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.