i }). Table 3 shows these results.

## Analytical Expressions and Results

### Analytical Expressions of the Patterns *M* _{ j }

In this subsection we present the analytical expressions of the angles and the vertex coordinates in patterns *M* _{ j }, where *j* = 1 ÷ 13 (these are the patterns which are generated by the methods stated in “Introduction” section).

### Method *M* _{1}

If we use method *M* _{1} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = arcsin\frac{7}{25}\) and \(\beta = \pi - 10arcsin\frac{7}{25}\). The coordinates of the 22 vertices are: *u* _{1,1} = (1, 0); *u* _{ n+1,1} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,1} = \left( {cos\left( {20\alpha + \beta } \right), sin\left( {20\alpha + \beta } \right)} \right)\).

As stated in “Approximate Methods For Constructing an Icosikaidigon” section, Table 2 shows the parameters corresponding to the comparisons between pattern *M* _{1} and the patterns *R* _{ i }, where *i* = 1 ÷ 22.

Table 3 shows only the means of the parametric values corresponding to the comparison between pattern *M* _{ j } and the patterns *R* _{ i }, *i* = 1 ÷ 22, *j* = 1 ÷ 13, as stated previously.

### Method **M** _{ 2 }

**M**

If we use method *M* _{2} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = arcsin\frac{2}{7}\) and \(\beta = \pi - 10arcsin\frac{2}{7}\). The coordinates of the 22 vertices are: *u* _{1,2} = (1, 0); *u* _{ n+1,2} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,2} = \left( {\cos \left( {20\alpha + \beta } \right), \sin \left( {20\alpha + \beta } \right)} \right)\).

### Method **M** _{ 3 }

**M**

Method *M* _{3} results into an icosikaidigon having 21 angles *α* plus 1 angle *β*, where (2*π* = 21*α* + *β*). The analytical equations of these angles are: \(\alpha = 2arcsin\frac{1}{7}\) and \(\beta = 2\pi - 42arcsin\frac{1}{7}\). The coordinates of the 22 vertices are: *u* _{1,3} = (1, 0) and *u* _{ n+1,3} = (cos *nα*, sin *nα*), where *n* = 1 ÷ 21.

### Method **M** _{ 4 }

**M**

If we use method *M* _{4} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles *β*, where (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = \frac{1}{2}arctan\frac{16}{25}\) and \(\beta = \pi - 5arctan\frac{16}{25}\). The coordinates of the 22 vertices are: *u* _{1,4} = (1, 0); *u* _{ n+1,4} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,4} = \left( {\cos \left( {20\alpha + \beta } \right), \sin \left( {20\alpha + \beta } \right)} \right)\).

### Method *M* _{5}

Method *M* _{5} results into an icosikaidigon having 21 angles *α* plus 1 angle *β*, where (2*π* = 21*α* + *β*). The analytical equations of these angles are: \(\alpha = 2arctan\frac{1}{7}\) and \(\beta = 2\pi - 42arctan\frac{1}{7}\). The coordinates of the 22 vertices are: *u* _{1,5} = (1, 0) and *u* _{ n+1,5} = (cos *nα*, sin *nα*), where *n* = 1 ÷ 21.

### Method *M* _{6}

Method *M* _{6} results into an icosikaidigon having 21 angles *α* _{ i } plus 1 angle *β*, where (2*π* = ∑ _{ i=1} ^{21} *α* _{ i } + *β*). The analytical equations of these angles are: \(\alpha_{i} = \frac{17\pi }{66} - 6arctan\left( {\frac{{2\sin \frac{\pi }{24}}}{{2\cos \frac{\pi }{24} + 1}}} \right) + \varepsilon_{i}\) and \(\beta = 126arctan\frac{{2\sin \frac{\pi }{24}}}{{2\cos \frac{\pi }{24} + 1}} - \frac{75\pi }{22} - 21\varepsilon_{i}\). In these equations we have assumed that the error made by the sculptor when visually determining the angular difference \(\Delta = \frac{\uppi}{132} {\text{rad}} = 1.\widehat{36}^{{^\circ }}\) is \(\varepsilon_{i} = \frac{0.1\pi }{180}\). Therefore, the analytical expressions are as follows: \(\alpha = \frac{17\pi }{66} - 6arctan\left( {\frac{{2\sin \frac{\pi }{24}}}{{2\cos \frac{\pi }{24} + 1}}} \right) + \frac{0.1\pi }{180}\) and \(\beta = 126arctan\left( {\frac{{2\sin \frac{\pi }{24}}}{{2\cos \frac{\pi }{24} + 1}}} \right) - \frac{75\pi }{22} - 21\frac{0.1\pi }{180}\). The coordinates of the 22 vertices are: *u* _{1,5} = (1, 0) and *u* _{ n+1,5} = (cos *nα*, sin *nα*), where *n* = 1 ÷ 21.

### Method *M* _{7}

If we use method *M* _{7} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles *β*, where (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = \frac{\pi }{4} - \frac{1}{2}\left( {arcsin\frac{7}{{2\sqrt {103} }} + arcsin\frac{21}{{2\sqrt {309} }}} \right)\) and \(\beta = - \frac{3}{2}\pi + 5\left( {arcsin\frac{7}{{2\sqrt {103} }} + arcsin\frac{21}{{2\sqrt {309} }}} \right)\). The coordinates of the 22 vertices are: *u* _{1,6} = (1, 0); *u* _{ n+1,6} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,6} = \left( {\cos \left( {20\alpha + \beta } \right), \sin \left( {20\alpha + \beta } \right)} \right)\).

### Method *M* _{8}

If we use method *M* _{8} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles *β*, where (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = arctan\frac{1}{{\sqrt 3 + 4\sin \left( {\frac{\pi }{6} - \frac{1}{4}arctan\frac{1}{1 + \sqrt 3 }} \right)}}\) and \(\beta = \pi - 10arctan\frac{1}{{\sqrt 3 + 4\sin \left( {\frac{\pi }{6} - \frac{1}{4}arctan\frac{1}{1 + \sqrt 3 }} \right)}}\). The coordinates of the 22 vertices are: *u* _{1,7} = (1, 0); *u* _{ n+1,7} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,7} = \left( {\cos \left( {20\alpha + \beta } \right), \sin \left( {20\alpha + \beta } \right)} \right)\).

### Method *M* _{9}

If we use method *M* _{9} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles *β*, where (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = arcsin\left( {\sqrt 2 \sin \left( {\frac{1}{2}arcsin\frac{\sqrt 6 }{4} - \frac{1}{24}\pi } \right)} \right)\); \(\beta = \pi - 10arcsin\left( {\sqrt 2 \sin \left( {\frac{1}{2}arcsin\frac{\sqrt 6 }{4} - \frac{1}{24}\pi } \right)} \right)\). The coordinates of the 22 vertices are: *u* _{1,8} = (1, 0); *u* _{ n+1,8} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,8} = \left( {\cos \left( {20\alpha + \beta } \right), \sin \left( {20\alpha + \beta } \right)} \right).\)

### Method **M** _{ 10 }

**M**

If we use method *M* _{10} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 20 angles *α* plus 2 angles *β*, where (2*π* = 20*α* + 2*β*). The analytical equations of these angles are: \(\alpha = arcsin\frac{\sqrt 3 }{6}\) and \(\beta = \pi - 10arcsin\frac{\sqrt 3 }{6}\). The coordinates of the 22 vertices are: *u* _{1,9} = (1, 0); *u* _{ n+1,9} = (cos *nα*, sin *nα*) where *n* = 1 ÷ 20 and \(u_{22,9} = \left( {\cos \left( {20\alpha + \beta } \right), \sin \left( {20\alpha + \beta } \right)} \right)\).

### Method *M* _{11}

If we use method *M* _{11} for constructing a hendecagon, after bisecting each angle we obtain an icosikaidigon having 18 angles *α* plus 4 angles *β*, where (2*π* = 18*α* + 4*β*). The analytical equations of these angles are: \(\alpha = arcsin\left( {\frac{8\sqrt 3 }{7}\cos \left( {2arccos\frac{2}{\sqrt 7 }} \right)} \right)\) and \(\beta = \frac{\pi }{2} - \frac{9}{2}arcsin\left( {\frac{8\sqrt 3 }{7}\cos \left( {2arccos\frac{2}{\sqrt 7 }} \right)} \right)\). The coordinates of the 22 vertices are: *u* _{1,10} = (1, 0); *u* _{ n+1,10} = ( cos *nβ*, sin *nβ*) where *n* = 1 ÷ 2; \(u_{n + 3,10} = \left( {\cos \left( {2\beta + n\alpha } \right),\sin \left( {2\beta + n\alpha } \right)} \right)\) where *n* = 1 ÷ 10; *u* _{ n+11,10} = (*cos*(*nβ* + 10*α*), *sin*(*nβ* + 10*α*)) where *n* = 3 ÷ 4 and *u* _{ n+5,10} = (*cos*(4*β* + *nα*), *sin*(4*β* + *nα*)) where *n* = 11 ÷ 17.

### Method **M** _{ 12 }

**M**

Method *M* _{12} results into an icosikaidigon having 14 angles *α* plus 8 angles *β*. The analytical equations of these angles are: \(\alpha = \pi - 4arctan\frac{\sqrt 3 }{2}\) and \(\beta = - \frac{3\pi }{2} + 7arctan\frac{\sqrt 3 }{2}\). The coordinates of the 22 vertices are: \(u_{1,11} = \left( {1, 0} \right);u_{n + 1,11} = \left( { cos n\beta , sin n\beta } \right)\) where *n* = 1 ÷ 2; \(u_{n + 2,11} = \left( {\cos \left( {\beta + n\alpha } \right),\sin \left( {\beta + n\alpha } \right)} \right)\) where *n* = 2 ÷ 4; \(u_{n + 3,11} = \left( {\cos \left( {2\beta + n\alpha } \right), \sin \left( {2\beta + n\alpha } \right)} \right)\) where *n* = 4 ÷ 5; \(u_{n + 4,11} = \left( {\cos \left( {3\beta + n\alpha } \right), \sin \left( {3\beta + n\alpha } \right)} \right)\) where *n* = 5 ÷ 7; \(u_{n + 5,11} = \left( {cos\left( {4\beta + n\alpha } \right), sin\left( {4\beta + n\alpha } \right)} \right)\) where *n* = 7 ÷ 9; \(u_{n + 6,11} = \left( {\cos \left( {5\beta + n\alpha } \right), \sin \left( {5\beta + n\alpha } \right)} \right)\) where *n* = 9 ÷ 11; \(u_{n + 7,11} = \left( {\cos \left( {6\beta + n\alpha } \right), \sin \left( {6\beta + n\alpha } \right)} \right)\) where *n* = 11 ÷ 12 and \(u_{n + 8,11} = \left( {\cos \left( {7\beta + n\alpha } \right), \sin \left( {7\beta + n\alpha } \right)} \right)\) where *n* = 12 ÷ 14.

### Method **M** _{ 13 }

**M**

Method *M* _{13} results into an icosikaidigon having 15 angles *α* plus 7 angles *β*. The analytical equations of these angles are: \(\alpha = - \frac{3\pi }{2} + 7arctan\frac{\sqrt 3 }{2}\) and \(\beta = \frac{7\pi }{2} - 15arctan\frac{\sqrt 3 }{2}\). The coordinates of the 22 vertices are: \(u_{1,12} = \left( {1, 0} \right);u_{2,12} = \left( {\cos \beta , \sin \beta } \right);u_{n + 2,12} = \left( {\cos \left( {\beta + n\alpha } \right), \sin \left( {\beta + n\alpha } \right)} \right)\); where *n* = 1 ÷ 2; \(u_{n + 3,12} = \left( {\cos \left( {2\beta + n\alpha } \right), \sin \left( {2\beta + n\alpha } \right)} \right)\) where *n* = 2 ÷ 4; \(u_{n + 4,12} = \left( {\cos \left( {3\beta + n\alpha } \right), \sin \left( {3\beta + n\alpha } \right)} \right)\) where *n* = 4 ÷ 6; \(u_{n + 5,12} = \left( {\cos \left( {4\beta + n\alpha } \right), \sin \left( {4\beta + n\alpha } \right)} \right)\) where *n* = 6 ÷ 8; \(u_{n + 6,12} = \left( {\cos \left( {5\beta + n\alpha } \right), \sin \left( {5\beta + n\alpha } \right)} \right)\) where *n* = 8 ÷ 10; \(u_{n + 7,12} = \left( {\cos \left( {6\beta + n\alpha } \right), \sin \left( {6\beta + n\alpha } \right)} \right)\) where *n* = 10 ÷ 12 and \(u_{n + 8,12} = \left( {\cos \left( {7\beta + n\alpha } \right), \sin \left( {7\beta + n\alpha } \right)} \right)\) where *n* = 12 ÷ 14.

## Results

Table 3 below ranks the methods in descending order according to the means of their respective Pearson’s adjusted coefficients of determination. From a statistical point of view, using the means only makes sense if they are very representative; this quality is measured with Pearson’s coefficient of variation *CV*. As a standard, from a statistical point of view, the mean is considered to be representative if *CV* < 25 %. After making the relevant calculations with the above-mentioned parameters, we can see that all the *CV*s obtained for the means of the adjusted coefficients \(\eta_{adj}^{2} \times 100\) are less than 0.063 %. This very high degree of representativeness enables us to establish a ranking in Table 3, which presents an optimal comparison between each pattern and the rose window. Table 3 shows the means of the parametric values corresponding to the comparison between pattern *M* _{ j } and the patterns *R* _{ i }, where *i* = 1 ÷ 22 and *j* = 1 ÷ 13.

On the basis of the above-mentioned geometric/statistical results, Table 3 shows that the most approximate comparison is the one made with pattern *M* _{12}.

## Conclusion

After this geometrically and statistically thorough analysis, we can say that method M_{12} is the one which best suits the design of this rose window, which is based on a 22-sided polygon.

*M*

_{4}(fourth position in the ranking of Table 3). As for the other group of methods (see “Methods That We Are Not Certain Were Known in the Middle Ages” section), in our paper we show that, despite the absence of historical records and graphic documents showing direct applications of method

*M*

_{12}in medieval architecture, a stone document has survived the geometry of which mirrors almost exactly the geometry offered by method

*M*

_{12}(Fig. 11). This stone document is the cathedral’s rose window itself. This geometric and statistical evidence, together with the constructive simplicity of this method, leads us to claim that Andrea di Cione used method

*M*

_{12}to design and build the rose window in Orvieto cathedral.

These numeric methods supplement the epistemological knowledge about the geometric techniques which were used in Gothic architecture.

## Notes

### Acknowledgments

We would like to thank Dr. Giuseppe della Fina, Director of the Claudio Faina Museum in Orvieto; Laura Andreani, Secretary-General of the Archivio dell’Opera del Duomo di Orvieto; and the firm Leica Microsystems.

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