When An Abelian Category Has Enough Flat Objects?

Is there a universal ratio, pattern or proportion between sizes of natural objects ?

• From objects that are microscopic in size like bacteria, virus etc, to objects of human scale like trees, animals, etc to extermely large objects like solar systems and galaxies   Are categories of objects man made, or are actual jumps in sizes between different categories observed, is there a substantial leap in size variation between categories when compared to the average size variation of objects in each category    Is there a proportion or pattern or ratio between sizes of each category

• Answer:

  No in the broadest sense, but yes if you restrict yourself to a more narrow category than absolutely everything. There is no mathematical function or theory which relate the size of bacteria to the size of trees or to size of solar systems. The different categories are simply governed by very different processes. As Benoit Mandelbrot once remarked, berries are round because it provides the smallest surface to a volume while planets are round because of the pull of gravity and there is no underlying secret to their geometrically similar shapes. There are however plenty of theory to explain the range of sizes within each category. Gravitation and nuclear science dictate the smallest mass a star needs to 'light up', while capillary pressures put a limit on the height of the tallest trees. A real classic is the maximum size of insects. These creatures breathe trough their exo-skeleton and it's a fact of geometry that if you scale up an object, the volume grows more rapidly than the surface. Large insects would simply suffocate. There are also interesting but less understood scaling laws relating for instance the size and metabolic rate of different but related animals. A possible explanation is the efficiency of transporting nutrients around in a large body. You can see that the categories, while created by humans, are in fact very useful. Take everything in and you're just banging your head in a wall, but divide the universe into categories of similar objects and all kinds of insights become possible. Holism is not all it was carved out to be.