Ashland’s Lithia Park

March 11, 2014 5 Comments

IMG_1410 ASituated beneath the Siskiyou Mountains is a 93 acre refuge called Lithia Park.  The park was built in 1915 by John McLaren and retains some of its original features.  It is in the town of Ashland Oregon and meanders along the sparkling Ashland Creek.  In recent years, crowds have become a major problem, so weekdays are best.  The lower part of Lithia Park features a pond of floating swans that marks the entrance to the Oregon Shakespeare Festival.  The upper part offers a waterfall, a pond of ducks, a band shell, used for concerts and a series of intriguing steps and bridges.  Lithia Park is a favorite spot for hikers, actors, who want to rehearse in the shade of trees and families seeking a beautiful picnic spot.  Children love to play in the shallow, clear water of Ashland Creek, and are heard often, laughing and cavorting.  I offer some photos of the upper part of Lithia Park.

An aside:  For visitors that are new to Southern Oregon, one must realize that the towns of Medford and Ashland are like different countries, so great are the differences.  Ashland has had a teacher’s college for years(Southern Oregon College).  It recently became Southern Oregon University.  However, it is a university in name only.  It offers no doctoral programs.  That was the agreement reached with other Oregon universities to avoid competition for students.  In general, Oregon has been a state of ecological awareness, but poor education.  At one time, it ranked 49th out of all the states, outranking only lowly Louisiana.  But the Oregon Shakespeare Festival brought in many people from the arts, and Ashland offers more educational opportunities than any other Southern Oregon town.  People throng daily to walk the streets and investigate the many shops the town offers.  By contrast, Medford is not a town of walkers, and does not attract many visitors.  West Medford is notorious for crime, poverty and drug use.  Type 2 diabetes is the illness of choice for women, since obesity is rampant.  Alcoholism is the illness of choice for men.  The youth prefer pot or meth, because of its availability and relative cheap cost.  Medford abounds in single mothers with multiple fathers, while Ashland has the region’s largest gay community.  The poverty is so bad in Medford, that during a recent teachers’ strike, the Medford Superintendent, Long, stated that more than half of the children were either receiving free lunches or were getting them at a discount.  This does not mean there are no poor people in Ashland.  Just that they are less visible among the teeming citizens.  Ashland is the most liberal community in Southern Oregon.  Medford, and all other communities in the region are far more conservative.  A friend of mine, when he arrived from the Bay Area, told me he thought of Medford as a different country.  And, in a way, he’s right.IMG_1235IMG_1243IMG_1248IMG_1249IMG_1306IMG_1269IMG_1341IMG_1285IMG_1355IMG_1408IMG_1234

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The Amazing Dodie Hamilton: One Of Medford’s Treasures

March 1, 2014 1 Comment

“My art is my life, and it has been since I was very young, scribbling away at drawings on every scrap of paper I could find…”DH 2DH 1

 

Glendora “Dodie” Hamilton has been a major part of the art community in the Rogue Valley since her arrival in 1982.  I have had the privilege of working with her on a number of projects, from small town histories to the study of form and she has always embraced my work with enthusiasm and a willingness to do something new.

Her indefatigable spirit led a Missouri girl to the distant state of California where she taught English and art for many years.  During her California stay, she attended many art courses and workshops, working primarily in oils, acrylics and pen and ink drawing.  When she came to the Rogue Valley, she began to focus on watercolor.  Her favorite subjects are the flowers and landscapes of the Rogue Valley, but she has also done children’s illustrations and abstract renditions of shapes.

Dodie is now in her nineties, but she continues to paint, offer art workshops and she remains an active executive member of the Art and Soul Gallery in Ashland.  Despite her age, she loves to travel and recently held two workshops in Mexico.   She also takes workshops with other artists.  The amazing Dodie Hamilton continues to surprise with her zest for learning, her willingness to share and her desire to explore new horizons.  Dodie, who now lives in East Medford, is indeed one of Medford’s greatest treasures.  Please visit her website at:  dodieart.comDH 9ADH 8ADH 15DH 14 DH 13DH 12

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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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