| The Parthenon Peristasis | |

Ates Gulcugil

Abstract
Two golden ratio models will be constructed for the peristasis of the Parthenon and their dimensions compared with the actual one.

Definitions
Φ: The golden ratio, 1.618...
Golden Numbers: Integer powers of Φ.
Interval: Distance between two, neighboring, parallel line segments.
Golden Interval: An interval which is a golden number.
Normalizing: Dividing each dimension of a structure by a selected one of its dimensions.
Aspect Ratio: Ratio of longer-to-shorter side of a rectangle.

The Parthenon Peristasis
The dimensions (in feet) of the Parthenon peristasis as measured by Francis Penrose are shown in the following diagram.

There are two different kinds of intervals in the peristasis: The corner spaces (total 8), and the intercolumnia (total 38). These are shown below.

Substituting the (scaled up by 101.361/101.341) intervals of the left hand side for the unmeasured right hand side, the mean values are figured out as:
Mean corner space: 15.448 ft
Mean intercolumnium: 14.090 ft

The peristasis with the mean values is shown below.

Golden Ratio Relations
If the Parthenon was designed around the golden ratio there must be a golden ratio relationship between the corner space and the intercolumnium. The following diagram shows the analysis of these two intervals. The intercolumnium, because it is uninterrupted, will be considered as a golden interval, (interval a). When normalized with respect to the intercolumnium the corner space is 1.096 (i). Since 1.096 is not a golden interval it must be composed of several golden intervals in order for the peristasis to be a golden ratio design. The distance of the corner column’s center to the near edge of the stylobate manifests itself as one of these intervals (interval d) (ii). This interval is also present on the right hand side of the corner column’s center. The remaining uninterrupted space will be called interval b (iii). So, the corner space is the sum of three intervals: d+d+b. It remains to find out which golden numbers the values of b and d are. The radius of the corner column, as given by Penrose, is 3.194 ft. Interval d must be larger than this value because the perimeter of the corner column does not touch the edge of the stylobate. The radius, 3.194, divided by interval a, 14.090, is 0.227. So interval d must be 0,236, which is Φ^-3. Interval b can now be calculated as 1.096 - 2x0.236 = 0.624, which is very nearly 0.618, Φ^-1 (iv). The three golden intervals of the peristasis will be taken as: a= Φ^0, b= Φ^-1 and c= Φ^-3 (iv).

The model peristasis with formulas and numerical values is as follows.

Comparison
If the golden ratio model is uniformly scaled up so as to make its width equal to the width of the actual peristasis (228.156 ft), its dimensions are as shown below.

In the following table, dimensions of the golden ratio model is compared with Penrose measurements.

| Penrose(ft) | GoldenRatioModel(ft) | Absolute Difference(ft) | Percent Difference | Corner Spaces | 15.448 | 15.370 | 0.078 | 0.50 | Intercolumnia | 14.090 | 14.101 | 0.011 | 0.08 | Width of Stylobate | 228.156 | 228.156 | - | - | Breadth of Stylobate | 101.346 | 101.246 | 0.100 | 0.10 | Aspect Ratio | 2.2513 | 2.2534 | 0.002 | 0.09 |

Percent Difference = 100 x Absolute Difference / Penrose value

Producing Golden Intervals
The following procedure, and diagram, shows a way by which the ancient architects might have produced the golden intervals necessary for their buildings.

i- A square of side length a is constructed. ii- Around midpoint of base, a circle with radius equal to the diagonal of the half of the square is drawn.intersecting the extent of the baseline. The distance of this intersection point to the near corner of the square is b. iii- a-b = c iv- b-c = d

Φb=a, Φc=b, Φd=c, Φ
Two adjacent golden intervals add up to the next larger golden interval. Similarly, the difference of two adjacent golden intervals is the next smaller one. More golden intervals can be produced using this knowledge.

Application of the Golden Ratio in the Parthenon Construction
It is probable that the 5x14 grid, shown below, formed by the lines joining the centers of the columns was used for determining the location of items on the stylobate.

This grid, marked on the stylobate by stretched strings, or with chalk, together with wooden (or metal) bars cut to the lengths of golden intervals were used to determine the locations of the walls, steps, column centers etc. Items placed in this manner would always be golden intervals away from each other enhancing the consistency of the whole structure.

A Simpler Method
The unifiying quality of the golden ratio is due to its addition property. That is, Φ^n being the sum of Φ^(n-1) and Φ^(n-2). It is this property that makes the golden ratio different from any other ratio like 1.75, or 1.50 etc. It is because of this property that in a temple built with the golden ratio one sees intervals which support each other by way of being the same or adding up to the same lengths. Ancient Greeks surely were enchanted by this and applied the addition principle to integer numbers to obtain a series of numbers what is today called, the Fibonacci Numbers.

The Parthenon Peristasis with Fibonacci Numbers
The intervals of the Parthenon peristasis in Fibonacci numbers are shown below.

The following diagram shows the Fibonacci model for the peristasis.

The scaled up Fibonacci model is as follows.

In the table below, dimensions of the Fibonacci model is compared with Penrose measurements.

| Penrose(ft) | FibonacciModel(ft) | Absolute Difference (ft) | Percent Difference | Corner Spaces | 15.448 | 15.434 | 0.014 | 0.09 | Intercolumnia | 14.090 | 14.092 | 0.002 | 0.0001 | Width of Stylobate | 228.156 | 228.156 | - | - | Breadth of Stylobate | 101.346 | 101.328 | 0.018 | 0.018 | Aspect Ratio | 2.2513 | 2.2517 | 0.0004 | 0.018 |

Advantages and Disadvantages
-Employing Fibonacci numbers instead of golden numbers as intervals has the advantage of using only one measure, thus doing away with golden interval preparations.
- In the beginning of the Fibonacci series, ratio of adjacent numbers is not equal to the golden ratio. But this ought to have no negative effect since addition property is there all the time.

Which Numbers to Choose
The choice of the Fibonacci numbers 21. 13 and 5 is because this set by far yields the best result according to the Penrose measurements. The following table compares the aspect ratios of several choices of sets with the actual one.

Set of Fibonacci Numbers | Aspect Ratio | Penrose Aspect Ratio | Absolute Difference | 55,34,13 | 2.2531 | 2.2513 | 0.0018 | 34,21,8 | 2.2540 | “ | 0.0027 | 21,13,5 | 2.2516 | “ | 0.0004 | 13,8,3 | 2.2580 | “ | 0.0067 | 8,5,2 | 2.2413 | “ | 0.0100 |

Conclusion
Although there is about fivefold increase of precision with the Fibonacci model compared to the golden ratio model, it’s not easy to say which one was used in the actual design. The Parthenon stylobate being measured with high precision does not guarantee that it was constructed with the same. The following tables, which show the discrepencies of the corner spaces and the columnia from the mean values give an idea of how much precision was exercised in the building. According to these results the golden ratio model is eligible to be used for the Parthenon.

CornerSpace (ft) | Mean CornerSpace (ft) | AbsoluteDifference (ft) | PercentDifference | 15.531 | 15.448 | 0.083 | 0,540 | 15.443 | “ | 0.005 | 0,030 | 15.468 | “ | 0.020 | 0,132 | 15.449 | “ | 0.001 | 0,009 | 15.478 | “ | 0.030 | 0,197 | 15.367 | “ | 0.081 | 0,522 | 15.478 | “ | 0.030 | 0,197 | 15.367 | “ | 0.081 | 0,522 |

Intercolumnium(ft) | Mean Intercolumnium (ft) | AbsoluteDifference(ft) | PercentDifference | 13.983 | 14.090 | 0,107 | 0,759 | 14.052 | “ | 0,038 | 0,270 | 14.124 | “ | 0,034 | 0,241 | 14.110 | “ | 0,020 | 0,142 | 14.079 | “ | 0,011 | 0,078 | 14.093 | “ | 0,003 | 0,021 | 14.088 | “ | 0,002 | 0,014 | 14.094 | “ | 0,004 | 0,028 | 14.066 | “ | 0,024 | 0,170 | 14.113 | “ | 0,023 | 0,163 | 14.089 | “ | 0,001 | 0,007 | 14.068 | “ | 0,022 | 0,156 | 14.124 | “ | 0,034 | 0,241 | 14.084 | “ | 0,006 | 0,042 | 14.060 | “ | 0,030 | 0,213 | 14.109 | “ | 0,019 | 0,135 | 14.103 | “ | 0,013 | 0,092 | 14.075 | “ | 0,015 | 0,106 | 14.111 | “ | 0,021 | 0,149 | 14.084 | “ | 0,006 | 0,043 | 14.090 | “ | 0.000 | 0.000 | 14.064 | “ | 0,026 | 0,185 | 14.084 | “ | 0,006 | 0,043 | 14.087 | “ | 0,003 | 0,021 | 14.114 | “ | 0,024 | 0,170 | 14.056 | “ | 0,034 | 0,241 | 14.062 | “ | 0,028 | 0,199 | 14.141 | “ | 0,051 | 0,362 | 14.078 | “ | 0,012 | 0,085 | 14.106 | “ | 0,016 | 0,114 | 14.113 | “ | 0,023 | 0,163 | 14.084 | “ | 0,006 | 0,043 | 14.115 | “ | 0,025 | 0,177 |

The dimensions of the Parthenon taken from the book: An Investigation of the Principles of Athenian Architecture by Francis Cranmer Penrose, 1888, published by The Society of Dilettanti
Web address: http://digi.ub.uni-heidelberg.de/diglit/penrose1888/0002?sid=a8d10fa70d18f77e68edbfbe99de1d7d

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