# Three Old Arches

In August 2022 the Daily Mail published a story which reported that a "scientific" study "has revealed that the walled cathedral city, in the northwest of England, has the highest percentage of buildings - 83.7 percent - that align with the 'Golden Ratio'." Venice came second, and London third. This was presented as evidence that Chester was the most beautiful city in the world.

Back in November 2021 Cheshire Live posted essentially the same story and provided some further information on "the most beautifull buildings in Britain" based on a very similar study. Chester had the second highest scoring building in the study, Three Old Arches, in Bridge Street which scored "99%". The 13th Century stone frontage at the street and row levels of No. 48 is also considered to be the earliest shop front still surviving in England.

The "judge" is this supposedly "scientific" study was a computer program which compared the dimensions of the front-facing profile of the building (as it appeared in Google Street View) with the "Golden ratio" and used this as a test of "beauty".

### The Golden Ratio

The ratio is the number x such that 1/x = x-1. It can also be written in the form X^{2} = x + 1. It also happens to be close to the ratio of the lengths of the sides of a standard proportions credit card (85.6 mm x 53.98 mm).

The Golden Ratio. is commonly represented by the Greek letter phi ( ϕ ). Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry. It is sometimes said that the 5th-century BC mathematician Hippasus of Metapontum discovered that the Golden Ratio was neither a whole number nor a fraction (and therefore an "irrational" number), surprising Pythagoreans. Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the Golden Ratio, and contains its first known definition. The Golden Ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers. Luca Pacioli named his book *Divina proportione* (1509) after the ratio, and explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea ('golden section'). German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the Golden Ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) Golden Ratio was stated as "about 0.6180340.." in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. In fact the ratio has an infinite number of decimals without any repeating pattern.

#### In Mathematics and Nature

The ratio has many interesting mathematical properties. For example, it is defined in a unique way by a very simple equation. Like the number pi it cannot be written down exactly as a fraction using whole numbers, but only as an approximation. However it can be calculated to any degree of precision required.

There is also a special relationship between the Golden Ratio and Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, ... etc, each number is the sum of the two numbers before it). When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio and gets closer as the numbers get bigger.

There are also some "fluke" instances of the Golden Ratio - for example the conversion factor 1.609344 for statute miles to kilometers is close to the golden ratio, being only about 0.5% out. However this is pure co-incidence, the English mile is related to how long a furrow is when ploughed by pulling it with an ox, and the meter was originally 1/10,000,000 of the distance from north pole to equator along the meridian through Paris.

The Fibonacci Numbers also are often said to appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone's bracts. This was first noticed by Kepler. The reason why this type of mathematics turns up in nature is that it is the result of a simple process which are repeated to produce a more complex structure. However finding the exact Golden Ratio in nature is difficult: while the shells of molluscs are often said to have a spiral influenced by the Golden Ratio the actual measurements of shells does not support this claim.

#### In Art

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as

**"rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."**

The idea that the Golden Ratio is somehow connected to visual beauty is frequently repeated in "popular science" books and articles. These frequently refer to the "Divina proportione" (Divine proportion), the aforementioned three-volume work by Luca Pacioli, which was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. Divina proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the Golden Ratio's application to yield pleasing, harmonious proportions, Mario Livio points out (in "The Golden Ratio: The Story of Phi, the World's Most Astonishing Number" (2002)) that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions that can be expressed as exact fractions of whole numbers, which is quite different to the Golden Ratio.

Psychologist Adolf Zeising brought a resurgence in the popularity of the Golden Ratio in the 1854, writing of a universal law for “beauty and completeness in the realms of both nature and art”, after noting the Golden Ratio’s appearance in plants. Publications since have labeled many of these claims as simply misconceptions. Another example of this myth is the claim that the Golden Ratio appears in the proportions of the Parthenon, part of the Acropolis in Athens. There is no evidence of this in Greek scholarship, and the idea that the Parthenon has proportions given by the Golden Ratio only dates back to the 1850s. Furthermore, the actual measurements of the Parthenon do not give proportions especially close to the golden ratio, unless you are selective with your choice of rectangles. In fact, the Parthenon takes its harmonious appearance from the clever deployment of lines that look parallel but in fact converge or curve, so it's virtually impossible to take measurements precise enough to give exact ratios. As the proportions of the Parthenon vary with its height it is simply not possible to find an overall proportion that agrees with the Golden Ratio.

Livio states:

**"literature is bursting with false claims and misconceptions about the appearance of the Golden Ratio in the arts (e.g. in the works of Giotto, Seurat, Mondrian). The history of art has nevertheless shown that artists who have produced works of truly lasting value are precisely those who have departed from any formal canon for aesthetics. In spite of the Golden Ratio's truly amazing mathematical properties, and its propensity to pop up where least expected in natural phenomena, I believe that we should abandon its application as some sort of universal standard for ‘beauty,’ either in the human face or in the arts."**

One possible explanation for the association of the ratio with beauty is that the dimensions of the human face are very roughly related to the ratio. The human brain has a highly developed mechanism (the Fusiform face area) which is involved in spotting human faces, and indeed can often be fooled into seeing faces in clouds, rock formations etc. Research has shown that an optimally attractive face has certain ratios of measurements which are close to the Golden Ratio, but not quite identical to it. The proportions found attractive also vary between different racial groups. Maybe it is simply co-incidence that attractive facial proportions are close to a mathematically unique number and research is still ongoing.

### In Chester

The building plots on the four main streets of Chester were originally so-called "Burgage Plots". These had a narrow street frontage which could be used for commercial purposes but could run back from it for some distance. Similar patterns of land use are found in many of the walled "burh" town layouts which date from the time of Alfred the Great. The basic unit of measurement was the perch which was 5.5 yards (5.03 m) and the plots can be identified today because they are in multiples of perches. In Chester a different unit may have been used but the point is that within any particular burh the unit would have been a standard one.

As for the height of buildings this was also somewhat standardised with an undercroft at street level, a row level above that and a further upper flow below a roof space which later became attics. There is a reasonably well-preserved example at Leche House. In Chester the presence of The Rows at an essentially standardised continuous level further constrains the height range encountered.

It is possible that one consequence of this common building design (row height and burgal width) has been to make the average proportions of the buildings in Chester similar, and already quite close to the Golden Ratio. Does this make Chester especially beautiful? - not of itself. The Golden Ratio is beautiful to mathematicians, but it is a number which simply falls out of mathematics, and in this case "beauty is in the AI of the beholder".