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 spaces).
Climbing 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.
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, 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.
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.
The first of these problems is familiar, but I have not heard of the others.
A vine does not, for example have to be a linear thing. Consider, that the vine grows like a mat, the growing tip is not a point, but a line. Then when it finds a branch sticking out, it can cover it so as not to fall off.
Hi, Wendy!
Quite so, and indeed there are things in this world that grow in two dimensions; but there are also things that grow in one.