Three (3 page)

“What kind of advanced math
involves art history!”

“I’ve been assigned to find
examples of symmetry breaking in works of art from two or more artists.  These
three books were recommended by one of the graduate teaching assistants as a
good place to start.”

“What’s symmetry breaking,
and what does it have to do with art?”

“In mathematics, symmetry
breaking’s defined as some event or representation that disrupts or alters an
otherwise symmetrical pattern.”

“Can you give me some examples?”

“Well, one simple example is
a book standing on edge on a table.  It’s unstable but symmetrical until it falls,
either face up or face down, thus breaking its symmetry.  Another example could
be the coloring of one of the daisies in Grandmother’s wallpaper with a red
crayon so as to break the repeated pattern covering her dining room wall.”

“I remember doing something
like that.  I was spanked for it but I thought it was a great improvement.”

They both laughed as Oliver
opened the first art book on French Impressionism.  “I don’t think I am going
to find anything here.  The basic theme of these paintings seems to me to be as
devoid of symmetry as possible.  The artists are experimenting with color and
form in order to create a mood.  Monet, Renoir, and even Seurat paint the same
basic landscapes as do their American counterparts in the Hudson River School,
and break only the traditional rules of realism by emphasizing impressions of
color or form.  I don’t think symmetry is broken in these paintings, just
tradition.”

“OK, what about this book of
etchings by M.C. Escher?”  Peter opened the book and began reviewing the plates,
working through the history of the artist’s work.  “Escher’s later work, based
in the intricate tile patterns at the Alhambra in Spain, is nothing but
symmetry, but I don’t see where he colors in any of his own daisies.  There is
absolutely no symmetry break in any of his highly symmetrical patterns.  But
his last etchings do break with reality; look here at Escher’s most famous
work, ‘Waterfall’.

Oliver looked at the image of
a continuously running closed loop of water that defied logic by continuing to
drop over a fall and drive a water wheel.  Reading the description of this work
that Escher completed in 1961 he said, “An Oxford mathematician named Roger
Penrose gave Escher the idea for this illusion.”

“Roger Penrose!  He’s my
professor’s idol.  He continuously references Penrose’s work in his lectures. 
This must be one of the symmetry breaking pictures.  Oliver you need to find me
at least one more.”

“Well we only have one book
left, the illustrations of Norman Rockwell.” 

“Please tell me you see
something in them; my class meets tomorrow morning!”

Oliver and Peter sat
side-by-side examining the various Saturday Evening Post illustrations, some
humorous, some patriotic, some sympathetic, but all moving in their depictions
of average people.

Oliver opened the book to a
picture called “Before the Shot”.  “Here’s a typical doctor’s office.  The
doctor’s preparing a needle with his back turned.  Standing on a chair next to
him is a little boy with his pants down, ready for the shot.  It’s a typical
scene, completely symmetrical, except that the little boy is carefully
scrutinizing the Doctors diploma hanging on the wall before him.  The humor is
in the broken symmetry.”

“You’re right; let me see if
I can find another one.”  Peter flipped pages until he came to a painting
called “The Critic”.  “Here’s a good one.  The young art critic, with art
supplies and an art museum guide book, is carefully examining a classical
painting: a perfectly normal and symmetric scene.  What breaks the symmetry and
gives humor to the scene is that the various characters in the paintings are
carefully examining the young critic.” 

“Maybe I should take up
mathematics, Peter; I had no idea it was so easy.”

Peter followed that comment
by throwing Oliver’s stuffed mascot at him.  “If you think group theory is
simple, come sit in on class tomorrow.”

“I would, but then I’d have
to tell your professor you had help in your assignment.”

This time Peter threw a bed pillow.

Their friendship strengthened
throughout their four years of undergraduate studies.  Peter had begun to date
a fellow Math major named Jane Sumner, and often Oliver and his date would join
them for a movie or an evening on the town.  Oliver had a number of girlfriends
but never one he wanted to spend the rest of his life with.   Peter, on the
other hand, had fallen deeply in love with Jane.  They were clearly meant for
each other, sharing many common interests besides mathematics.  Oliver saw how
they each appreciated almost everything they shared: a work of art, an exotic
food, the lake-side walk or jog, a sunset.

Upon graduation, Peter and
Jane became engaged and Peter asked Oliver to be the best man in their wedding.
 It was a simple but festive affair attended by many of their fellow students
as well as family members.  Later that year Oliver entered Harvard Divinity
School.  He had begun to feel more and more like he had been called to the
ministry.  Growing up attending a small Methodist Church in a working-class
community, aptly named Cutten, on the outskirts of Eureka, he had always
admired the ministers who not only knew the scriptures, but ministered to those
in need.   Meanwhile Peter, with his young wife at his side, began work on a
Doctorate in Advanced Mathematics at Princeton.  Oliver continued to keep in
touch with Peter and Jane during the following year.  In that period of time, Oliver
distinguished himself at Harvard Divinity School and found himself drawn into
the richness of the literature of all the world religions.  However, he began
to feel pulled away from the ministry, as though the calling he felt two years
before was no longer to a church, but to a classroom.

By the time baby Alice was
born to Peter and Jane, Oliver had realized his interest in religion was more
academic than spiritual and had shifted into a Doctoral program in Philosophy
with an emphasis on World Religion.  This shift also represented a water-shed
in Oliver’s spiritual life.  He had gradually become disillusioned by religion
because so much of war and strife in the world seemed to have its origins in
religious intolerance. 

He had begun to feel
uncomfortable in church, even when he stood with Peter and Jane at Alice’s
christening at the Catholic Church near their home next to the University of
Chicago campus, where Peter now worked as a Postdoctoral Fellow.  Despite these
feelings, Oliver took his promised responsibilities of Godfather seriously.  Little
Alice called him Uncle Oliver and drew him even closer to Peter and Jane.  His
parents had recently moved into a nursing home in northern California, and
although he visited them frequently, he knew they were both beginning to lose
their ability to remember.  On some occasions he found himself a stranger to them. 
The pain of slowly losing touch with his own family was tempered somewhat by
his becoming a closer part of Peter’s

Fifteen years after Alice’s
birth, when Jane lost her battle with breast cancer, he was there giving
comfort to Peter and Alice even though his soul was filled with bitterness; the
prayers of Peter and Alice seemed to have been in vain.  Now, suddenly and
without warning, Peter was gone.  Again, Oliver felt a darkness move between
his soul and God, his faith shivering in the shadows of grief.  As he walked
down the O’Hare concourse to the exit he felt fear and anger as well as grief,
and was unable to overcome this inner darkness and doubt.

Finding his second car, left
the week before at the park-and-ride, he drove through the city to Peter’s
house near the campus.  Oliver was fearful of what the next few days would
bring.  He had not been in the position of a grief counselor or minister since
his few attempts while in divinity school. 
Now I’m going to be called on to
maintain my composure so I can comfort and counsel Alice.
  The chill of his
anxiety was momentarily overcome with warmth as he tearfully embraced Alice and
found that she was there in Peter’s living room with Father Patrick Ryan, the
priest who had christened her twenty years before, and had clearly been a part
of her life ever since. 

Now, at age twenty, Alice had
become her own person, a distinguished student of architecture who was being
approached by the top firms in Chicago.  As a young girl she had shown a talent
for art and mathematics.  She adored her father and the two often interacted in
the same enthusiastic manner that Peter and Oliver had shared at Princeton. 
Alice even displayed some of the same electrically charged bursts of
inspiration Oliver had always seen in Peter.

Once in college she decided
to major in Architecture and Structural Design.  Her professional life was just
developing and she had decided she could wait a few more years before she
became serious about marriage.  She had been dating Chuck, a fellow designer,
but had given him no indication of wanting to settle down.  He, on the other
hand, was in love with her but also intimidated by her talent and intense work
ethic.  She had shared her feelings with Oliver on a few occasions, and he had
tried to be a fatherly listener without giving fatherly advice, something she
greatly appreciated.  At the present time her relationship with Chuck was on
hold while he was spending a year in Paris, studying classical architecture.

Alice continued to weep as
Oliver held her, her rosary hanging loosely in her right hand.  Father Pat
stood and gently touched Alice and Oliver on the shoulder and said, “Let us all
pray together.”  He then began to pray the Lord’s Prayer.  Alice followed weakly
but Oliver was unable to pray, his darkness and distrust welling up within
him. 

At the end of the prayer
Oliver, noticing the fatigue in Alice’s face, her eyes swollen from crying,
said to her, “I’m here for you, and I’ll help in any way I can in the coming
week.  You’ve had a long day and need to get some sleep.  I‘ll stay here a
while longer after you have gone to bed and talk with Father Ryan.”

Alice nodded agreement and
left the living room, heading up stairs. Oliver then turned to Father Ryan. 
“Thank you for being here for Alice and for me; we’ve both been hit hard by
this sudden tragedy. Peter had always been so full of life and energy.  Had he
shown any recent signs of health problems?”

 “Peter was fine according to
his doctor, who just conducted his annual physical three weeks ago.”   

“Did he give any indication this
morning he was ill?” 

“According to Alice, at the
breakfast table he was excited and upbeat about his most recent work on the
fundamental nature of matter.  The only thing he said to me at the hospital before
he died made no sense, but might hold some clue to his condition.  I’ve been
trying to find some meaning in it for the past hours.  I had begun last rites
at Peter’s bedside, and had just made the sign of the cross, when he uttered, ‘You
must find the three.’  I said nothing to Alice about this last request for fear
of further upsetting her, but thought you might see some hidden meaning in it.”

Oliver, as puzzled as Father
Pat, asked, “The three what?” 

“It could have been his team
members, Elizabeth, Khalil, and David, but they were the ones who found him,
accompanied him to the hospital, and had been in the waiting room when he
died.  But in case this had something to do with his work, I asked Elizabeth to
retrieve his laptop from his office to see if there was something there he was
concerned about.” 

“What did she find?”

“Nothing so far, the computer
seems to be locked in a ‘blue-screen’ mode.  It won’t respond to any commands. 
Elizabeth Ward, as you know, is the team member with probably the best
knowledge of computers in the mathematics department.  After her preliminary
search, she has concluded the laptop had been attacked by some worm or virus
and that it would take her a day or more to determine what, if anything, could
be retrieved.” 

“You mean his most recent
work has been erased by some deliberate cyber-attack!?”

“They don’t know yet if this
virus was deliberate, or if it entered his computer through some innocent
download from the internet, but that is one thing Elizabeth is looking into.”   

Oliver looked at Father Pat
with a worried expression.  “I hate to even suggest this, but if it turns out
his work was deliberately attacked, maybe his death was not from natural
causes.” 

“You think he might have been
murdered?!  Who would possibly do such a thing?” 

“I don’t know, but
considering his importance to the entire country as well as the world
scientific community, the authorities should be asked to conduct a careful
autopsy to determine the exact cause of death.” 

Father Pat replied, “Alice
and I have already made such a request out of concern for the sudden nature of
his death, regardless of the cause.  As a result, we’ve decided to delay memorial
services for a few days, depending on the duration of the examination.  As you
can understand, the word of his death has already spread to the news media, and
condolences and inquiries are flooding Alice’s email.  Oliver, you need not
worry yourself about these final arrangements for Peter’s service, Alice and I
have begun working them out and there will be time for us to handle his affairs
as well as the service.”