And A Little More Russian Humor

February 19, 2014 2 Comments

The following examples of Russian humor were taken from the Russian magazine, Laughter All-Around.

1.  –I heard that you decided to back to your husband.

— Yes.  I could no longer stand to see him enjoying himself so much!

2.  Little Elsie was finishing her prayer.

–And one more thing, Dear God.  Please send some clothes to those poor naked girls that Daddy looks at in his magazine.

3.  –Just imagine what a catastrophe!  Yesterday my three-year-old son threw fifty pages of my new novel into the fire!

–What!  He read it already?

4.  –Mommy!  Why did the wolf eat the grandmother instead of Little Red Riding Hood?

–Go to sleep, my precious one…Maybe he wanted dried fruit.

5.  A married couple was taking a stroll through the forest.  The wife says:

—  What a wonderful spot to rest and have a snack!

—  You know.  You may be right.  Fifty million ants can’t be wrong.

6.  From a theater review:

–Art demands victims and the hall was soon filled with them.

7.  A wife, who had just gone fishing with her husband, speaks to her friend:

–It turns out I did everything wrong!  I spoke too loud.  I didn’t bait the hook properly.  I didn’t cast the line in the right place.  I didn’t get a strike the right way.  And what’s more:  I caught more fish than he did!

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Robert Brooks: The Wizard of Shape, part 3.

February 15, 2014 Leave a comment

RW:  Dr. Brooks, could you explain or give examples of geometrical shapes derived from a sphere?

RB:  Let me give you an example of the torus, which resembles a doughnut.  This was part of a lecture I gave in my class, which dealt with the Classification of Surfaces.  This is one of the great theorems of topology.  To set up the statement of the theorem, take a sphere(see Figure 3).  Now take a torus(see Figure 4).  Then take a two-holed torus(see Figure 5).  And now down below you have a projective space(see Figure 6).  It’s not oriented, so you can’t draw it properly.  And then take the Klein bottle, which you also can’t draw(see Figure 7).

RW:  Right.  The Klein bottle is the surface that continues into itself without end, somewhat like a three-dimensional Mobius strip.

RB:  Yes.  Let’s return to the projective space(see Figure 8).  We’ll put a little handle on it(see Figure 9).  You can take other spaces, and put more handles on them.TOPO 4A

RW:  So you’re taking the basic sphere and adding holes.

RB:  Well, think of it as adding handles.  A torus is a sphere with one handle, and the two-holed torus is a sphere with two handles.  To understand what I mean by handle, just think of what’s attached to your luggage.  So, if you think of your luggage as a sphere, and attach a handle to it,  you get Figure 10a.  And then if you start shrinking the sphere down and pull the handle, you get Figure 10b.  Eventually, your shrinking and pulling will result in Figure 10c.TOPO 5A  And so the shape in Figure 11(a-c) is the same as pulling the two handles on the sphere. TOPO 6A And then you take the real projective plane(the space of all lines going through the origin.  It is obtained by identifying opposite points on a sphere.  It is also called the cross cap.) and the Klein bottle, and you put handles on them.  By taking a square, and folding the appropriate ends such that the arrows are lined up and pointing the same way, you can obtain the shapes we have been discussing:  torus, Klein bottle, sphere, and real projective plane(see Figures 12a-12d).TOPO 7A

First, I will define a surface as an object such that any sufficiently small piece of it can lie in the plane.  Second, I will assume that the surface has no boundary.  For instance, if you puncture a sphere, it goes out like a balloon, and you can imagine it continuing like an infinite plane.  The third thing that I’m going to insist on is that the surfaces are compact, which means whenever you have infinitely many points, they have to pile up.

RW:  So by compact we’re placing a limit?

RB:  Yes.  Then the question is, “What are all the possible surfaces, if you call two surfaces the same if you can obtain one from the other by deforming them through stretching?”(See Figures 13-19). TOPO 8A And the theorem states that these shapes form a complete list.  Okay, take the sphere, and add a number of handles on it.  Take the projective plane, add a number of handles on it.  Take the Klein bottle, and add a number of handles and that will form all possible surfaces.  Okay, in the terms that I specified in this theorem are all the two-dimensional shapes which don’t have any boundary and are compact.  This is it.  It’s beautiful, you know.  I don’t know if you count this as an abstract theorem.

RW:  Well, it’s a generalization, which is an abstraction of sorts; moving from a specific example to some kind of rule.

RB:  From that point it is an abstraction.  You go to the proof, and, in step one, you list all the parts that such a surface might be built out of.  There are seven parts.

RW:  It’s interesting when we look at these shapes that some of them look very familiar.  The upside-down version of the Y-shape looks like a pair of pants on a store dummy.

RB:  Actually, the technical term for this shape is “pairs of pants.”

RW:  So your point is that there’s something very concrete about all of this.  We can recognize some things as ordinary objects.  Yet, it’s also true that we are aiming for certain generalizations about these objects.  This seems to me to be somewhat abstract.

RB:  I think, in terms of the metaphor you used about looking into the soul of an object, that the soul needn’t be some abstract thing.  It could be made out of bits and pieces of these shapes.  On the one hand, it’s abstract in the sense that you’re looking into the essence of things.  On the other hand, it’s concrete;  you can sit down and play with these shapes with your children.

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Robert Brooks: The Wizard Of Shape, Part 2.

February 13, 2014 Leave a comment

“A man’s reach should exceed his grasp

or what’s a heaven for?”–Robert Browning

RW:  Perhaps you could tell us something about the origin of topology.

RB:  The founder of topology is considered to be Henri Poincare, who had a very strong philosophical outlook towards mathematics and science.  He was led to topology through differential equations, and, in particular, by the “N-Body Problem”, which depicts gravitation.  This situation raises the following questions:  What’s going to happen to these “N”(number) of bodies in the long run?  Are they going to settle into some definite pattern?  Are they going to collide with each other?  Are they going to fly off?  The prospect of trying to solve such a system of equations is just preposterous.

RW:  Is it because there are too many variables involved?

RB:  Yes.  Poincare also wrote a beautiful essay on probability in which he gave the following example:  Imagine you have some large number of molecules of gas A in a bottle, and you ask, “What’s going to happen to this gas in the bottle?”  Then, you make up some experiment in which you have two different gases in two different parts and you remove the partition.  Now some people believe(Albert Einstein:  “God does not play dice.”) if you could find the location of all the particles and work out the equations, you would know how the system evolved.  But the system is so complicated and the equations so difficult, all you can do is guess.  Poincare wanted to show that you get the right answer precisely because you don’t know all of the factors!  It’s the kind of paradox that he always seems to be looking for in mathematics.

On the one hand, you have a determinist point of view, “Well, if we can just sit down, and calculate all these things, we’ll know the right answer.”  On the other hand, Poincare wrote that just by stepping back from the situation, and relying on insight without getting distracted by all the little things, that you get a very clear picture of what’s going on.  And that’s what topology is all about.

RW:  Now in topology are we talking about the position of points, and the possible arrangement of shapes?

RB:  Well, I tend to think about topology as being certain mental habits;  certain ways of thinking about things.  Let me go to these shapes hanging on my door.TOPO 1ABoth shapes are built out of the same fundamental pieces, in this case a figure “H”.  So the H’s are the building blocks of these surfaces.TOPO 2ATOPO 3AThe blocks are put together in a fairly clever way, so that on the one hand,they’re the same, and on the other hand, they’re different.  They’re the same in that they make the same sounds, but they’re different enough to make different shapes;  one’s longer, and the other’s fatter.  So we put similar pieces together in different ways, and say. “What can happen so that they’re the same in some ways and different in other ways?”

RW:  But topology has this other characteristic that Michael Guillen points out in his book, Bridges to Infinity, that basically deals with the souls of geometric objects, those aspects of their geometry that don’t change under certain kinds of transformations.

RB:  Yes.  It goes back to the image of Poincare and the molecules of gas.  You step back from the millions and millions of little interactions that you have no control over and look into the soul of these objects.  That’s the beauty of topology.

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A Digression: The Dark Side Of My Childhood, Part 1.

February 4, 2014 8 Comments

AT THE SCHOOLYARD

Oh, it is dark in the schoolyard!

The walls are stained with young blood.

How many innocent souls enter here

only to end up writhing in pain.

For this institution chokes its victims,

and leaves them as broken toys.

This was one of my earliest poems(The earliest poem was a love poem to a girl in the 5th grade.) and expresses my feelings toward incarcerated education.  The story probably began long ago when the Los Angeles Board of Education had to come up with a plan to keep thousands of children occupied and entertained for six hours, five days a week.  It seemed like an impossible task.  But after countless meetings, suddenly Dr. Doctor stood up, and shouted:  “I have it!  We’ll create teachers!”  And so it was.  At first, teachers were scattered sparsely across the LA basin.  But hormonal impulses took over, and soon teachers were begetting other teachers, who beget others.  For, as Leonard Bernstein pointed out in his Mass, “God said that sex should repulse, unless it leads to results.  And so we crowd the world full of consenting adults.”  Eventually, there were enough teachers to take care of the thousands of children and schools were created to take care of both teachers and children with police guarding the gates to prevent possible escapes.

“Don’t feed the birds, feed your own little selves!”

The schools were run on the Soviet plan;  strong centralization with a small group at the top making decisions for all, virtually no freedom in expressing ideas that weren’t sanctioned from above.  However, the Board did keep its promise; the teachers were entertaining, although not a lot of fun.  My 6th grade teacher was a true servant of the system.  He even taught us The Communist Manifesto.  This man wore thick, dark glasses and we assumed he had problems with his vision.  That this wasn’t the case, was proven many years later when he was arrested for making pornographic films of children.  My 2nd grade teacher was a heavy, strong, obese woman, who shook you if you misbehaved.  She would shake a child so hard that her cheeks would turn red and she would end up gasping for breath.  The only reason we could think of for her unduly exertions was she suspected a child might have money in his/her pockets.  But the California Gold Rush ended at her door.  Her career came to an abrupt end when she locked a child in a closet and forgot about it.  My 3rd grade teacher used to rap a child’s knuckles with a thimble or ruler, cawing:  “Take your medicine!  Take your medicine!”  She was a wiry old lady with crow’s eyes and a suspicious disposition.  When she went to the great beyond the following year, not one child shed one tear.

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The Obstinate J Rogue River Float 1962

February 1, 2014 2 Comments

We stayed at the Obstinate J Ranch from 1961-1979.  Our cabin was called Steelhead Point, and abounded in mosquitoes, and yellow jackets, which entered whenever we opened our hinged door.  Below us, the Rogue River flowed through a wonderful trout spot, and below that, there was an interesting rapid, which ended in a large hole and several steep waves.  The rapid disappeared after the 1964 flood, and I remember Obstinate J co-owner, George Pearson, driving his tractor in the middle of the river in a vain effort to bring the rapid back.  But the memories remain:  cooking barbecues along the river, finding my first calcite crystals lodged in a basalt boulder, watching numerous eddies twirl struggling leaves, starry, clear nights, Saturn Rock, beyond which you dared not go, and the many floats down the lower rapid.  In the video below, Dad rowed Grandpa Johnny, my sister Nancy and me through the rapid.

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