Hi, yesterday we explored symmetry by asking
what remains the same when something is transformed.
Today, I want to turn that question around. How
much can one small change alter the shape of
something? Think about a familiar U -shaped parabola.
You know, the kind of curve that you might see
when a ball is tossed through the air. We can
change one small part of the equation behind
that curve and make the parabola taller and narrower.
Another small change makes it wider and flatter.
A different change moves the entire curve to
the right or left. And placing a single negative
sign in front of the equation turns the whole
shape upside down. Now, on one hand, this is
basic math. and it may seem routine and not interesting.
But when you take a step back, I find that in
some ways, this really is remarkable. The changes
within the equation may look small, a number,
a sign, or a symbol, but they can transform the
entire shape that appears. There is not just
one parabola, there is an entire family. of parabolas.
And this is one of the beautiful connections
between algebra and geometry. Algebra communicates
through numbers and symbols. Geometry communicates
through lines, curves, and shapes. They may appear
to be two different languages, but a graph allows
them to speak to one another. The equation tells
us what is happening beneath the surface. The
graph allows us to see the result. Without the
graph, it may be easy to overlook the importance
of one small change. But once we see the shape,
its effect becomes obvious. We can play with
different parts of this equation and literally
see the family members emerge, all with different
shapes. I wonder if something similar happens
outside of math. In his book Creativity, Inc.,
Pixar co -founder Ed Catmull describes the table
that they use for important meetings. It was
long, narrow, and large enough to seat everyone.
In one sense, it served its purpose perfectly.
But its shape was influencing the conversation
in a subtle but very real way. The leaders typically
sat near the middle where they could easily see
and speak with one another. But people with less
authority sat towards the ends, further from
the center of the discussion. Now, nobody had
deliberately decided that the people at the ends
were less important. Yet, the shape of the table
communicated that message. And Catmull finally
recognized the problem. after attending a meeting
around a smaller square table. Eye contact was
easier, the conversation flowed more naturally,
and everyone felt closer to the center. So, Pixar
replaced the long table with a square one. Now,
the people had not changed, the purpose of the
meeting had not changed, but changing the shape
of the table help change the shape of the conversation.
And I find this example powerful because the
table seemed like a neutral object. Yet beneath
the surface, it was influencing who felt important,
who felt included, and who felt free to speak.
Sometimes the structures shaping our relationships
are so familiar that we no longer notice them?
Who sits at the center? Who is pushed toward
the edge? And what might change if we gave everyone
a place where they could be seen, heard, and
most importantly, valued?