> In the case of a WiFi mesh network, you'd really want a set of omnidirectional antennae. An omnidirectional antenna with a 1/4 mile radius would need a lot more power to operate. Further, in a place like new york, you have to start worrying about 3 dimensions (skyscrapers are tall!) and obstructions (buildings can be difficult for radio signals to pass through).

The last part (buildings) is incredibly important to remember. WiFi signals—which are typically in the 2.4GHz or 5GHz bands—are extremely susceptible to attenuation in this manner, due to the higher frequency.

Note that, in part, this is why carriers are so hungry for the sub-GHz bands (i.e. 700MHz in the US, 850-900MHz in Australia, etc) for their 3G/LTE+ deployments. The lower frequencies exhibit significantly better building penetration, offset by the need to deploy more cells in order to get similar bandwidth capacity (considered a worthwhile trade-off).

An omni that could broadcast 1/4 mile (400m) in Manhattan at 2.4GHz would need to have a VERY high transmit power. It would likely be illegal, and it would also be dangerous.