## A Graph of Tradeoffs

Regarding the problem of designing a vacuum cleaner, architect Christopher Alexander spells out the tradeoffs using a completely connected 2-coloured graph:

Four desirable properties = nodes, and 2-way connections between them = edges. The edges are “coloured” with {+, }. Most of the desired properties come at the expense of one another and so are coloured .

In the language of Jacobians, a minus on the edge between performance and simplicity means both that  $\large \dpi{200} \bg_white \frac{\partial \rm{\ simplicity}}{\partial \rm{\ performance}} < 0$ and that $\large \dpi{200} \bg_white \frac{\partial \rm{\ performance}}{\partial \rm{\ simplicity}} < 0$. Increasing performance decreases simplicity, and increasing simplicity decreases performance.



You can let your imagination run with this thought-form. Besides mathematical generalisations (3-way connections, more edge types, partly connected graphs), lots of other kinds of things can be fit onto the nodes of a completely connected weighted coloured graph. If concepts as broad and multifarious as “economy”, “simplicity”, and “performance” fit into the form of a graph, ……

• politics: {parties with opposed interests, parties with mutual interests} (redundant edges if parties are both opposed and aligned)
• economics: {competitors, partners}
• artistic composition: as I said before, all elements of a composition are connected in a complete graph … lots of edge types
• philosophy: {opposed concepts, symbiotic concepts} (add edges for more options)
• sex: {correlated properties of a desirable mate, uncorrelated properties of a desirable mate}

By the way, C Alexander is the inspiration for the Gang of Four’s book on Design Patterns. Just ask a software engineer: you can program a model of anything.

Arming your imagination with bolas and trebuchets,

Isomorphismes

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