Alan,

As far as I understand, semi-lattice is created by adding a few random 
links. This is very similar to adding a few random links into a regular 
graph leading  to a small world graph. It is in this sense that I was 
impressed by Alexander's deep insight into the problem; see my remarks 
in this paper (p. 4441), http://fromto.hig.se/~bjg/flowcapacityPHYSA.pdf

Maybe I have not fully understood your question yet. According to my 
reading or understanding of Alexander's theory of centers, he stressed 
that hierarchical order, recursively defined. The first three of the 15 
geometric properties (levels of scale, strong centers, and boundaries) 
are something about the hierarchy.

Cheers.

Bin
  On 12/26/2012 3:55 PM, Penn, Alan wrote:
> Bin,
>
> I know that they are both hierarchies. My question was why? The part of a Pattern Language that always convinced me that he had got it wrong was his focus on hierarchical order - larger elements has smaller elements, recursively, something that seems completely anti urban as well as rather anti social.
>
> By the way the semi-lattice and the tree are both precisely defined structures in mathematical terms, which if I remember correctly is what Alexander was talking about. A semilattice is associative, commutative and idempotent which induces a partial ordering. What I remember Alexander liked about the semilattice was the partial ordering: while for a tree an element could only be in one subset, in a semilattice an element could be in the overlap between a number of different  subsets. This is certainly an attractive property.
>
> Alan
>
>
> On 26 Dec 2012, at 12:07, Bin Jiang <[log in to unmask]> wrote:
>
>> Alan, let me try to explain why hierarchical order on behalf of Alexander.
>>
>> A city is not tree, but a semi-lattice, which I interpret as a network. However, both a tree and a network share the hierarchical order; I had this view expressed in this short paper (p. 8 below figure 2): http://arxiv.org/abs/1212.0940. Still a bit odd to you...?
>>
>> Bin
>> On 12/26/2012 11:22 AM, Penn, Alan wrote:
>>> Hear, hear! I wonder why Alexander is so into hierarchical order? He famously spotted that the city was not a tree on account of hierarchy, but then reinserted order in the form of the semi lattice... By the time of Pattern Language hierarchy seemed to have taken over entirely. It always seemed a bit odd to me...
>>>
>>> Alan
>>>
>>> Sent from my iPad
>>>
>>> On 26 Dec 2012, at 01:55, "David Seamon" <[log in to unmask]> wrote:
>>>
>>>> Bin,
>>>>
>>>> One last comment. If I were you, I would not set too much store in Salingaros's work. I realize he is a mathematician, but my sense is that he constructs his "objective" equations and scalings on the basis of personal prejudices. He has written a number of "over the top" commentaries on modernist and deconstructivist architecture that are mostly opinionated, subjective tirades. For some reason, Alexander has been taken in by his work and in fact there is a highly questionable quantitative ranking of the "best architecture in the world" as an appendix drawing on Salingaros's "degree of life" index in the first volume of Alexander's NATURE OF ORDER (p. 469).
>>>>
>>>> Salingaros claims to have come up with a quantitative means for measuring  the "degree of life" in a building. From what I can see, his approach is a priori biased against any modernist work, thus the no. 1 building with the most "degree of life" is--wait for it!--the Alhambra, with the Taj Mahal no. 2, and the Dome of the Rock, no. 3.  As one might guess, modernist work scores much lower, and the first modernist work to appear on the list is, oddly, Piano's Pompidou Centre at no. 17 (Fallingwater is no. 19, the Sidney Opera House no. 20, and the Seagram Building  no. 21). The bottom three buildings on the list of 24 are Saarinen's TWA terminal, Kahn's Salk Institute, and Corbu's Ronchamp chapel.
>>>>
>>>> To me, this whole business of attempting to measure the ineffability of wholeness and environmental robustness is silly and impossible. I don't know why Alexander included this work in vol. 1, but sometimes his good sense gets lost.
>>>>
>>>> David Seamon
>>>>


-- 
--------------------------------------------------------
Bin Jiang
Division of Geomatics, KTH Research School
Department of Technology and Built Environment
University of Gävle, SE-801 76 Gävle, Sweden
Phone: +46-26-64 8901    Fax: +46-26-64 8758
Email: [log in to unmask]  Web: http://fromto.hig.se/~bjg/
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Computers, Environment and Urban Systems: An International Journal

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