The structures where life occurs just cannot be made in any other way — it is a categorical and mathematical must.

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• Any part of the world we build will have life if it is created by structure-preserving transformations, and will not have life if it is not created by structure-preserving transformations.claim0.845
  The central thesis of the chapter.
• Such a process will not—in general—create living structure.claim0.842
• Life and living structure will appear in the world inevitably because the mathematical way space gives rise to structure necessarily reinforces wholeness as part of its most normal evolution.claim0.842
  Alexander's strongest ontological claim: living structure is not probabilistically improbable but mathematically necessary given the principle of unfolding wholeness
• There is nothing except the living structure of the world, and this living structure is all we need to reason with.claim0.838
  Radical assertion that function reduces to living structure, eliminating the need for external goals.
• A structure is truly generated, and perceived as having life, only when it has unfolded from a nice, beautiful sequence of differentiations.claim0.838
  The most profound claim of the chapter: the niceness of the sequence is directly perceptible in the built form and is the ultimate source of living quality.
• Nature always produces living structure because it follows a process of unfolding wholeness; human designers can create non-living structure by violating this process.claim0.830
  Core distinction between natural and designed configurations, explaining why properties are ubiquitous in nature but rare in bad design.
• The existence of living structure, as defined in The Nature of Order, requires modifications in our physical picture of the world, not only in the picture we have of architecture.claim0.828
  The closing claim of the chapter's mid-book appendix, asserting that the theory of centers has implications for physics.
• If it is a living process, it will create beautiful geometric structure. If it does not create beautiful structure, it cannot truly be a living process.claim0.827
  Beauty and geometry are the talisman by which a living process is known.