#### Sunday, April 1, 2012

# Sunday Spinelessness - A marsupial snail

I never thought I'd become a fan of land snails. As I've said before, I started my PhD with the quaint idea that you could study a group of organisms for years and still regard them as little bags of genes with no particular importance beyond their ability to help you answer questions. Perhaps that's true for some people and some animals, but not me and snails. I'm now a card-carrying member of the land snail fan club, and take every opportunity to remind people of the amazing lives these creatures lead.

Snail shells are beautiful. You don't need to know anything special about biology or maths to see that:

But, as is so often the case, the more you learn about snail shells the more beautiful they become. I'm not much of a mathematician. To be honest I find a lot of maths to be a horribly complex, and seemingly arbitrary, and I could never really follow it past basic algebra. Still, every now and again I'm struck by the beauty of a system that can explain parts of reality with such ease (and by envy for those who can see so much deeper than me). The mathematical description of snail shells is one of those cases in which the maths is easy enough for me to understand, and so I can appreciate the elegance.

The simplest way to model a snail's growth would be to say it adds its shell at a constant rate. In that case, we could know the size of a shell at any given time (x) using the exponential function e

The exponential function can tell us how big a shell gets, but of course, shells don't simply grow, they also spiral at the same time. If we want to model both the growth and the spiral pattern of a snail's shell we need to leave our familiar "x,y" system of placing points (called the Cartesian coordinate system) and think in "polar coordinates".

Just as any point in a two-dimensional space can be identified by its distance from another point along horizontal and vertical axes (x and y), it can also be identified by its angle and distance from another point. Think about a point at x=3 and y=2, you can just as easily, and just as uniquely, identify that point with polar-coordinates:

Snail shells are beautiful. You don't need to know anything special about biology or maths to see that:

But, as is so often the case, the more you learn about snail shells the more beautiful they become. I'm not much of a mathematician. To be honest I find a lot of maths to be a horribly complex, and seemingly arbitrary, and I could never really follow it past basic algebra. Still, every now and again I'm struck by the beauty of a system that can explain parts of reality with such ease (and by envy for those who can see so much deeper than me). The mathematical description of snail shells is one of those cases in which the maths is easy enough for me to understand, and so I can appreciate the elegance.

The simplest way to model a snail's growth would be to say it adds its shell at a constant rate. In that case, we could know the size of a shell at any given time (x) using the exponential function e

^{x }(e being the base of the natural logarithm, which you can think of as the base unit for any pattern of continuous growth). You can probably remember the exponential function from high school maths, it's the one that gets big quickly:The exponential function can tell us how big a shell gets, but of course, shells don't simply grow, they also spiral at the same time. If we want to model both the growth and the spiral pattern of a snail's shell we need to leave our familiar "x,y" system of placing points (called the Cartesian coordinate system) and think in "polar coordinates".

Just as any point in a two-dimensional space can be identified by its distance from another point along horizontal and vertical axes (x and y), it can also be identified by its angle and distance from another point. Think about a point at x=3 and y=2, you can just as easily, and just as uniquely, identify that point with polar-coordinates:

Using polar coordinates it's very easy to write an equation that describes the growth of a snail shell:

r = e

Here "θ" (theta) is an angle relative to the starting point, "r" is the amount of growth the spiral has made by the time is swings around to that angle and

I know that shape, that's a

With a little bit of tweaking you can make a paua (

Just changing one parameter in a pretty simple equation is enough to produce spirals that fit most snails' shells. In fact, spirals like these ones, which are called logarithmic spirals, pop up in nature all the time - from the arms of galaxies to the nerves in your eyes. Logarithmic spirals have some pretty cool properties, the most interesting of which is that not matter how large they grow they ever change shape. A snail that grows according to these equation will be the same shape from the day it's born to the day that it dies.

If you know a bit more maths you can extend these models into a third dimension and, with one more parameter, create flat disc-like shells or tall conical ones. I think it's truly amazing that you can get a good approximation of snail shells using so few parameters - but it's worth remembering mathmatical constructs are just models we use to examine reality. David M. Raup got a bit carried away with the mathematical description of shells in the 1960s, and created what he called the "museum of all shells" by exploring the three dimensional shapes you could make by tweaking just three parameters in a model of shell-growth. But Raup's virtual musuem doesn't include all the shells that snails can grow. Biology is weird, and any "law" that a biologist might claim to have discovered will have an exception. None of the shells above

Worm snails grow in way that is radically different from most of their close relatives, but more subtle deviations from logarthmic spiralling are just as interesting. Remember these guys?

As the shell get's bigger, the opening to the umbilicus gets smaller...

...and smaller.

Of course, the original "wide" umbilicus is still part of the older shells. In effect, this pattern of growth creates a cavity within the shell which has lots of space at the top, but a very narrow opening. Amazingly,

Land snails usually don't do much for their young. A few snails lay extra large clutches, so that the first of their offspring to emerge will have eggs to eat before they set off on their lives. Others hang on to their eggs, either within their shells or withing their body.

Most of the larger shells from this species show this sort of damage. When you zoom in on the damage you can see a slighltly irregular pattern.

These holes are creatued by immature snails emmerging within the brood pouch and eating their way out of their parent's shell. Such damage doesn't seem to kill the snails - they effectively wall-off the first few whorls of the shell once they are large enough, so there is no animal within the part of the shell that is broken.

I can tell you a lot more about these snails. Allen Solem described the "brood pouch" and a little of their ecology in 1968, but he worked from old shells and no one has studied their behaviour

The shell photographs onto which I've super-imposed the sprials are all Creative Commons Licensed courtesy of Te Papa 1,2,3.

r = e

^{k.θ }Here "θ" (theta) is an angle relative to the starting point, "r" is the amount of growth the spiral has made by the time is swings around to that angle and

*k*determines the "tightness" of the spiral the shell forms. I was playing around with Wolfram Alpha in preparation for this post, drawing spirals with different values of*k,*when I came across this spiral at*k*= -0.2:*Wainuia*shell!With a little bit of tweaking you can make a paua (

*k*= -0.6) or something close to a tightly-turning charopid (*k*= -0.1).Just changing one parameter in a pretty simple equation is enough to produce spirals that fit most snails' shells. In fact, spirals like these ones, which are called logarithmic spirals, pop up in nature all the time - from the arms of galaxies to the nerves in your eyes. Logarithmic spirals have some pretty cool properties, the most interesting of which is that not matter how large they grow they ever change shape. A snail that grows according to these equation will be the same shape from the day it's born to the day that it dies.

If you know a bit more maths you can extend these models into a third dimension and, with one more parameter, create flat disc-like shells or tall conical ones. I think it's truly amazing that you can get a good approximation of snail shells using so few parameters - but it's worth remembering mathmatical constructs are just models we use to examine reality. David M. Raup got a bit carried away with the mathematical description of shells in the 1960s, and created what he called the "museum of all shells" by exploring the three dimensional shapes you could make by tweaking just three parameters in a model of shell-growth. But Raup's virtual musuem doesn't include all the shells that snails can grow. Biology is weird, and any "law" that a biologist might claim to have discovered will have an exception. None of the shells above

*quite*fit the spiral I've super-imposed on them, and some snails grow shells that radically deviate from logarithmic growth . My favourite example of such a radical departure are the "worm snails", marine snails that cement the apex of their shell to a rock then grow an almost un-coiled tube of a shell.Worm snails grow in way that is radically different from most of their close relatives, but more subtle deviations from logarthmic spiralling are just as interesting. Remember these guys?

*Libera fartercula*are one of a great deal of snails that change shape as they age. Very young shells have a very broad opening (an umbilicus) on the underside:...and smaller.

*L. fratercula*is a marsupial snail. Over the course of its growth this species*creates a pouch within its shell, which it then lays its eggs in, protecting them from would-be predators who can't get inside the narrow opening.*Land snails usually don't do much for their young. A few snails lay extra large clutches, so that the first of their offspring to emerge will have eggs to eat before they set off on their lives. Others hang on to their eggs, either within their shells or withing their body.

*Libera fratercula*takes parental investment to a much greater level. Here's an older shell:I can tell you a lot more about these snails. Allen Solem described the "brood pouch" and a little of their ecology in 1968, but he worked from old shells and no one has studied their behaviour

*in situ*to be able to measure the impact of this strange adjustment to snail-life has on parents.The shell photographs onto which I've super-imposed the sprials are all Creative Commons Licensed courtesy of Te Papa 1,2,3.

Labels: maths, photos, sci-blogs, science, snails, sunday spinelessness

### 2 Comments:

Great post! In fact, the logo of the Society for Mathematical Biology is a Nautilus, because it's shell has the properties you describe (I don't know what k is though).

Hi David - I highlighted your post on my blog:

http://theartofmodelling.wordpress.com/2012/04/03/snails-shells-the-logarithmic-spiral/

http://theartofmodelling.wordpress.com/2012/04/03/snails-shells-the-logarithmic-spiral/