The three realms of the Commedia are built on borrowed structures, and knowing which ones changes how the poem reads. Hell is sorted by an ethical scheme the poem credits to Aristotle: incontinence in the upper circles, then violence, then fraud, on the principle that the sins which most misuse reason lie deepest. Purgatory is organised instead by the seven capital vices of the confessional tradition, one terrace each, with love misdirected, love deficient and love excessive offered as the explanation of all seven. Paradise is the Ptolemaic cosmos, with the souls appearing in the planetary spheres although they all in fact dwell in the Empyrean beyond them, a point the poem states outright rather than leaving the reader to work out.
Purgatory is the newest of the three. As a defined place in church teaching it was young when Dante wrote, and giving it a geography, a mountain in the southern ocean with an ante purgatory for the late repentant and terraces that clean rather than punish, did more to settle it in the European imagination than any council did. What most readers now picture on hearing the words hell and purgatory comes from a poem rather than from a doctrine.
The counting is part of the architecture too. There are a hundred cantos, thirty three in each realm with one extra at the beginning, and the poem is written throughout in a three line stanza whose rhymes interlock so that no stanza can be lifted out without breaking the ones on either side of it. A reader is meant to notice that the shape of the verse and the shape of the cosmos are making the same claim.