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http://www.faculty.biol.ttu.edu/rice/shell.pdf Looks at biologically defined quantities, such as shell production rates, aperture maps and relates them to morphometrics such as Raup’s parameters. Also: evidence of logistic growth of shell size in time: http://scholarspace.manoa.hawaii.edu/bitstream/handle/10125/11049/?sequence=1 Working on maximizing volume for given surface area.
Issues last time: forgot Jacobian We added the Jacobian but when maximizing volume for given surface area, the provided volume, r and theta values are very large. We think our problem is that we are not using the correct surface area equation. We were only considering the area of one side of our curved box. Found area for outside surface, inside surface, and top of our box. Is this enough for surface area? Jacobian question: we know we need to add r into our integral. But is our r =r1 +aTheta. Where a is our rate of change of r and r1 is our initial r value (we decided r1=1) After maximizing we obtain negative volumes and {a,b} -> 0 Want to discuss on Thursday: Hudson River Conference, Christine found book: Optima for Animals The former way solving the function of volume and surface was not clear so we discussed alternative way to maximize v and minimize s. One thing that needs to be corrected is the surface equation of square. The updated equation is in the file: “3/6square” in Google drive. Chris points out that the maximizing problem is just Lagrange Multiplier problem. I used equations v[h_,a_,b_] and s[h_,a_,b_] and assumed the cubic square surface(which is 4). Maximize[v[h_,a_,b_], s[h_,a_,b_]<=4,{h,a,b}] The result is indeterminate. I also tried when h=1, also indeterminate. I also tried MaxValue and got infinity. My thought was that if we can get a=0 and b=0, it means the volume is maximized as a cube. If we get other surface area, as long as a=b, the volume is maximized. But this could be misleading. Hudson River Conference April 26th Abstracts due March 10th http://www.marist.edu/compscimath/mathdept/jhrsdumc.html
Celebrating Collaboration April 12th http://www.smith.edu/events/collaborations.php Limitations I argue that the snail must progress in a tubular fashion. I feel that the aperture of the snail must be small enough that the foot fills it completely to seal the opening. However, the opening must always be large enough that the snail can potentially escape and leave the shell. (I assume this, since if a hermit crab steals the shell, the snail needs to make a get-away). Therefore, the opening must always be within a range (of not too large, nor too small) throughout the entire shell addition process. Therefore, the shell must progress in a tubular fashion. For example, it can’t form a large sphere or dodecahedron and live in there, since the process of making this shape would not obey these rules and at some point in the shell making process the opening would have to be too large or too small.
Surface Area vs Volume The volume of a cylinder is maximized when the hight ($h$) is the same as the diameter ($2r$). As the snail grows and the height increases, the diameter also increases. This seems like this would minimize surface area and maximize volume. To compare, say the foot region has area $B$ and grows height $h$. This will produce the same new volume for any foot shape since it is approximately base times height (neglecting any curving taking place as the shell twists). A circular foot is more effective since for a circle and square of equal area, the square has side length $\sqrt{\pi}$ and the circle has radius $1$. An increase in height by $h$ results in a surface area increase of $4*\sqrt{\pi}*h$ for the square opening and $2*\pi*1*h = 2*\pi*h $ increase for the circular opening. Both of these are positive values, so for comparison I will square them and obtain $16*\pi*h^2$ and $4*\pi^2*h^2$, respectively. Now, $16*\pi*h^2 > 4*\pi^2*h^2$ since $4 > \pi$. Therefore, the surface area of the added material to the circular opening is less. One can use a similar argument for any regular polygon of $n$ sides which form an opening of equal area to a circle. The circular opening will lead to less surface area of added material of height $h$, but will provide an equal amount of new volume. The snail must start with some sort of home and to add to it, it must progress in some sort of tubular direction. Otherwise, it is closing itself in or increasing the opening to be become too large for the foot to close it. Furthermore, the most efficient way to progress in a tubular fashion is to have a circular opening, as I argued above. Strength Another interesting aspect that I thought of was the strength of the shell with respect to its geometry. I found that the arc (in other words a circle) is the strongest structural shape. The sphere is the strongest $3$-D shape in nature since it distributes stress evenly along the arc instead of concentrating it at a point. I would think that this would also be a large benefit of having a progressing circular tubular shape (i.e. a cylinder) since it is based off of a circle and likely uses this to reinforce the shell strength. This is especially important if the snails want their shells to be hard for crabs to crack. The circular shape redistributes the force of the crushing pinchers of the crab. Now, given that the shell must progress in a tubular manner that is circular in shape, and explanation for the spiraling of the shells is that this also adds extra strength to the tubular structure and reinforces other areas of the shell. There are two variables Surface and Volume which need to be optimized at the same time. Surface needs to be minimized and volume needs to be maximized. Fact: Maximizing the volume for a given surface area is the same as minimizing the surface area for a given volume. (able to prove) Does it mean we only care about one side? Such as controlling same volume and find the best shape of minimizing surfaces? Isoperimetric inequality: The most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere. (like the design of water bottle or soda bottle. Minimize the surface and production cost)
I have two ideas:
Snail shell can have other shapes. Why this shape? This is related to questions raised before. Other shapes cannot minimize surfaces. We can test on different shape: cylinder, cone, prisms, polyhedrons, platonic solids…..We can use the ratio: S/V, the smaller the ratio, the larger V, smaller S. I have done some of the calculation. Another question is that different generation has different size. How to generalize the overall shape of snail? Like a cone composed of many curving cylindrical shapes? 2. For each generation, there must be a ratio between parameters of the shape, like radius and height. This ratio maximizes the volume. This implies self-similarities. Snails need to keep the ratio to maximize its volume and minimize surface. I will think more about PCA and self similarity as well as reading papers posted. The current question we arrived at in our first meeting: “Does self similarity – and in particular planarity of aperture – maximize volume increment per shell area increment?” General reference to shell shapes and patterns http://algorithmicbotany.org/papers/shells.sig92.pdf http://www.mi.sanu.ac.rs/vismath/lucca/ Famous paper by Raup (on our shared google folder): https://drive.google.com/a/smith.edu/#folders/0B6fiafNScUxbV3hOeGdhV0d5ZTA
This is the product of a previous 301 group, now on Wofram demonstration: http://demonstrations.wolfram.com/ShellParameterSpace/
This looks like the state of the art in shell modeling (Gorieli et. al.): people.maths.ox.ac.uk/moulton/JMB-Growth_kinematics.pdf http://eprints.maths.ox.ac.uk/1476/1/finalOR01.pdf
Measuring self similarity – connection to principal component analysis? I did a Principal Component Analysis and self-similarity search on google. These are some of the articles I got: Ozkurt, T.E., Akgul T., Baykut S.: Principle Component Analysis of the Fractional Brownian Motion for 0 <H< 0.5. On the Use of Principle Component Analysis for the Hurst Parameter Estimation of Long-Range Dependent Network Traffic – Melike Erol1, Tayfun Akgul2, Sema Oktug3, and Suleyman Baykut4 Strang’s linear algebra (related to a MIT MOOC) has a chapter on linear algebra as applied to statistics, and in particular to Principal Components Analysis This looks like a decent survey on PCA: http://www.snl.salk.edu/~shlens/pca.pdf Isoperimetric inequalities The question we ask: “What does self-similarity optimize?” and the one we zoomed in in our first meeting “Does self similarity – and in particular planarity of aperture – maximize volume increment per shell area increment?” is similar in flavor to the very classical isoperimetric inequality: given a fixed surface area, what is the shape of the surface that encloses the largest volume (you guessed: the sphere). That’s true in all dimensions. I’m hoping we won’t have to deal with such technicalities! http://en.wikipedia.org/wiki/Isoperimetric_inequality#CITEREFFederer1969 An actual proof: http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf Today we measured snails for the high spire/low spire scaling experiment and fed them to the crabs. We then cleaned out the Burton aquarium and the SR201 aquarium. After lunch we changed the algae in the snail containers for the CP and HH snails. Lastly, we took the snails out of the dye and put them into their containers with new water and fresh algae. This morning, we imaged all of the snails for the shape experiment on the microscope. We then finished the measurements for the high spired shape experiment snails and put all of them in to dye overnight. We scored the snails from the 3rd feeding trial for the crab high and low spire scaling experiment. We cleaned around the lab and organized/stored the snails from the crushing experiment. |