Octagon

The octagon is the square, bisected. Inscribe a square in a circle, bisect each of the four arcs by striking equal arcs from the ends of each side and joining the crossings to the centre, and the eight resulting points are the vertices of a regular octagon. Continue the process and you get 16, 32, 64 and every further doubling.

Under Gauss-Wantzel, which permits a power of 2 multiplied by a product of distinct Fermat primes, 8 is 2 cubed: a pure power of two, needing no Fermat prime at all. Doubling the side count of any constructible polygon amounts to bisecting arcs, an operation the compass performs unconditionally, so the constructible polygons come in chains: 3, 6, 12, 24; 4, 8, 16, 32; 5, 10, 20, 40.

The proportions of the regular octagon involve sqrt 2, as its parentage would suggest. The ratio of the distance across the flats to the length of a side is 1 plus sqrt 2, about 2.414, the number sometimes called the silver ratio, which stands in roughly the same relation to the octagon that phi does to the pentagon. The interior angle is 135 degrees, which does not divide 360, so regular octagons do not tile the plane alone. They do tile with squares filling the gaps, a pattern found from Roman mosaic floors onward and still the standard tiling of a great many bathrooms.

Archimedes' method for bounding pi is this same work extended. He inscribed and circumscribed regular polygons in a circle and repeatedly doubled their side counts, reaching 96 sides, which placed pi between 223 over 71 and 22 over 7.

The octagon's chief architectural job is the transition from a square room to a round dome. A dome needs a circular base; a room usually has four walls. The octagon is the intermediate figure, and two devices produce it: the squinch, an arch or corbelled bracket thrown across each corner, and the pendentive, a curved spherical triangle doing the same job continuously. Squinches are the older device and are characteristic of Iranian and Central Asian building. Pendentives are the great Byzantine solution, brought to full scale at Hagia Sophia in the sixth century.

This is a real and practical instance of the phrase squaring the circle, and it should be distinguished sharply from the mathematical problem of that name, which is impossible. Here nothing has to be exact. An octagon splits the difference geometrically and the masonry absorbs the remainder.

The Dome of the Rock in Jerusalem, completed in 691 or 692, is an octagon with a double ambulatory encircling the rock, and it is the earliest major surviving Islamic building. Octagonal plans recur in Islamic architecture at every scale thereafter, including the eight-pointed star formed by two overlapping squares, a motif so widespread that it is often simply named after the seal or khatam.

In Europe the form runs from the Tower of the Winds in Athens, a water clock and weather station of the second or first century BC carrying a personified wind on each of its eight faces, through Charlemagne's Palatine Chapel at Aachen around 800, to the fourteenth-century octagon lantern at Ely Cathedral, built to span the crossing after the Norman tower there collapsed. In each case the eight-sided plan is doing structural work as well as symbolic work, and it is usually a mistake to separate the two.

Christian baptisteries were built with eight sides, and the reasoning was written down. Ambrose of Milan built the baptistery of San Giovanni alle Fonti in the 380s, the building in which he is said to have baptised Augustine in 387, on an octagonal plan with eight niches. An inscription preserved in a later manuscript states the reason: the hall was rightly eight-sided, because salvation came at dawn on the eighth day, when Christ rose. The eighth day is the day after the seven of creation, the day standing outside the week, and baptism was understood as dying and rising with Christ. Octagonal fonts and baptisteries followed across Europe for the next thousand years, from Pisa and Florence down to parish churches, and the eight-sided font remains standard in English churches today.

Buddhism counts the noble eightfold path: right view, intention, speech, action, livelihood, effort, mindfulness and concentration. It represents the teaching as an eight-spoked wheel, the dharmachakra, which is the form on the flag of India.

Chinese cosmology arranges the eight trigrams of the Yijing, the bagua, around an octagon, in two standard orders traditionally ascribed to Fuxi and to King Wen. The bagua mirror is hung in octagonal form over doorways in popular practice, and the arrangement organises directions, seasons, family relations and natural forces all at once.

The arithmetic helps. Eight is the first cube after one, it is two doubled twice over, and it falls naturally after seven wherever seven has already been used to count out a completed cycle. That is the pattern lying behind the eighth day, and behind the sense of the octagon as a figure of renewal rather than of completion.