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book:geometry-the-imaginationGeometry & the Imagination
Book by Hilbert and Cohn-Vossen cited for the eleven properties of the sphere, illustrating simplicity that allows maximum relationships.
Extracted from this book
Claims (10)
- A cleaning process goes on in parallel with differentiating, keeping the structure spare and simple so that there is always room for more relationships; this is the only way profound, well-adapted structure can be built.Describes the dual process of complexity buildup and simplification, leading to the beauty of primitive art and nature.
- A complex object will only be successful if it is generated.Generalization from organisms to software and buildings; used to argue against mere assembly.
- ALL the well-ordered complex systems we know in the world, all those anyway that we view as highly successful, are GENERATED structures, not fabricated structures.The central thesis of the chapter; supported by examples from nature, artifacts, and human settlements.
- An ordinary American house contains a density of about 1 decision per cubic foot, thus about 20,000 possible mistakes when the house is not generated.Estimate based on labor hours and physical pieces; used to motivate economic cost analysis.
- For a community of 150 middle-class American houses, the total loss of value due to fabricated mistakes is on the order of a staggering $30 million.Economic calculation extrapolating mistake costs to a whole community.
- The fabricated plans D-G, even when designed with good intent or individual variation, lack the essence of living structure and are not generated structures.Judgment on the specific examples, emphasizing that simulated variety does not equal generated structure.
- The Japanese tea bowl is a generated structure in regard to its shape, glazes, and decoration, leading to its beauty and comfort; the wine glass is fabricated and therefore stiff and lifeless.Aesthetic comparison used to illustrate that generatedness is perceptible even in small objects.
- The real essence of generated structures lies in the structure-preserving transformations which move the structure forward through time.Core claim that distinguishes generated from mere evolutionary adaptation; the transformations are necessary for unfolding.
- The value difference between a house with 10,000 mistakes ($150,000) and a mistake-free one ($600,000) could be on the order of $450,000.Cost estimate illustrating the economic cost of fabricated structures; even a low-cost Indian house could see a tenfold value increase.
- You can only get the profound multiple structure of centers by unfolding each bit from the previous state, allowing the next layer of structure to appear from the previously established layers.Explains why time and sequence are essential for generated complexity.
Hypotheses (2)
- If indeed the programs are so complex, then it is likely that they, too, will be potentially subject to hundreds of thousands, perhaps millions of egregious mistakes of adaptation.Extends the mistake analysis from buildings to software, predicting that complex programs without generating processes will be full of adaptation failures.
- Truly successful programs can only be generated; and the way forward in the next decades will be through programs which are generated through unfolding, in some fashion comparable to what I have described for buildings.Predicts that the only path to highly adapted software is to apply the principles of generated structure.
Neighborhood — ranked by edge-count
Chapters (1)
chapter
- The chapter contrasts generated structures (complex, adapted, alive) with fabricated structures (designed, dead, full of mistakes), and argues that only generated structures can achieve deep complexity and avoid costly mistakes.