People don't notice change when it's gradual. Sometimes, however, small, incremental changes add up in a way that isn't noticed until a change in degree becomes a change in kind.
So it is with Internet connectivity and perimeter security. In the beginning companies had "the Internet connection," a single pipe to the outside world. With one connection, there was a clear and easy-to-manage perimeter. And so, perimeter security flourished.
Over time, companies have added more and more connectivity, gradually changing their network architecture from a single "funnel" into the Internet to a mesh-like network with near-ubiquitous connectivity. What started as a series of changes in degree has become a change in kind. Companies no longer maintain "a connection" to the Internet; they are fully meshed, and a significant percentage of their work happens "out there." As a result, perimeter security also has changed -- from being fundamental to being almost obsolete.
The closest real-world parallel to perimeter security is the medieval castle with its walls and moat. With its single gateway, the castle forces all traffic to a single choke-point where access controls can be applied. Contrast the castle with a modern city, where there are thousands, if not tens of thousands of entry and exit points. It might be possible to set up roadblocks at every point, but it would be totally impractical. Not only would such a security scheme be ineffective, but it also would cripple the city as trade, re-supply and people-flows would grind to a halt.