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.