After Octavian came to power, how did the Roman Republic become the Roman Empire?

After Octavian took over as an un-voted leader by the senate, Rome lost it's Republican values and went to a Monarchy. This is why Historians refer to it as an Empire.

consider an orange peel: if you want to try and lay it flat, you have to stretch it, we know exactly how things are being stretched or squashed at any given point. we have many different map projections because each has different patterns of some projections can even preserve certain features of the earth without