R - how to create a pre-determined number of identical square polygons to use for a choropleth map?

I am trying to plot crime data density (again!) over a city map, say of NY. As a well known problem there are plenty of examples on this (http://www.obscureanalytics.com/2012/12/07/visualizing-baltimore-with-r-and-ggplot2-crime-data/). These methods plot the crime density through isoclines, while I need to represent it through identical density squares of a pre-determined area (and the area / side length may change from one iteration to the other). The reason for representing crime density through squares is that the square are the hotspot areas to be patrolled. The size will influence the overall amount of police people required.

This is what I am trying to achieve:

1. I would like to be able to create identical square polygons with pre-determined size to overimpose to the map (is it a raster? apologies but I've just started to learn to spell GIS!)
2. I would like to use the above squares as items to colour as in a choropleth map (i.e. different colouring in relation to frequency of crime in the area), probably using ggplot2 or similar.
3. This should allow me to see how the density of crimes per square kilometre varies changing the size (i.e. the area) of the square.

I do not have a clue if it is possible to use R to create regularly shaped squares polygons of a pre-defined size to use for this (as the code snipped below attests). Any help or links to examples are welcome.

I would be glad to get some indication on alternative ways to calculate the density. I have used the stat_density2 (part of ggplot2) but maybe there are better / faster ways?

This is where I got to:

library(rgdal)
library(raster)
library(sp)