Just as we have cut the earth into a grid of latitude and longitude (and knowing that each "block" of 1 degree latitude by 1 degree longitude gets smaller and smaller as we approach the poles), we similarly cut our cities and rural areas into a finer mesh from that same grid. Much of this arises from the various large scale ordinance surveys that took places in the Americas, Australia, and India. There are of course grids dating much earlier, to Miletus and Mohenjo Daro among many others. Not all grids are aligned with longitude and latitude, sometimes they align with local landscape features, but most of the modern ones are. (Where grids of different alignments come together, interesting spaces are created). Not all grids are squares, most are more like rectangles.
So why should we have 90-degree rectilinear grids?
The arguments in favor are that it:
- simplifies construction and makes it easier to maximize the use of space in buildings,
- simplifies real estate by making the life of the surveyor easier,
- simplifies intersection management by reducing conflicts compared to a 6-way intersection,
- is embedded in existing property rights and so impossible to change.
We in the modern world need not be bound to the primitive tools of the early surveyor, the primitive signal timings of the 1920s traffic engineer, or the primitive construction techniques of early carpenters. And while for existing development we might be locked into existing property rights, for new developments that doesn't follow.
The arguments against the rectilinear include that it:
- is among the least efficient way to connect places from a transportation perspective,
- reduces opportunities for interesting architecture,
- wastes developable space by overbuilding roads.
There are many designs for non-rectilinear street networks. Ben-Joseph and Gordon (2000) (Hexagonal Planning in Theory and Practice (Journal of Urban Design 5(3) pp.237-265)) summarize a number of the 19th and 20th century designs. Most are simple aesthetic choices, as in Canberra, the planned capital city of Australia, and don't seem to relate to deeper urban organizational issues.
Rudolf Müller proposed The City of the Future: Hexagonal Building Concept for a New Division. Müller's plan offsets the 60-degree streets so that they come together in 4-way rather than 6-way intersections (though they are still at 60-degrees and not bent to make 90-degree intersections). This ensures that the cells in the plan are not bisected by roads, and that they are instead hexagonal blocks. This plan loses a lot of areas to ornamental parks in the middle of streets.
The circuity increase associated with a 90-degree rather than 60-degree network is obvious. Circuity (the ratio of Euclidean to Network distance) would be minimized if roads were at 0-degree angles. The downside is that this Euclidean network where everyone traveled in a straight-line would literally "pave the earth". Leaving aside the downsides for the environment of being so-paved, the more critical trade-off from a transportation perspective is construction costs. More roads are more expensive. So a network design trades-off travel costs accruing over time with the up-front construction and long-term maintenance costs. The optimal network design depends on the land use pattern it aims to serve. (And the land use pattern depends on the network design.) The City of Alonso or Von Thünen, with all jobs downtown merely requires a simple radial network to connect it. A polycentric or fully dispersed (homogeneous) city with everything spread uniformly across space begs for more cross-connections.
There is a large literature on the network design problem. One useful paper: Pierre Melut and Patrick O'Sullivan (1974) A Comparison of Simple Lattice Transport Networks for a Uniform Plain, Geographical Analysis 6(2) pp. 163–173, says:
The objective is to compare construction and transport costs for triangular [60-degree], orthogonal [90-degree], and hexagonal [120-degree] regular lattices as transport networks serving a uniform, unbounded plain. The lattices are standardized so that the average distance from the elementary area to the edge is the same for each. This standardization results in equal construction costs for the three networks; thus, the comparison can be made in terms of route factors [circuity], which favors the triangular lattice over the other two.
Because the circuitous network is less efficient, more network pavement and track and vehicle mileage must be provided to enable the same amount of transportation. This wastes spaces that could be better allocated to non-transportation purposes.
The lattice itself comprises a single level in a hierarchical system. Selected links in a lattice can be reinforced to make them faster, attracting traffic. This process of reinforcement is natural with investment rules that favor more heavily trafficked routes and explains the hierarchy of roads. If it is based on simple reinforcement of existing links rather than creation of new links, that hierarchy will not affect the topology of the network.
Ask MetaFilter has an interesting thread on Comparing perimeters of arrays of hexagons vs. squares - geometry tiling resolved . A key point is that arranging hexagons into a square-like shape has a higher perimeter than arranging squares into a square-like shape.
__ __ __ __ __ / \__/ \__/ \__/ \__/ \ \__/ \__/ \__/ \__/ \__/ / \__/ \__/ \__/ \__/ \ \__/ \__/ \__/ \__/ \__/ / \__/ \__/ \__/ \__/ \ \__/ \__/ \__/ \__/ \__/ / \__/ \__/ \__/ \__/ \ \__/ \__/ \__/ \__/ \__/ / \__/ \__/ \__/ \__/ \ \__/ \__/ \__/ \__/ \__/ Diagram 1. Sample hex map
I think your problem is this - to minimize the perimeter of n hexagons, when you add each new hexagon to the previously-existing group, you have to add it in such a way that touches the most neighbors possible. You would never add a hexagon that touches only on one face if you could add it somewhere else where it touches two faces or three faces, right? If you look at diagram 1 here (which is hexes in a grid shape), you see several hexes at the four corners which touch only on two faces, while there are areas on the outer surface at the top and bottom where those hexes could be placed where they would touch on three faces instead of two. So simply moving those four corner hexes would reduce the perimeter without changing the surface area.
Yet we know the hexagon is efficient, it replicates the closest packing of circles. (Take a penny, surround it with pennies so that they are all tangent. The central penny touches six others.) Thus following the closest-packing argument, the hexagon as geometrical shape is not sufficient for efficiency, we must also arrange those shapes into an efficient pattern, in this case, something more like the Glinski Chess Board:
Much of the inspiration for thinking about hex-maps comes from the gaming community, where such maps have been used since the 1961, when a Hex map was used for the Avalon Hill game Gettysburg. It has since become a standard that is widely used to represent directions of movement in games.
So, although we talk about "grids" as being necessary for connectivity, we can get even more connectivity if we think about a variety of different geometries. It would be a shame if we got locked into grid geometries for new developments when there are so many alternatives to be had.