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With the current revival of interest in astronomical intensity interferometry, it is interesting to revisit the associated theory, which was developed in the 1950s and 1960s. This paper argues that intensity interferometry can be understood as an extension of Fraunhofer diffraction to incoherent light. Interference patterns are still produced, but they are speckle-like and transient, changing on a time scale of $1/Δν$ (where $Δν$ is the frequency bandwidth) known as the coherence time. Bright fringes average less than one photon per coherence time, hence fringes change before they can be observed. But very occasionally, two or even more photons may be detected from an interference pattern within a coherence time. These rare coincident photons provide information about the underlying transient interference pattern, and hence about the source brightness distribution. Thinking in terms of transient sub-photon interference patterns makes it easy to see why intensity interferometry will have large optical-path tolerance, and be immune to atmospheric seeing. The unusual signal-to-noise properties also become evident. We illustrate the unobservable but conceptually useful transient interference patterns, and their observable correlation signal, with three simulated examples: (i) an elongated source like Achernar, (ii) a three-star system like Algol, and (iii) a crescent source that roughly mimics an exoplanet transit or perhaps the M87 black hole environment. Of these, (i) and (ii) are good targets for currently-planned setups, while (iii) is interesting to think about for the longer term.