When a Knot Is Not a Knot
A stressed-out person may be
described as “tied in knots,”
unable to relax or make good
decisions. Mathematicians, in contrast,
find knots fascinating, and some of
them devote their lives to the math of
knots, including computer modeling to
understand all their twists and turns.
It may sound like an armchair interest
with no practical value, but read on.
One of the great pioneer knot explorers
was August Ferdinand Möbius
(1790–1868), a descendent of Martin
Luther. Homeschooled and gifted in
mathematics, Möbius is best known
for his study of a twisted belt of material
called a Möbius strip. The unusual
figure somewhat resembles a lazy eight
or infinity symbol.
The Möbius shape has several fascinating
properties, and the reader
is encouraged to experiment with a
ribbon of paper to see it for himself.
When constructed, the two surfaces
of the paper ribbon become joined to
form a ring with just a single side,
strange but true.
By twisting and tearing a Möbius
strip in various ways, you can create
several new novel forms of ribbon. The
technical details of Möbius ribbons are
analyzed in the field of mathematics
called topology, which has significant
uses in modern industry, computers,
and even particle physics.
The Möbius shape has had several
practical applications since its discovery.
For example, a large Möbius band makes
a good conveyor belt. It will last twice
as long as a conventional belt because it
wears evenly along its entire length, both
topside and underside. Farm machinery
also may use the Möbius concept of a
twisted belt for longer wear.
Some continuous-loop audiotapes
employ the Möbius shape to double
their recording time. The Möbius shape
is also popular in sculpture and graphic
art, especially the work of M. C. Escher
(1898–1972), who was intrigued by its
nearly magical, otherworldly effects
that seemed to defy basic geometry.
Though Möbius is credited with discovering
this unusual shape, it already
occurs in nature. For example, some
of the vast circulating ocean currents
follow the path of a Möbius loop. The
metallic compound niobium selenide,
NbSe3, assumes the twisted curve shape
when grown in its crystalline form. Certain
organic molecules and segments of
DNA likewise can adopt a Möbius pattern
or other types of “molecular knots.”
These result in materials with new physical
properties and possible applications
in miniaturized electronic devices.
Mathematicians define knots differently
from the knots we see in everyday
life, such as shoelaces or sailing knots.
A mathematical knot has no loose ends
to tie up; instead, it is made up of a
closed loop that may be entangled in
various ways. If the loop can be untangled into an open circle, it is known as
an “unknot.” (Unknots have their own
Several creatures are adept at knots:
The moray eel can tie itself into a
knot when it needs to hold its body in
place on the seafloor or in a cave.
The hagfish, similar to the eel, can
tie itself into a knot, which slides down
the body, cleaning the skin surface.
Weaver birds tie knots in nest material
using their beaks and feet.
In contrast to these knot-makers,
octopuses have special instincts to
avoid knotting their eight limbs.
Analysis of the Möbius shape and
related animal behavior reveals the
deep mathematics that underlies all of creation. This elegant and complex
language was established during the Creation Week.
Job 12:7–9 challenges us to speak
to the earth, animals, birds, and fish,
“and they will teach you.” That is, God
expects us to survey and study His
world, including the birds, fish, and
other animals, to learn more about our
Creator and how He operates. All of
creation declares the hand of the Lord,
including the knots we find in nature.
See For Yourself . . .
A sheet of paper, scissors, masking tape, and a pen or pencil
Anyone can construct this simple object with such surprising properties.
Begin by cutting a sheet of paper into four strips, two inches wide. Tape the
sheets together into a single, long strip.
Twist one end of the strip 180 degrees. Now bring the two ends together
and fasten with tape. This should make a twisted, somewhat curled-up loop.
Next, draw a line along the outside of the loop. This can be rough and
freehand. You just want to leave a visible mark along the ribbon until you
arrive back at the starting point. The result is unexpected: the entire strip
appears to be lined, both inside and outside. The conclusion is that the loop
of paper, despite its front and back surfaces, somehow has only one side!
Next, cut a small tear along the pencil line. With scissors, cut along the
entire loop, continuing through the tape. When finished, the result should be
a new single loop that is twice as large as the original!
Start again with a fresh strip of paper. This time give one end a full
360-degree twist before taping the two ends together. As before, cut the strip
roughly along the center for the entire length. The result should be two linked
loops, similar to a paper chain.
You should enjoy trying other variations, such as making two full turns of
one end before taping the loop. Depending on how many times the original
ribbon is twisted before taping, you will discover a series of unusual results.
See, mathematicians can have fun, too!
department at Grace College, Winona Lake, Indiana. He
is currently president of the Creation Research Society
with hundreds of members worldwide. His website is
https://answersingenesis.org/mathematics/when-knot-not-knot/ This article originally appeared on answersingenesis.org