Robert Brooks: The Wizard of Shape, part 1.

During my adult life, I became interested in discovering why forms were the way they were, the laws governing the formation of forms, the applicability of forms, and the inks between different forms.  I realized that for the purpose of my study I would have to come up with an appropriate definition of form.  I chose to define form as a perceived structure or concept represented by a definite pattern.  Applying my definition to the study of forms, I saw that forms would fit into three major categories:  NATURAL FORMSCREATED FORMS, and THEORETICALLY-DERIVED FORMS.  Natural forms pertain to those forms which exist in the physical world independent of human beings.  Cloud patterns and ocean waves are natural forms.  Created forms are those forms which arise from the human imagination.  These forms include poetry, architecture, and sculpture.  Theoretically-derived forms involve forms which arise through logic and reasoning.  Cycloids and lemniscate are examples of such forms.

Topology belongs to the last category.  It may be called the “aerobic” branch of mathematics, because it looks at the properties of shapes after they are twisted and stretched.  Topology is a qualitative form of mathematics, involving concrete shapes that ten-year-olds could play with.  Stephen Barr’s Experiments in Topology, offers numerous examples of topological fun.  One such, is twisting a two-sided strip so that it has one edge, resulting in the Mobius Strip.    For those that want to read about the creative, imaginative side of topology, I recommend Clifton Fadiman’s two collections of short stories and verse:  Fantasia Mathematica,  and The Mathematical Magpie.

When I started selecting interviewees for my book The Magicians of Form, The late Robert Brooks, then a Professor of Mathematics at the University of Southern California, was my first choice.  Dr. Brooks taught a course in topology, and had an ability to make the complex simple.  His warmth and enthusiasm put me at ease, and I found myself even more interested in the subject matter of topology.  What follows is an excerpt of an interview that took place in his office at USC.IMG_5961

                              “I think the thing that motivated me was the thought that ‘They’re holding something back from us’.”

RW:  Dr. Brooks, perhaps you could say something about your early interest in mathematics.

RB:  I think I wanted to be a mathematician since I was in the 4th or 5th grade.

RW:  Does that mean you had a natural aptitude in solving mathematical problems?

RB:  Well, I like to think I have a natural aptitude.  But let me tell you a story…  This was in the 1st grade, and we were doing primitive addition, and learning to add several digit numbers together.  Then we began to learn carrying, and it dawned on me that all the numbers we had been given to add up until that time, had been kind of “cooked up”, so you didn’t have to carry.  I was a little upset that no one had pointed that out to me; and I said to myself, “I wonder what else they’re holding back?”  And I must have spent about two weeks adding random numbers together.  Then I came to the conclusion that the only thing you had to know in adding two numbers together was carrying, and then you could any two numbers, no matter how many digits they contained.  But I felt I had to prove that.

This was the first problem I remember thinking seriously about.  I recall working on it for a long time, and I ended up giving up.

RW:  So you had the desire to go from the specific to a general rule, to an overall proof/

RB:  I think the thing that motivated me was the thought that “They’re holding something back from us”, and I wanted to be on top of what was going on.

RW:  You had a certain lack of trust in the whole procedure.

RB:  Absolutely!  And I think one thing that’s so appealing to me about mathematics is its real immediacy;  that you’re basically on your own with the materialand if there’s something thereyou’ve got to find it.

RW:  So you’re the pioneer?

RB:  You’re just about everyone in this business.  You’re the pioneer, you’re the explorer, you’re the critic.  In many cases, you’re the audience.

RW:  Then it’s really your world.  You’re immersed in this abstract universe that you’ve created.

RB:  That’s right….  But topologists have a certain disdain for abstraction.  Topologists want to show what’s there.

The Journey Continues…

“Assume 1 exists.  Now prove 2 exists.”–Thomas D. Hedden’s cynical concept of mathematics

Numbers always fascinated me.  I remember going down to the Rogue River bed to construct numbers out of stones.  I liked particularly the shape of 4.  In fact, when I turn the TV off, I always leave it on channel 4.  4 has seemed to me the “perfect” number.  And when I was a child, I wanted to see 4 and feel the stones that formed it.  3 has been a troublesome number; I had difficulty forming 3s in my writing and the oddness of the number disturbed me.  The mirror twins of 6 and 9 were also troubling.  I enjoyed looking at 1, but felt I didn’t understand it.  0 was a special number, but my multiplication cards somehow excluded it.  I did my times tables from 1×1 to 12×12.  There was no 13.  I’ve felt there is something malevolent about 13, and perhaps the makers of the multiplication tables did as well.  To this day, I mark all my checks that end in 13 VOID!, and shred them.  My accountant doesn’t like the way I handle checks, but how can you deal with such an irrational person as I?

The cabins at Diamond Lake fascinated me.  I would follow one cabin marked 12, and try to construct the whole number sequence.  I remember looking through bushes, circling trees and hills, in an effort to complete the sequence.  78 record albums were also a source for numbers.  In my grandparents’ home at Amesbury near Griffith Park, I recall seeing records of South Pacific on the floor.  I saw 5, 9, 10, 4, but the others were missing.  I was quite disappointed.  However, I made up for the loss when Grandma bought me the complete Columbia album of South Pacific many years later.  Motels, of course, often proved an exercise in futility, because some would start with 100 or some other number.  Still, I considered the number of the motel I stayed at quite special and have the key with its number to one of them!

Multiplication by 0 was intriguing;  What is President Obama x 0?  But division by 0 was even more bewildering.  Common sense tells us that if we divide an object by nothing, we are not dividing, so the result should be the object unchanged.  Look!  I will divide this chocolate cake by nothing.  See this knife!  I will hold it up in the air.  Now everyone, dig in!  But in mathematics, if 5/0=5, then through cross multiplication, 5×0 should equal 5, but that contradicts the way we originally defined multiplication by 0.  What can we do?  Simple! We’ll take the easy way out and say that division by 0 is undefined.  I must admit that multiplying a chocolate cake by 0 is something I cannot fathom!

Xs and Ys hurt my eyes.  Ys and Zs no more please!– childhood verse about algebra

I remember algebra only too well.  The subject inspired my first story, The Tale of the Brilliant Xaquenta Qualzifaz Xitg and the Birth of Algebra.  Xaqenta falls in love with Yakshwe Reginald Yorkes and they meet at a special place called the origin.  At that point, they decide to form a family.  There was a communicative law, so that all members could recognize one another better and an associative law to improve family relations.  The eternal optimist, a small man on stilts called absolute value, was never far away.  But the story faded into fractal dust, and so it has remained.  At the time, I decided there were other worlds to explore. And so, armed with my protractor and a straight edge, I was ready for whatever shapes and symbols I might encounter.

The Journey Begins…

“Grandma, when you die, will they bury you?”


“Very deep.”


“Then I’ll just dig you up again!”  —–Kornei Chukovskij From Two To Five

The journey, which would eventually lead to The Magicians Of Form, started in childhood.  For the book represents a synthesis of the many conversations I had with my Dad and Grandma  Lillian about forms that I encountered throughout my life.  In retrospect, I believe there was an unseen path that was guiding me to complete that book.  Little did I know it, but these apparently innocuous discussions held the seeds of a definite future purpose.

To understand the determination and courage needed to finish the volume, I have to look back to a now distant world:  a world before abstract reasoning had taken firm hold, and banished me from an all-inclusive world.  A world in which sensations, colors, sounds, and forms enticed with a vividness, excitement, and spontaneous directness that become dulled in adulthood.

To go to that special place, I need to summon memory as my guide.  Fragments of thoughts and images fly into my mind:  pine cones scattered along a path, a night sky covered with sparkling stars, the rough red of jasper, sand painting, sticker albums, wooden puzzles of a bus, and Old King Cole, a record player on the ground spinning music, farm lotto, water-colored flowers, The Golden Book of Children’s Verse, and, one verse in particular:  “When I grow up, I will carry a stick, and be very dignified.  I will have a watch that will really tick.  I will have a tall house that is built of brick.  And no one will guess that it’s just a trick, and I’m really myself inside.”, The Big Ball Of StringThe Big Jump And Other Stories, and Gillespie And The Guards( in which a child outwits adults in power), The Five Chinese Brothers(in which every brother has a special skill to keep him from harm), arithmetic problems with shiny colored dots, glasses of lemonade, scoops of chocolate ice cream, dragging a watering can to create my own river in the sandy beach,  Grandma’s Archie the Chipmunk bednight stories, making a miniature golf course out of my parents’ lawn, climbing walnut trees, listening to Walt Disney’s The Grasshopper and the Ants, dancing to Tchaikovsky’s Overture Miniature from the Nutcracker Suite and watching the falls of Lone Pine Creek…

“How high is high?”

Grandma said I asked this question when I was four-years-old.  It was the start of many questions I had about the surrounding world.  My special path was unraveling before me.  The hour glass of time was running.  The journey begins…

How My Great-Grandfather, Irving I. Turner, Taught Me A Valuable Lesson In U.S. History.

IMG_5936Every summer, it was a family tradition to visit Grandpa Turner before our Oregon departure.  He lived in a modest apartment on Vantage Street in North Hollywood, California.  When you entered, your nose was assaulted by cigar smoke, which seemed to permeate every piece of furniture in the living room.  His saltine crackers were in their usual plastic container.  Sculpted dogs of various breeds and sizes greeted you from a shelf.  The TV was the essential component, for grandpa was almost always watching some program when we visited.  He especially liked “the fights” and Perry Mason.

Grandpa lived to be 100, surviving a car accident and metastatic cancer of the stomach, which he was told was an ulcer.  The cancer in the stomach was removed and never grew back again.  That was about fifteen years before he died.  He never had a heart attack and maintained excellent health for most of his life.  He liked simple foods, an excellent Havana cigar and good conversation.  He was a real estate broker for many years and was honored by the business community in an article that Grandpa was very proud of.  When I visited him in a rest home, I told him he should be lucky to have a family that cares about him.  He replied with scorn:  “Family!  That’s my family!”, pointing to a picture of himself on the wall.  At that time, when he was 99, his mind began to fail him.  He kept repeating that Grandma Lillian was a “rich widow, kicking up her heels, referring to Grandpa Johnny’s death the previous year.  All in all, he was a character.  However, I enjoyed speaking with him as the following dialogue shows:

“Grandpa.  You’ve been around a long time and have seen many Presidents come and go.  Who was your favorite?  Who made the best impression?”

“They were all a bunch of bastards!”

I now draw a curtain of silence over the whole scene.

A Few Thoughts For The New Year

I began the New Year with a vision of a fence that extended infinitely in both directions.  Beyond the fence was a spacious green field filled with dandelions.  There was a gate in front of me with a latch, but I didn’t have the key.  This vision became an inspiration for a book about writing.

Bassoon Ecstasy: Nadina Mackie Jackson And Milan Turkovic

Antonio Vivaldi, noted baroque composer, wrote 39 bassoon concerti, supposedly, for his female students, but no proof exists that such was the case.  Il Prete Rosso(The Red Priest) seems to have had a special affection for this instrument unlike any composer before him.  Some critics, however, have stated that Vivaldi wrote one bassoon concerto 39 times!  It is Nadina’s purpose to show that this is not true and she succeeds!  As Nadina writes:  “The extraordinary collection of 39 bassoon concerti by Antonio Vivaldi provides a journey of a lifetime, and I intend to visit every stop on this incredible ride…”  In the first volume of a purported five volume series, she proceeds to do just that.

From the first notes, Nadina plays with abandon, tonguing at lightening speed in the fast passages, while giving the quiet passages all her reflection and introspection.  Notes are scattered in all directions only to reunite with stronger bonding.  She reinvigorates Vivaldi’s works with a fury and intensity rare in my hearing.  Nicholas McGegan’s ensemble provides her ample support, but she is the star, promenading and displaying the bassoon in its many facets.  I can’t wait to hear subsequent volumes!IMG_5892

Can you imagine four bassoons playing together?  When one of these bassoonists is Milan Turkovic, listening becomes an undisguised pleasure.  Milan is one of the world’s most renowned bassoonists and a consummate artist.  Known for impeccable tone and a sly wit, he never disappoints.  His attempts at slides give certain pieces a jazz feel, which Milan delights in.  He has recorded Mozart’s famous concerto several times and each interpretation is unique and revelatory.

This disc is great fun to listen to and I recommend it highly.  The four bassoons give a unique sound to “Summertime” and “Flight of the Bumblebee”, among many others.  A great way to spend a lazy weekend!IMG_5895

“Like The Wind Across Your Face, Lucas Boy.”

The previous year is already gone, replaced by a new year.  Thus, it seems fitting to quote marshal Micah Torrance’s description of the passage of time from The Rifleman:  “Like the wind across your face, Lucas boy.”  He is an older man talking to a younger man, Lucas McCain, about the fleeting nature of time, a major theme in the show.

The series shows the special relationship between widower, Lucas McCain and his son Mark.  We see how Mark changes, from a ten-year-old to an adolescent and how this affects the relationship with his father.  He learns to accept people’s differences(another major theme), and learn there is a time for different things in life.  But, even in the early episodes, Mark takes an active role in defending his father and saving his life.  The Rifleman really deals with Mark’s education in the broadest sense, from going to the confines of the schoolhouse to learning about survival amidst rugged terrain.  His interaction with his father gives the show its dynamic, as they confront outlaws, outcasts, people uncertain of what they want and people who are too proud to admit they may be wrong.

In the last episode, we see Mark with a potential girlfriend, though it is clear Mark is not ready for commitment.  If the series had continued, Mark would have become an adult, which was not what The Rifleman was about.  As an audience, we are visual witnesses to the characters’ swift aging, and we are forced to accept Micah’s statement about the sudden rush of time.