![]() That year Cornell mathematician Daina Taimina also worked out how to model such surfaces using crochet, which was a big deal because it’s actually hard for humans to construct these forms.Ī crochet coral reef based on hyperbolic geometry. These carbon nano-foams were discovered in 1997 by physicist Andrei Rode and his colleagues at the Australian National University. Hyperbolic surfaces can also be built at the molecular scale from carbon atoms. In the film Avatar, there is a fabulous CGI grove of giant hyperbolic blooms that curl up when touched. There are hyperbolic structures in cells, hyperbolic cacti and hyperbolic flowers, such as calla lilies. Wherever there is an advantage to maximising surface area – such as for filter feeding animals – hyperbolic shapes are an excellent solution. Along with corals, many other species of reef organisms have hyperbolic forms, including sponges and kelps. Yet critters who’d never studied non-Euclidean geometry had meanwhile just been doing it. “For God’s sake please give it up,” said the Hungarian mathematician Wolfgang Bolyai to his son János Bolyai, urging him abandon to work on hyperbolic geometry. When mathematicians discovered this aberrant geometry in the early 19th century they were nearly driven mad. It’s a geometric analogue of a negative number. The surface of a sphere (like a beach ball) has positive curvature, and a hyperbolic plane has negative curvature. A flat, or Euclidean plane has zero curvature. One way of understanding different geometries is in terms of their curvature. Margaret Cagyle, Institute For Figuring, Author provided All the pencil lines that appear to be curved were drawn with a ruler so they are actually straight. This image shows straight lines drawn on a paper model of a hyperbolic plane. And that, in turn, led to the revolution that produced the kind of maths now underlying general relativity, and thus the structure of the universe. Yet while nature has been playing with hyperbolic forms for hundreds of millions of years, mathematicians spent hundreds of years trying to prove that such structures were impossible.īut these efforts led to a realisation that hyperbolic geometry is logically legitimate. ![]() In hyperbolic geometry the plane is not necessarily so flat. These organisms are biological manifestations of what we call hyperbolic geometry, an alternative to the Euclidean geometry we learn about in school that involves lines, shapes and angles on a flat surface or plane. Yet outside our boxes, nature teams with frilly, crenellated forms, from the fluted surfaces of lettuces and fungi to the frilled skirts of sea slugs and the gorgeous undulations of corals. But 'Mountains' is the only example I can think of where the specifics of height were important enough to be mentioned.We have built a world of largely straight lines – the houses we live in, the skyscrapers we work in and the streets we drive on our daily commutes. And, as tectonic drift once had Antarctica near the equator, the mountains might have formed the base of a skyhook, although that idea post dates the story by a long while - but mountainous space launching cannons, a-la Jules Verne, do not, and HPL seems to have had an interest in such things. Schemes for huge, inflatable, launch towers reaching up to 20km have been put forwards. The old ones were highly adapted to space travel, and even described as venturing back into space to defend their colony here in the deep past - so I wondered, although daydreams might be a more apt term, if one purpose of some of those extremely high 'mountains' might have been mass launch platforms, to get them above most of the atmosphere before launching with whatever system they used. It could be that they are much higher than any purely natural mountain's because of this - I think I recall that, with modern building materials, structure in excess of 15km high are theoretically possible if money's no object. Nice to make your acquaintance Nate The Antarctic mountains that are the titular 'mountains of madness' are described as having a major artificial component, or heavy modification.
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