There seem to be a wide variety of shapes in the biological world . In 2d we have various shapes of leaves. In 3d we have anthills , trees , fruits and animals with different shapes. All of them are closed surfaces , mostly convex. The jack fruit has thorns on its surface . Why is this so? What is being optimized?How can we fit a mathematical function to describe these shapes? This is a general exploitative question and all are welcome to suggest views.
This is a good question. From shapes and sizes of living things in nature, I will add behaviors as well, why they behave differently according to seasons, is survival and continuity in existence. As most of you pointed out, for instance the shape and sizes of trees in desert areas (tiny in an optimal way) are clearly indicative of a purpose driven natural process to optimize resources of water beneath and survive in the desert. For the actual sequential processes, the Fibonacci sequence is one vital process in which branches of brocolis, cabbages and others multiply. These acts and process are purely mathematical and the very reason we say nature is mathematical. Galileo here is a right person to mention, "The universe can not be read until we know its language in which things are written, which is mathematics"
These are really interesting and useful problems. We have a lot of methods to describe carves in 2D and sometimes in 3D. But for various of shapes of the natural objects, The mathematical system nowadays will be helpless. I think there should be some new mathematical systems focusing on these specific problem of shapes. But who knows!
This is a complex question. Shapes of real biological objects are not necessarily the results of optimization of some (simple analytical) functions. There are two major aspects in biological evolution: individual dynamics of development (for example, branching of a tree) and evolutionary selection. Indeed, selection can be viewed as some optimizer, however, in more dimensional space that includes not only geometry, and only among the objects that are results of individual growth.
At the level of cells shapes are simpler that at the level of body. Because body is an interaction of several organs, and their size depends on the functionality (recall the difference in the shapes of male and female bodies). And individual growth can lead sometimes to so nice structures as fractals (mentioned above). See the example of broccoli.
One can often see spiral shapes in sea shells. It looks like a good form for dynamic development, rather than some optimum (for example, a ball is optimal to resist water pressure and is often used in technology).
Even discrete quantities (like number of legs) take different values: 6 or 8 for most insects, 4 for mammals, 5 for sea stars, etc.
Dear Narasim,
Yes, Fractal Geometry can describe all such shapes.
For example, the ant colony has the shape of the anthill .
Hope you find the following links are useful.
https://www.pinterest.com/pinarea/fractals/
www.sci-insect.com/an-aluminum-fire-ant-colony/
https://books.google.com.lb/books?isbn=1118099524
best wishes
This is an interesting question. In addition to the very helpful answers already given, there is a bit more to add.
Tessellate an image of biological object using a Voronoi function in either Matlab or Mathematica to learn about the geometry of the object.
This is a good question. From shapes and sizes of living things in nature, I will add behaviors as well, why they behave differently according to seasons, is survival and continuity in existence. As most of you pointed out, for instance the shape and sizes of trees in desert areas (tiny in an optimal way) are clearly indicative of a purpose driven natural process to optimize resources of water beneath and survive in the desert. For the actual sequential processes, the Fibonacci sequence is one vital process in which branches of brocolis, cabbages and others multiply. These acts and process are purely mathematical and the very reason we say nature is mathematical. Galileo here is a right person to mention, "The universe can not be read until we know its language in which things are written, which is mathematics"
Hi Prof. Narasim,
Anthills typically come in various shapes and sizes. One of the most pronounced shape and easiest to recognized is the dome-shaped anthill. Thinking mathematically, it is relatively easy to compare and visualize the surface of a dome-shaped anthill as a kind of elliptic paraboloid. It can also be obtained using the paraboloid of revolution. The following article may be helpful in understanding the shape of an anthill dome.
Article Optimal shape of an anthill dome: Bejan’s constructal law revisited
Thanks to all the commenters . Would welcome views on enclosed pdf.
Cheers
Try the following link
Article Biological shape and visual science (part I)
Worth to mention is also the superformula: https://en.wikipedia.org/wiki/Superformula
Dear Antonio Quercia
Thank you very much. Indeed this is what I had in mind. ...Now it would be interesting to do the inverse .ie. given shape find nest parameters of Formula .
Cheers
Additional material:
C. Blanc, C. Schlick, “Ratioquadrics: an alternative model for superquadrics,” The Visual Computer, vol. 12, pp. 420–428, 1996.
A. Jaklic, A. Leonardis, F. Solina, Segmentation and Recovery of Superquadrics, Springer, 2000.
- Badges
- Science topic
- Similar topics
- Mathematics