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My friend sent me these pictures of a pool that has been abandoned for a long time, and we are curious about the reason behind the peculiar growth of algae in this pattern. The needle-like towers of algae seem to resemble the mathematical equation $z = \frac{1}{x^2 + y^2}$.

My theory is that this shape is kind of an inverted version of a water droplet. While a water droplet is pulled down by gravity when it is in the air, these algae appear to be trying to grow upwards despite the force of gravity. However, I must admit that I am uncertain and lack confidence in this hypothesis. Have you ever encountered a similar pattern elsewhere?

Peculiar growth pattern of algae in a pool

Peculiar growth pattern of algae in a pool

The graph for the aforementioned function

[UPDATE]

The topics and points you mentioned were all interesting, especially the concept of the constant-stress hanging rod and the idea that a function with singularities is not used to model day-to-day physical phenomena. However, what I was hoping for was some evidence to support the naturalness and authenticity of this shape. When I first saw the shape, I couldn’t help but wonder if it was yet another hidden celebrity shape in nature that I wasn’t aware of. However, the consensus seems to be less exciting: a variety of factors have contributed to the development of this stable shape for the algae, and these small Eiffel Towers likely do not possess any remarkable geometrical properties.

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    $\begingroup$ OK... why do you think it looks like $\frac{1}{x^2 + y^2}$ and not some other singular non-negative function like $\frac{1}{\sqrt{x^2 + y^2}}$ or $\frac{1}{x^4+y^4}$? $\endgroup$
    – hft
    Jul 20, 2023 at 19:21
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    $\begingroup$ You kind of seem to be asking: why does it go up... And the answer is probably the same reason why anything in water goes up; it is less dense than water. $\endgroup$
    – hft
    Jul 20, 2023 at 19:23
  • $\begingroup$ Also, the algae clearly can not behave like $\frac{1}{x^2 + y^2}$ for all $x$ and $y$ since there can be no real singularity (the algae doesn't shoot out of the water into infinite space...). So, why not suggest a function like $e^{-(x^2+y^2)}$ instead? $\endgroup$
    – hft
    Jul 20, 2023 at 19:24
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    $\begingroup$ I've voted to reopen because the question seems quite clear: Can this natural phenomenon be modeled—with biophysical justification—using a Gaussian fit, or perhaps with some other constitutive equation? An additional photo was provided to better present the shapes of interest. The concern with a singularity for one particular form seems excessive; all models are wrong, but some are useful within the limitations of their applicable scope. $\endgroup$ Jul 21, 2023 at 2:30
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    $\begingroup$ I think you are much more likely to get a good answer from some biologist working on this. It might even just be that the pool originally had some other plants that would have vertical stems, and the algae had strangled them to death, leaving these towers behind. There are too many possibilities for us to assert definitively which mechanism caused this behaviour. $\endgroup$ Jul 21, 2023 at 4:37

3 Answers 3

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A great many curves (described by exponentials, logarithms, powers, trigonometric functions, etc.) might be fit to the contour of any particular natural phenomenon. Alternatively, there are published "atlases" of surfaces of revolution:

Undirected curve-fitting and shape-matching can provide a useful closed-form analytical expression but may not provide insight into the underlying processes.

As an alternative, we might propose a hypothesis based on biophysical principles and see where it takes us.

Although I've never studied algae growth specifically, based on a research background in mechanical engineering, materials science and biophysics, it occurs to me that the features resemble at least two other distinct features, and a commenter has helpfully described a third scenario:

(1) The constant-stress hanging rod loaded by self-weight only.

(All images from quick online searches of the relevant keywords.)

A classic statics problem is to design a rod or wire, hanging from its own weight, such that its internal stress is everywhere constant. Here, the cross-section area must increase when moving up because the applied weight progressively increases. This problem has enormous importance in the prospective design of a space elevator, where we find that existing materials would require an impossibly large cross section for the required height.

To summarize the problem, for a height-dependent radius $r(z)$, the weight below any vertical position $z$ is $W(z)=\int_0^z \rho\pi r(z^\prime)^2\,dz^\prime$, with material density $\rho$ ($z^\prime$ is a dummy variable). Equivalently, the extra weight per vertical slice $dz$ is $dW=\rho\pi r(z)^2 dz$. We require that $\frac{W(z)}{\pi r(z)^2}$ is a constant stress $\sigma$.

What function solves $\frac{dr(z)}{dz}=\frac{\rho r(z)}{2\sigma}$? The exponential $r(z)\sim\exp\left(\frac{\rho z}{2\sigma}\right)$, and this provides an exponential/logarithm surface of revolution.

What does this have to do with algae growth? Assuming the algae is buoyant, upward movement is spontaneous, and thus we might expect vertical stringers of algae to form. But the vertical and bending stresses arising from gentle water motion could prompt proliferation in a mechanosensitive manner, in the same way that plants strengthen in the wind. Perhaps vertical assemblies of algae efficiently evolve to a stable level of constant stress, similar to our bones:

So the first hypothesis is that the algae features taper because of an intrinsic or externally imposed requirement for near-constant stress. This allows equations to be derived as described above. If the underlying biological behavior is confirmed and if the features are similar to the predictions, then we might be prepared to present an argument for causality, subject to strict and rigorous examination through experiment.

(2) Dendrite growth during solidification or precipitation

In a supersaturated solution or supercooled melt, protruding dendrites spontaneously form because any region that randomly pokes out a bit is exposed on more sides to material with a driving force to join it:


What does this have to do with algae growth? Algae that emerges slightly from the mat has more access to light and nutrients. The resulting spontaneous growth may always be upward due to buoyancy, but with accelerated growth and mounding around the source. (I don't know of a suitable constitutive equation, but the evolving shape could be numerically evaluated through kinetics simulations of light and nutrient absorption and growth.)

So the second hypothesis is that the tapered shape arises from instability-driven growth of randomly protruding algae.

(3) The deformation of an elastic membrane from a point force. (Thank you to @ChristopherJamesHuff for mentioning this in the comments.)

Waste gas from metabolic activity could collect under an algae mat and nudge it upward from buoyancy. Like the other two scenarios, this features positive feedback to draw out a tapered shape: Gas flows to the highest point, and so any random perturbation is unstable. The corresponding approximate (inverted) appearance is as follows, with the clamped boundary condition representing where the mat hasn't (yet) detached:

Although a closed-form equation may not exist for this shape due to the nonlinear nature of the mat elastically and permanently deforming (while proliferating), the process could be numerically simulated, as above.

The third hypothesis (contributed) is then that the tapered shape arises from gas-buoyancy-driven detachment and ascent of portions of the algae mat.

Now, one or more or none of these hypotheses may find experimental support. My point is that it's interesting to note that a shape in nature looks like a certain mathematical function, but it's arguably more rewarding to connect this—or another—equation with some reasonable biophysical justification.

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    $\begingroup$ It's closer to the first, but more of a deformation of an elastic sheet: the algae forms a cohesive mat, and the algae and things living in the muck under it produce bubbles of gas. Once a bubble tents the mat a bit, other bubbles will converge toward it and increase the buoyancy pulling that part of the mat up. Eventually that part will tear loose and float up to the surface, or the bubbles will be lost and it'll fall back down. $\endgroup$ Jul 21, 2023 at 17:28
  • $\begingroup$ The topics you mentioned were all interesting, especially the concept of the constant-stress hanging rod. However, what I was hoping for was some evidence to support some naturalness and authenticity for this shape. When I saw this shape, I couldn’t help but wonder if it was yet another hidden celebrity shape in nature that I wasn’t aware of. However, the consensus seems to be less exciting: a variety of factors have led to the development of this stable shape for the algae, and these small Eiffel Towers likely do not possess any remarkable geometrical properties. $\endgroup$
    – Tripasect
    Jul 23, 2023 at 1:56
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Why did these algae grow like this in the pool? Are these curves the gravitational equivalents of the bell curve?

Please recall, a "bell curve" is a Gaussian, which in two dimensions has the form: $$ f(x,y) = e^{-(x^2+y^2)}\;, $$ or more generally: $$ f(x,y) = Ae^{-B(x^2+y^2)}\;, $$ where $A$ and $B$ are constants. Such a gaussian or "bell curve" is a non-negative and non-singular function.

My friends sent me these pictures of a pool that has been abandoned for a long time, and we are curious about the reason behind the peculiar growth of algae in this pattern. The needle-like towers of algae seem to resemble the mathematical equation $z = \frac{1}{x^2 + y^2}$.

A function like $ \frac{1}{x^2+y^2} $ has a singularity at $x=y=0$ so cannot describe the patterns you are seeing (at least for all x and y).

To me the patterns seem to widen a lot more than $1/(x^2+y^2)$, which might be better represented by something like $$ \frac{1}{\sqrt{x^2+y^2}} $$ at least over a region far away from $x=y=0$.


Ultimately, it seems like you would like to know why the algae grow upwards towards the surface of the water and kind of "neck up" in a similar way that a pour of water "necks down."

The answer as to why the algae grows "up" is because the algae is less dense than water and so, just like seaweed, it floats up towards the surface, but it is rooted in some way at the bottom so it cannot all float up. Presumably there is some dirt or other mechanism at the bottom of the pool by which the algae is fixed to the bottom.

The exact functional form of the algae will be difficult to determine. However, physicists have researched other sorts of semi-fluid shapes influenced by gravity, such as the formation of sand piles. Looking for this sort of literature might provide a more precise answer.

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  • $\begingroup$ Thank you for your explanation. I don't quite understand your objection to functions with singularities. Why is it considered impossible for the algae to grow indefinitely? What if they did grow infinitely, assuming the pool was an infinite space and atoms were infinitesimally small? Why are functions with singularities completely dismissed from the discussion? $\endgroup$
    – Tripasect
    Jul 20, 2023 at 19:55
  • $\begingroup$ Also, do you happen to know of any relevant literature sources? $\endgroup$
    – Tripasect
    Jul 20, 2023 at 20:07
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    $\begingroup$ Nothing great, but with respect to sand piles you can check out: en.wikipedia.org/wiki/Angle_of_repose and with respect to algae maybe have a look at: e-education.psu.edu/egee439/node/694 and other references therein. Sorry this may not be exactly what you are looking for. $\endgroup$
    – hft
    Jul 20, 2023 at 20:42
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    $\begingroup$ A singularity does not mean "growing indefinitely", it means "currently infinitely tall at one point" - clearly not the case. Generally that sort of function can not describe any physical structure or process, except perhaps a black hole. Whatever the form of the function is, you can work around this by simply adding a constant to the denominator. $\endgroup$
    – monguin
    Jul 22, 2023 at 20:40
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    $\begingroup$ @Tripasect He means something like $f(x,y)=\frac{1}{x^2 + y^2 + 0.1}$, where the constant $0.1$ keeps the function from going to infinity. Instead of infinity, the max value of the function is $10$ (the max is at $x=y=0$). Of course you can adjust the constant to whatever you like, but as long as it is some positive number it will stop the function from "blowing up" to infinity. $\endgroup$
    – hft
    Jul 23, 2023 at 1:47
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Algae are plants, they perform photosynthesis, this results in the production of oxygen which sometimes collects as bubbles on outside the algae. Photo examples at https://www.reddit.com/r/PlantedTank/comments/156wp4u/it_would_seem_that_i_have_inadvertently_created/

Any derivation of a function model to describe the structures you observed should include the upward force of the oxygen bubbles.

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  • $\begingroup$ Sounds like an astute observation. $\endgroup$
    – Tripasect
    Jul 23, 2023 at 2:20

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