Comments on: Spare ribs http://bit-player.org/2009/spare-ribs An amateur's outlook on computation and mathematics. Fri, 30 Jul 2010 22:22:18 +0000 http://wordpress.org/?v=2.6.3 By: Jeremy http://bit-player.org/2009/spare-ribs#comment-2088 Jeremy Thu, 12 Mar 2009 03:57:01 +0000 http://bit-player.org/?p=309#comment-2088 That's a good point, but I wasn't really thinking about it from a dynamical point of view. More from the point of view that we have a cactus of a particular shape, and we set up a static solution to the system on that background. But you're absolutely right if we want to do dynamical cactus growing, it would have to be a problem over R x S^1. It would seem to be a much harder problem to generate a whole cactus this way :). Although perhaps a model from first principles would be easier like this? Since you could go out and measure biochemical potentials and everything, and put them directly into the dynamics. That’s a good point, but I wasn’t really thinking about it from a dynamical point of view. More from the point of view that we have a cactus of a particular shape, and we set up a static solution to the system on that background.

But you’re absolutely right if we want to do dynamical cactus growing, it would have to be a problem over R x S^1. It would seem to be a much harder problem to generate a whole cactus this way :). Although perhaps a model from first principles would be easier like this? Since you could go out and measure biochemical potentials and everything, and put them directly into the dynamics.

]]> By: brian http://bit-player.org/2009/spare-ribs#comment-2087 brian Wed, 11 Mar 2009 12:15:21 +0000 http://bit-player.org/?p=309#comment-2087 @Jeremy: It's a good analogy, and there are certainly connections between the stripes and spots seen in biological patterns and those observed in magnetic domains and other physical systems. But there's a complication to be kept in mind when you're looking at the cactus ribs: Once a rib is formed, it can't move around or change its geometry. In effect, the two-dimensional pattern we observe is a graph through time of a one-dimensional process. The cactus grows only at the apical meristem—the tip of the trunk or of an arm. So the process that generates the ribs has to be carried out in a one-dimensional ring of cells surrounding the apical meristem. Within that ring, morphogen-secreting clusters of cells might move back and forth with respect to their neighbors; clusters might split or fuse; new clusters might arise. All of these events leave behind a distinctive trail of ridges and grooves. But you don't have a situation comparable to a magnetic garnet crystal or a convecting layer of liquid, where true two-dimensional patterns are free to swirl around. @Jeremy: It’s a good analogy, and there are certainly connections between the stripes and spots seen in biological patterns and those observed in magnetic domains and other physical systems. But there’s a complication to be kept in mind when you’re looking at the cactus ribs: Once a rib is formed, it can’t move around or change its geometry. In effect, the two-dimensional pattern we observe is a graph through time of a one-dimensional process. The cactus grows only at the apical meristem—the tip of the trunk or of an arm. So the process that generates the ribs has to be carried out in a one-dimensional ring of cells surrounding the apical meristem. Within that ring, morphogen-secreting clusters of cells might move back and forth with respect to their neighbors; clusters might split or fuse; new clusters might arise. All of these events leave behind a distinctive trail of ridges and grooves. But you don’t have a situation comparable to a magnetic garnet crystal or a convecting layer of liquid, where true two-dimensional patterns are free to swirl around.

]]> By: Jeremy http://bit-player.org/2009/spare-ribs#comment-2086 Jeremy Wed, 11 Mar 2009 06:38:19 +0000 http://bit-player.org/?p=309#comment-2086 I think this is just like magnetic domain walls in a ferromagnet in a weak external field. The individual subsections of the cactus are trying to locally line up with whatever "potential" they feel, but this doesn't always line up with the global structure, since the "potential" at one point doesn't know about far away points. This explains the picture of the pathological cactus, too. Small neighborhoods of it look okay enough, and follow the local rule, but the local pictures match up poorly to form a crazy global picture. I'll bet you could even model this on a computer with some kind of nearest-neighbor potential. I'd be willing to bet a small number of parameters could give you all the behavior you see in those pictures. Then you can study phase transitions and topological order in cacti! I think this is just like magnetic domain walls in a ferromagnet in a weak external field. The individual subsections of the cactus are trying to locally line up with whatever “potential” they feel, but this doesn’t always line up with the global structure, since the “potential” at one point doesn’t know about far away points.

This explains the picture of the pathological cactus, too. Small neighborhoods of it look okay enough, and follow the local rule, but the local pictures match up poorly to form a crazy global picture.

I’ll bet you could even model this on a computer with some kind of nearest-neighbor potential. I’d be willing to bet a small number of parameters could give you all the behavior you see in those pictures. Then you can study phase transitions and topological order in cacti!

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