### Fourth set of review questions for the first hour exam

Sketch an ordinary old-fashioned light bulb, and then use bunches of arrows of different relative lengths to indicate directional amounts of curvature in different directions at four or five different representative locations scattered around the lightbulb surface.

Mark the part of the light bulb surface which consist of "saddle points" in the sense that curvatures are positive in one direction but negative in other directions.

Sketch a saddle, indicating the difference of curvatures in perpendicular directions.

On the surface of a cleaving embryonic cell, are there any saddle points? Hint: yes!

On the surface of a cleaving cell, are there any places where the surface curvature differs with direction?

In the surface cortex of a cleaving cell, where do mechanical tensions need to be stronger in certain directions than another?

If surface curvatures in a cleavage furrow are measured to be 20 times as large in one direction (pole to pole) as they are in the (circumferential) direction, perpendicular to this, then how much stronger does the contractile tension of the cortex have to be, and in which direction?

Sketch any of the following shapes, and draw bunches of radiating arrows to indicate the amounts of surface curvatures in different directions:

A pear, an apple, a lemon, a watermelon, a banana, an onion, a mushroom, a doughnut, a string of sausages, a chicken egg, a football (American football), a cylinder, a cone, an embryonic brain with several lobes, an eyeball (including especially the cornea), the cornea of somebody with astigmatism, the lens of the eye, an individual red blood cell (of a mammal; red blood cells are shaped like footballs in other vertebrates), branching blood vessels

Starting with the light bulb, figure out how each of these shapes could be produced (and be made mechanically stable) by variations in tension of a flexible surface. These tensions will need to differ as functions of location, and as functions of direction at each location.

This is not as difficult as it may seem at first.

Start with the light bulb: Where the curvature is less, therefore the tension must be... what?

Where the curvature is the same in all directions, then how do you expect tension to vary with direction?

On what part of the surface would tensions in perpendicular directions work in opposition to each other?

On the surface of a cylindrical pipe, the circumferential tension is twice as strong as the tension in the longitudinal direction.

If you dip an irregularly bent wire into soapy water, and a soap film forms than spans your approximately circular wire ring, the C + c will be equal to what at each point on the surface of the soap film.
(Hint: this is because T = t for soap films).

Up to this point, assume zero pressure difference between on side of the soap film and the other.

But now imagine dipping an irregularly broken off end of a metal pipe into a soap solution, and producing a single layer of soap film across the ends, so that this soap film's curvature varies from point to point, and also varies with direction. THEN, if you blow air into the other end of the pipe, what rule will be obeyed by all the local curvatures? dP = ?

What if you stretched a thin piece of rubber across the end of such a pipe, causing its tension to be twice as strong in one direction as in the perpendicular direction (call the stronger tension T and call the weaker tension t. Then for this rubber sheet dP (meaning difference in pressure) equals what at every point, in terms of amounts of surface curvature in different directions?

If you blow two soap bubbles, one with twice the diameter as the other, and you merge these side to side (so that there is an area of soap film that is part of the wall of both bubbles), then will the big bubble bulge into the small bubble, or will the small bubble bulge into the big bubble? If the radius of curvature of the big bubble is ten centimeters, and the radius of curvature of the small bubble is two centimeters, then what can you deduce or calculate about the radius of curvature of the area of soap film that is the boundary between the two bubbles? If anything.