<\/a>Ever since reading Greg Egan’s novel Diaspora<\/i><\/a> (1998), part of which takes place in a five-dimensional universe, I’ve occasionally tried to imagine aspects of life in higher spaces (which is tricky, as I lack the knack of visualizing in such spaces). A related question is phyllotaxis. Plants have a selective pressure to minimize the amount of shadow that its higher leaves cast on those lower. One way they do this is by arranging them in a helix, spaced at a fraction of a turn which is a rational approximation (1\/2, 2\/3, 3\/5, 5\/8, . . . ) to the most irrational number<\/a>, namely the golden section. But in four-space, for each of the polyhedral ribbon-helices mentioned above there is presumably an optimum spacing for leaves along the ribbon’s center line. In both three-space and four-space, that optimum depends in part on how much the helix rises in each turn.<\/p>\n The spider’s problem is one dimension higher. She has to fill a region with a zigzag so that no point in that region is more than the radius of a fly from the nearest sticky part of her path. Corners may be outside that region, but every corner must be anchored by a pre-existing line or other fixed object. Spiders in our universe often start with three anchor lines and build a disc of web within that triangle. The analogous region in four-space is the intersection of a tetrahedron, formed by six anchor lines, with a sphere tangent to those lines.<\/p>\n The first of these problems is familiar, but I have not heard of the others.<\/p>\n","protected":false},"excerpt":{"rendered":" Ever since reading Greg Egan’s novel Diaspora (1998), part of which takes place in a five-dimensional universe, I’ve occasionally tried to imagine aspects of life in higher spaces (which is tricky, as I lack the knack of visualizing in such … Continue reading
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\nClimbing vines, for one thing. A vine in three-space is effectively one-dimensional, because the phase-space that it has to fill – the set of directions in which it could fall off the column – has only one dimension: a circle (expanded in the third dimension to form a helix). But in four-space, that phase-space is the surface of a sphere; so the vine is a ribbon to wrap it, probably in a pattern with polyhedral symmetry. The pattern varies with species; I think infinitely many are possible, but there is an important constraint: the length of one circuit of the ribbon cannot exceed the height of the supporting column, or the length that can be supported by the vine’s lesser stiffness.<\/p>\n