Spatial Scales

In this model, 1 Independent region is the "largest" pattern and consists of only one other pattern; the 'distribution of towns':

The "distribution of towns" itself is again composed of four other patterns:

• "3 City country fingers"
• "4 Agricultural valleys"
• "6 Country towns"
• "7 The countryside"

These patterns are all components of the pattern "the distribution of towns", which in turn construct the "independent regions":

This continues all the way down to 'atomic' patterns, that are not constructed of anything else (shown in red):

The atomic patterns are 94 sleeping in public, 245 raised flowers, 246 climbing plants, 253 things from your life. They do not seem to have anything particularily in common - arguably all of these could have been split into smaller patterns or left out if the book wanted to go into more or less detail.

The spatial scales lend an undeniable hierarchy to Alexander's patterns: Independent regions can not be part of Ring Roads, and ring roads can not construct marriage beds. Thus one would expect that we should be able to get hold of a more or less hierarchical structure to the network of patterns, with 1 Independent Regions at the top, and very small local things at the bottom:

Deviations

This assumes that a pattern must spatially fit inside another pattern to construct it - e.g. a 'marriage bed' would be smaller than the 'couple's realm' it is part of, and both are smaller than a 'house for a couple'. One would expect that if A > B, and B > C, then we could also rely on A > C. A beautiful property of this network is that this is not always the case.

First, hierarchy in spatial scale are fuzzy and not necessarily hold up on a local scale. Without doubt "climbing plants" and "independent regions" are on opposite sides of a spectrum, but whether a garden seat is larger than filtered light is rather philosophical.

The fuzzines on a local level allows bidirectional (A constucts B and B constructs A) and circular connections:

Second, some patterns are very abstract. Take, for example 10 Magic of the City - a lot of things on various scales contribute to it, and there is no obvious reason why it shouldn't be constructed by, or used in the composition of patterns on all scales. Here is the local environment of the pattern 'Magic of the City':

Fuzzy Hierarchy Graphs

What we find then is an interesting property of the network: From afar, there is an obvious emergent hierarchy, but in the details, that hierarchy doesn't hold up at all. Is there a way to define where in the hierarchy each pattern lies?

First of all, Alexander did order his chapter numbers roughly according to scale (starting with "independent regions"). A reasonable next step would be to start with the one pattern that we know is at the "top" of the hierarchy - 1 Independent regions - and look at how many steps away each other pattern is from it (only going "downhill" in the direction of the links):

alternatively, we could do the same, but looking at the (average) distance from the atomic patterns:

This gives a very different pattern, because 94 Sleeping in public appears roughly in the middle of the book, and is connected to a lot of mid-scale patterns. Mainly what this shows though is how sensitive this measure (starting from nodes in the network that have only incoming or outgoing connections, but not both) is to the "edges" of the observed system - Alexander might just as well have not included 94 Sleeping in public, or perhaps included an atomic pattern somewhere else, and the resulting observed hierarchy might would have changed drastically.

Another way to look at this would be to consider the average distance to all nodes "downstream" compared to the average distance to all all nodes "upstream":

¹

This shows us one large cluster (with stray point to the left), and separate groups: on the left, patterns that do not influence others; they include the patterns influenced most strongly by the other patterns. They are:

• 1 independent regions
• 2 the distribution of towns
• 3 city country fingers
• 4 agricultural valleys
• 5 lace of country streets
• 6 country towns
• 8 mosaic of subcultures
• 12 community of 7000
• 14 identifiable neighborhood
• 26 life cycle
• 27 men and women

The second group, on the bottom, are those that have no pattern constructing them; they are among the closest to influencing other patterns on average; namely:

• 94 sleeping in public
• 245 raised flowers
• 246 climbing plants
• 253 things from your life

Overall I find this interesting as a kind of network that has a fuzzy kind of linear hierarchy. Rather than - as is common - looking at nodes in terms of how "central" they are to the network, there is a vague idea of "upstream" and "downstream" given by the constellation of directed links. One could define some sort of metric for directed networks that would be maximised for nodes connected in a straight line, and minimised for circles or fully connected graphs.