First set of review questions for the second exam: Symmetry

Questions marked with a * are not easy, but you will gain a lot from thinking carefully about them. I won't ask many such difficult questions.

Regarding symmetry:

1) If you can take a wall-paper pattern and either move it 2 inches to the right, or to the left, or move it 4, 6, 8 etc inches to either the right or left, and it would look the same, then what kind of symmetry does it have? (Let's assume it looks different if you move it one inch, or 3.5 inches, or any distance but multiples of 2 inches). Hint: displacement

2) If this pattern also looks the same after being rotated 60 degrees, 120 degrees, or 180 degrees, then what additional kind of symmetry does the pattern have?

3) If the pattern is all one color, and looks unchanged when moved an inch, or 1/2 inch, or any small distance, and is also apparently unchanged when rotated by any number of degrees, then does it have more or less symmetry than the patterns mentioned above?

4) When an animal develops regularly spaced stripes, in what way has the symmetry of its skin changed?

5) Suppose an animal develops regularly spaced spots: does it then have more, or less, symmetry than a striped animal? (increased or decreased?)
5.5 What combinations of symmetries do Venetian blinds have?

6) Turing's "Reaction Diffusion" mechanism (or "system") is a way to reduce/increase the d----------t symmetry of a spatial pattern?

7) If we have two diffusible chemicals, "X" and "Y", or "A" and "B", or "I" and "P", and one of them promotes increasing the concentration of itself and the other, at rates proportional to its own local concentration, while the other chemical promotes the removal or loss of both substances (at a rate proportional to what?), and if the destroying chemical diffuses several times faster than the other chemical, then what will happen?

7A) What kind of symmetry is "broken" by such a mechanism?
7B) The pattern produced depends on magnifying random fluctuations, but is not random? ** Why not?
7C) Could such a reaction-diffusion system also break reflection symmetry?
7D)* If these chemicals were confined to a shorter length, or smaller area, or smaller volume, would the number of peaks of concentration become less, stay the same, or become larger?
7E) Suppose the variable "A" was not a chemical concentration, but the population density of moving cells, then could it still participate in a pattern generating mechanism? Hint: what could B be?
7F) How would it change the pattern produced by a Turing mechanism if B became able to diffuse faster than previously?
7G*) What if these reactions were going on in a material through which A can diffuse with equal speeds in all directions, but B can diffuse faster in one axis (the dorso-ventral axis) than in the axis perpendicular to this (anterior-posterior)?
7H*) If you were considering the hypothesis that Zebra Fish skin color patterns are caused by some variation on Turing's mechanism, and your method was to look for mutant fish with abnormal color patterns, then make a list of all (as many as you can think of) the different color patterns that could result from mutations in the genes for A and B. For each, sketch the pattern and state the effect of the mutation on A or B, either their properties, effects, or initial distributions in the embryonic fish.

8) When the anterior-posterior axis of an amphibian oocyte is caused to form by the location where the sperm enters, then what change in symmetry has been caused? (And likewise for nematode oocytes).

9*) If you fertilize an amphibian or nematode oocyte exactly at the animal pole, and nevertheless it develops a normal anterior-posterior axis (with only one plane of reflection symmetry), then what does this suggest about the underlying mechanism by which a single anterior-posterior axis is chosen from the infinity axes of reflection symmetry that had existed until then?
(Hint: maybe it can be initiated either by small stimuli, like sperm entry, or when those are not available then maybe random fluctuations can initiate symmetry breaking, as in Turing's mechanism, or as when a punching dummy has its internal weight raised too high?)

10) Which symmetries, or combinations of symmetries are possessed by the following? Starfish? Jelly-fish? Propellers? Flowers? Daisies, Sun-flowers, Orchids, Lilies, Trees? Blastocysts? Gastrulas? Paramecia? Diatoms? Honeycombs? Apples? Bananas? Mitotic spindles? Flagella? Microtubules? Actin fibers? Muscle sarcomeres? Snails? Clams? Limpets? Barnacles? Feathers? Hairs? Claws? Teeth? Lungs? Glands? Lenses of eyes? Vertebrae? Arteries? Muscles and fibers in the walls of arteries? Bamboo? Fern leaves?! * Cauliflower? ** Mulberry leaves! Morulae? Knives? Forks? Spoons? Scissors? {Some of these are rather subtle and difficult; but please give each some thought.}

11) What (abnormal and also normal) planes of reflection symmetry are possessed by the bodies of conjoined twins? (so-called "Siamese Twins")

12*) Suggest reasons, in terms of embryological mechanisms, why conjoined twins are always (usually?) mirror images of each other? Think about whether embryological control mechanisms would interact, including either chemical gradients or mechanical forces, and what would be the effects of interactions between them.
(Hint: maybe what first becomes doubled are the elements of these control mechanisms?)

13*) When human identical twins develop by forming two primitive streaks in the same inner cell mass, then one of these twins will (Usually? Always?) have situs inversus viscerum (aorta on the right side of the heart, etc & everything a mirror image shape and position). Suggest why.

14) When a force, or balance of forces have spherical symmetry, then what shape will they tend to remold a cell into?

15) If you see a mass of cells changing from other shape into a sphere, then what do you tentatively conclude about the forces responsible?

16) If a gene could make an internal force less symmetrical, then could that cause cells and tissues to change from one symmetry to another?

17) Becoming less symmetrical means losing (?) or gaining (?) elements of symmetry? Either way, what alternative kinds of mechanisms can be used to choose which specific planes or axes, or other symmetry elements, will be gained or lost? (Sperm entry? Gravity? Random flucuations? Anything else?) What if you just poked it at some time of special sensitivity?

18*) To prove the occurrence of an actual Turing mechanism, would you need to isolate and identify the actual chemicals? Or can you invent experimental criteria, such as how the resulting patterns are altered by water currents or by more viscous fluids, or barriers to permeability, or something else.

19*) What the symmetries are approximated by the path of a meandering river? (Take a look on Google Earth, or the next time you fly somewhere.)
[Hint: glide reflection, plus something else, that involves narrowing widths and shorter wave-lengths.

20) Coiled snail shells have a combination or rotational and what other symmetry?

21) What phenomenon, discovered by Hans Driesch, indicates that developmental mechanisms of echinoderms have dilation symmetry? Does Driesch's entelechy have dilation symmetry!?

22) Try comparing the symmetries of brick locations in brick walls or sidewalks: how many different combination of symmetries can you find?
How many different ones are possible, in principle? (it's in the teens, I think; not infinity).