Answers to Questions About Curvature, Stress, and Symmetry|
1) A sphere; and also a flat plane, have the same curvature in every direction at all points.
2) A bigger circle has a smaller curvature.
b) The lower part, near where you screw in the bulb, has zero curvature in the up & down axis.
c) The saddle points are at the location drawn here.
5) It will be a sphere.
6) Its surface will bulge outward where you weaken the rubber; & be flatter where you strengthen it.
7) Then it would become cylindrical.
8) Because the tension is twice as strong in the circumferential direction, then it will rip like this:
9) The cells and collagen fibers are aligned mostly in the circumferential direction (around, not along)
10) That would allow blood pressure to inflate the artery surface into a bigger and bigger hemisphere.
11) The strength is in the wrong direction, but not weaker; it might even be stronger, but burst!
12) The organ or the organism will become spherical.
13) A stable balance of forces minimizes free energy only for the special case that all forces are reversible (like elasticity), but NOT if any of the forces expend energy, like actomyosin contraction.
14) Then some force must not have spherical symmetry, either because it changed, or a new force.
15) Weakening of tension allows the brain area to be inflated by pressure. Cell enlargement and division are a possible cause of this weakening, but not that part of the tension due to collagen or active contraction of cells. Hemispheres bulge where tensions in all directions have become equal.
17) A cylinder.
18) A cone.
19) The narrowest part of a cleavage furrow consists of saddle points; so do the places between your fingers, closest to your palm.
20) Because there are directions in which the surface curvature is positive, and other directions in which the curvature is negative, then there must be two boundaries between positive and negative, with the curvature being zero in these boundary directions.
21) P=C*T and P is zero, and T is positive, so C must be zero
22) P=C*T everywhere at every direction; T is constant in every direction (for soap films) and (P=pressure difference) P is zero everywhere, then the curvatures in perpendicular directions must average out to zero. (You might not expect soap films to be smart enough to figure that out!)
23) On such a surface, ALL the points are saddle points. (Soap films are smarter than you think!)
24) The soap film bends to that curvatures are equal and opposite in perpendicular directions, at every point. A computer can calculate what shape this will be, but soap calculates it faster. In WW II, soap films and rubber sheets were often used to make certain calculations, instead of doing math!
25) Unlike soap films, rubber sheets are capable of having different tensions in different directions;
so P = T*C + t*c, where T and t are tensions in maximum and minimum directions, and c & C
are curvatures in those same two directions (and if the maximum and minimum directions for curvature are not either the same or perpendicular to the directions in which tension has its maximum and minimum, then this equation is an over-simplification). For cylinders, these maximum and minimum directions are the same for both variables.
26) The curvatures of the ellipse are largest at the ends and smallest at top and bottom.
27) The curvature is larger for the cornea, and smaller for the rest of the eyeball;
and is constant within each area, so that both are shaped like parts of spheres. Except right at the boundary of the cornea, curvatures everywhere are isotropic (the same in all directions at each point).
The edge of the cornea consists of a ring of saddle points, where amounts of tension suddenly change.
A safety-valve called the canal of Schlemm is located here, and glaucoma results from its failure.
28) The sides of the inner tube toward the center are saddle points. Circumferential (around the direction of the wheel) curvature varies in a gradient from minus values facing the inside, and positive values facing outward. Curvature around the tube itself is constant everywhere.
29) The small bubble has a larger C, so it must also have a larger P!
Half the diameter would mean twice the curvature, would mean twice the inside pressure.
30) Let's call the pressure inside the smaller bubble Ps, and use Pb to mean the pressure inside the bigger bubble, and regard P as zero in the surrounding space.
Ps=C1*T for the part of the small bubble not facing the other bubble.
Pb=C2*T for the part of the big bubble not facing the smaller bubble
Ps-Pb= some positive number = C3*T
C1*T-C2*T=C3*T, and T is a constant everywhere and in every direction, for soap films.
Therefore C1-C2=C3 (I think? But I just did this in a big hurry!)
The point is that forces exerted by materials on each other can create (and maintain) predictable, consistent geometric shapes. In fact, the materials won't STAY in those geometric shapes unless the various tensions and pressures obey the correct equations; so how likely is it that some completely different set of molecular signals tells each part where to form, AND what tensions to have, as opposed to the genes creating the shapes by adjusting the mechanical tensions? I am biasd on this!!
31) Only if the tensions have the same strength everywhere, including along the boundary.
32) That would mean the pressure in the smaller balloon is NOT stronger than the pressure on the bigger balloon; so the tension must be MUCH stronger in the surface of the bigger balloon
33) The parts of the surface the bulge out more must have started out with weaker tension, but after inflation they have stronger tension. Parts that are hemispherical, or semi-spherical (Mickey's nose?)
Must have equal strengths of tension in all directions at every point within such a region. etc.
34) P=C*T, therefore in directions that C is zero, then P is zero, no matter what how big or small T is.
35) The circumferential curvature is very big at the small end, and very small at the big end.
The air pressure is exactly the same in all parts of any balloon, including this one; but it feels like there is more pressure at the fat end, and the rubber is stretched MUCH more tensely there.
36) The elastic substratum method (so called "traction force microscopy") was invented here at UNC, and Lev Beloussov in Moscow has used a) asymmetrical cell shapes, and b) gaping of small wounds, to map tensions in living embryos, and polarized light can make forces visible in living animals.
37) Not enough research has been done on this, despite the mechanical abnormalities of cancer cells.
38) Computer simulations can be used to predict shapes from forces; the reverse can also be done, by the same logic that you had to develop when trying to answer these questions. Someday, but not yet, surgeons will reshape anatomical structures based on such predictive reasoning and methods.
NOW! Do you begin to see the POWER of the Dark Side of Forces?