Biologists often speak of embryos "developing planes of symmetry", sometimes meaning right-left mirror image symmetry, but other times meaning geometrical properties that are really examples of the loss or lack of symmetry (such as when animals are said to have "3 planes of symmetry", anterior-posterior, etc. Biologists also speak many plants and some animals as having "radial symmetry" and regard this as a consequence of being sessile (non-motile), or of having sessile ancestors, but they do not make any systematic analysis of why this should be true, much less whether it is true.
Actually, for just about the last 100 years physicists, mathematicians and even chemists have been delving quite deeply into abstract questions of the classification of different kinds of symmetry, and the relationships between the symmetry of causation and the symmetry of effects. Thier conclusions have been very different from the intuitive, unexamined assumptions that biologists are used to!
In particular, henomena that nearly all biologists think of as "the formation of symmetry" are really examples of the destruction of symmetry. in other words, the removal of loss of symmetry elements.
Questions for you to consider on your own:
Can you give some examples of animals' bodies, single celled organisms, organs of plants and animals, etc. that have the following symmetry properties?
1) A single plane of reflection symmetry.
2) Two perpendicular planes of reflection symmetry.
3) Three mutually perpendicular planes of reflection symmetry.
4) Five planes of reflection symmetry.
5) Six planes of reflection symmetry.
6) 7 planes, 8 planes, etc.
7) When biologists say that something has "radial symmetry", what do they really mean?
8) What kind of symmetry does a cone or a hemisphere have (in terms of reflection symmetries)?
[ a good term for this is "axial symmetry" ]
9) When a developing egg that had previously had axial symmetry develops into an embryo that has only a single plane of "bilateral" (reflection) symmetry, is it more accurate to say that symmetry has been created, or that it has been destroyed.
10) Can something have 3 non-equivalent planes of reflection symmetry? Why or why not?
What about other odd numbers of planes?
11) Can something have 2 equivalent planes of reflection symmetry? What is a specific example?
12) Do such things have any other planes of reflection symmetry, different from these?
The letters N, S and Z do not have reflection symmetry. What they have is called rotational symmetry. Each has two-fold rotational symmetry about an axis.
13) Can you think of shapes, biological or otherwise that have 3, 4, or 5-fold rotational symmetry?
Notice that the letters H, I, Y and O also have rotational symmetry (how many fold, in each case?)
The point is that N, S, Z, and such things as propellers etc. all have rotational symmetry without also having reflection symmetry. Shapes like, say, the letter F are sometimes said to have one-fold rotational symmetry!! After all, if something is shaped such that when you rotate it 120 degrees, that makes it look the same, than we say that it has 3-fold rotational symmetry; and when rotating it 180 degrees has this effect, we say it has 2-fold rotational symmetry, so what SHOULD we call something that only looks the same if you rotate it 360 degrees? This is not quite as stupid as it sounds.
15) Could propellers serve their functions if they did not have rotational symmetry?
16) The sequence of patterns below were generated by a cellular automaton; what combinations of symmetries do they all have?
17) What about these ones?
and these ones:
18) Can you formulate any hypothesisabout the relationship between the symmetry of an initial pattern and the symmetries of the subsequent patterns that are generated from it by repeated iterations?
19) In what respects does the following series cause you to modify your initial hypothesis?
20) Besides the symmetry of the initial pattern, what else governs the symmetries of patterns generated by a cellular automata (Hint: What is different about the sequence below?)
21) In the cellular automaton known as "The Game of Life", random initial patterns usually degenerate into a combination of stable patterns, or sometimes patterns that oscillate stably. Using the program provided on the disk, find out whether these "end patterns" tend to be symmetrical or not, and what kinds of symmetry they tend to have.
In their earlies stages of development, the eggs of some marine algae such as Fucus and Pelvatia, have spherical symmetry. Their ymmetry then becomes reduced to axial symmetry by the formation of a rhizoid (root-like) protrucion at one end. The location where the rhizoid will be formed can be controlled by any of several environmental factors, incuding pH gradients, the direction of light etc. and is a main subject of research in the laboratory of this department's former chairman, Ralph Quatrano.
22) Can you see any logical relations between such symmetry breaking phenomena and the rules that you formulated in relation to the symmetries of the cellular automata, above?
Most animal eggs (but NOT Drosophila, by the way) start out having axial symmetry, but then reduce this to some small number of planes of reflection symmetry (8 of then, 6 of them, or in vertebrates only 1 of them).
[In echinoderms you might guess it would be 5 such reflection planes, but actually their embryos develop one plane of bilateral symmetry, and then later expand this back up to 5 planes.]
As with the algae mentioned previously, these cases of symmetry breaking can be influnced by external cues, in particular the direction of gravity and the ocation where the sperm enters the egg.
In bird eggs, the plane of bilateral symmetry becomes fixed such that the head end develops along whichever side of the embryonic disc was lowest at some crucial time of determination. In the case of frog eggs, the head end develops on the side toward where the sperm entered. When Ancel and Vittemberger tried fertilizing eggs exactly at the animal pole (so as to deprive them of this direcional cue, by having the sperm enter at a point on the axis of symmetry), the developing embryos still developed normlly with a head and tail, with the bilateral axis them being determined by the direction of gravity - unless this also was parallel to the axis, in which case the choice was said the be made "randomly".
22) Discuss what the word "random" might mean in this case. If you don't know what the next "fall back" variables are, is that the same as the directional choice being random? But what if there are a large number of different alternative "fall backs"? Is that "random". Or is it an example of a catastrophe in Thom's sense?
There is a lot of recent work from the laboratory of John Gerhard at U.C. Berkeley concerning the cytoplasmic rearrangement processes that occur following this decision process
23) The capsids of viruses usually have a very high degree of symmetry: the great majority have either icosohedral symmetry or helical symmetry. Can you see why having such high degrees of symmetry allows them to minimize the number of different capsid proteins that their genes need to code for, in order to contain any given volume of genetic material?
Another type of symmetry is called magnification or dilation symmetry. When magnifying the size of some structure or pattern leaves it looking "the same" in the sense of being superimposible on the geometry it had before magnification, this is called dilation symmetry. A coiled snail shell is an example. In this example, notice that the shape combines rotational and dilation symmetry, and you can also make the shape look "the same" by certain combinations of maginfication and rotation. If you had a color pattern in which the stripes and spots got bigger and bigger, as well as further and further apart, as one progressed in a certain direction, then you could say that this pattern has a combination of displacement symmetry and dilation symmetry, with this much dilation being "worth" that much displacement.
In fact, you can even invent your own new kinds of symmetry appropriate for whatever subjects interest you.
Do you have an opinion as to why biology has not yet developed such a close interrelationship with mathematics? Are living things inherently less quantitative? Or do biologists make enough quantitative measurements? Or are biological phenomena inherently not so consistent in their properties, from one instance to another? Or do biologists just not know enough mathematics. Alternatively, do you think that perhaps the right kinds of mathematics for biology have yet to be invented? Does biology need kinds of mathematics that are less quantitative and more qualitative?
Do you think we need a closer relationship between mathematics and biology? Are biological phenomena just so much more complicated than physical ones, so that the mathematics needed to deal with them is just too complicated and difficult for anyone to manage? The great physicist, Richard Feynman said that (long ago) physicists took all the kinds of problems that were easy enough for them to solve, and called this set of problems "Physics". At the same time, they took the sets of problems that were too difficult for them to solve, and called them "Chemistry", "Biology", "Psychology" etc. Could there be some truth in this, do you think?
If you look on the journal shelves in the library, you may be surprised to discover how many journals there are entirely devoted to attempts to apply mathematics and computers to biological problems (as well as many articles of this type in other journals). There are also hundreds of books on the subject.. On the other hand, if you try to read these articles, you may be disappointed by their apparent complexity, not to mention their lack of applicability to real biology. My own opinion is that mathematical biologists/biological mathematicians tend to operate on their own, with far to little interaction with, or influence on the rest of biology. Someone has written: "Of 4 possible categories in the application of math to biology, only one is missing: Good math applied to good biology!".