My previous post (a while ago now sorry – Christmas got in the way!) was about patterns in nature, but if you live in a city you can see patterns all around you too. Humans have taken inspiration from nature since the beginning of time: just as there’s geometry in nature with patterns in crystals, plants and skies, humans have also used geometric patterns to style their habitats throughout the ages…leading me to discuss the application of maths in another of my favourite topics: beauty in architecture & urban environments!
In Egyptian times, you can see how geometry was first brought into man-made structures, using the strength of triangles to create pyramids that still survive to this day:
Another shape with great strength is a cylinder, or in architecture: a column or pillar. Columns and arches make the strongest structures – they transmit, through compression, the weight of the structure above to other structural elements below. The Egyptians used columns inside buildings for the practical purpose of holding up the roof inside a building, but the Ancient Greeks, followed by the Romans, also loved to use them on the outside as well – introducing the use of geometric shapes for aesthetic purposes as well as practical. Greek architecture is characterised by the extensive use of columns on the interior and exterior of buildings, in buildings like the Parthenon. They also continued to employ other geometrical shapes, for example triangular pediments:
In Roman architecture, their use of pillars and arches is exemplified in their aqueducts, the Colosseum and in temples (Maison Carrée, one of the best preserved Roman temples):
The design of most classical columns incorporates entasis (the inclusion of a slight outward curve in the sides) plus a reduction in diameter along the height of the column, so that the top is as little as 83% of the bottom diameter. This reduction mimics the parallax effects which the eye expects to see, and tends to make columns look taller and straighter than they are. Different types of column use this feature to different extents:
- Doric columns are the oldest and simplest, with neither a base nor a detailed capital. It was used in the bottom level of the Colosseum and the Parthenon, and was therefore considered to be able to hold more weight, with a typical height-to-thickness ratio of about 8:1.
- Ionic columns are often fluted (grooves carved up its length), and features a scroll ornament at the top 4 corners (often used for academic buildings). The height-to-thickness ratio is around 9:1. These were used on the second level of the Colosseum.
- Corinthian columns (associated with the Greek city-state of Corinth) were used on the top level of the Colosseum and hold up the least weight, with the slenderest ratio of thickness to height (about 10:1). They are much more decorative at the top.
There are also examples of patterned geometrical shapes being used in comparatively recent times, in the symmetry of Baroque (St Paul’s Cathedral), Gothic (Milan Cathedral), and Tudor architecture:
Fast forwarding to today, geometric patterns are just as prevalent despite them no longer being as necessary for strength (thanks to advances in structural engineering and materials), but they continue to please the eye. Buildings with strong geometrical features such as the Gherkin and Kings Cross station in London, Abi Dhabi airport, the Sydney Opera House and art deco Chrysler Building are shown below:
Humans have also used patterns and symmetry in creating larger-scale urban environments, for example in designing city plans and transport networks. Hexagons are a classic model for spatial city designs, allowing for maximised accessibility of facilities, ideally located equal distances apart, in a hierarchical structure connected by roads:
Planning also occurs within cities: streets can be optimised by proportion of space provided for buildings. This comparison of 5 common street designs shows cul-de-sacs to have optimal packing:
Cities and streets have evolved more haphazardly and organically in reality – but still often along common patterns, with grids evidenced in San Francisco and New York, and radial patterns seen in Palmanova in Italy and Bath in England:
Finally, any good city needs plenty of green space! Did you know, 47% of London’s space is made up of parks and other green spaces? Of course, there are also examples of patterns in use in parks (they get everywhere!), with some beautiful landscaping and formal garden designs below: