FEATURE ARTICLE
Twisted Math and Beautiful Geometry
Four families of equations expose the hidden aesthetic of bicycle wheels, falling bodies, rhythmic planets, and mathematics itself.
Eli Maor, Eugen Jost
Epicycloids and Hypocycloids
Whereas the cycloid is generated by a point on the rim of a wheel rolling along a straight line, a related type of curve arises from a wheel rolling on the outside of a second, fixed wheel. The resulting curve is an epicycloid (from the Greek epi, meaning “over” or “above”). Alternatively, we can let the wheel roll along the inside of a fixed wheel, generating a hypocycloid (hypo = “under”). The epicycloid and hypocycloid come in a great variety of shapes, depending on the ratio of the radii of the two wheels. Let the radii of the fixed and moving wheels be
R
and
r,
respectively. If
R/r
is a fraction in lowest terms, say
m/n,
the curve will have
m
cusps (corners), and it will be completely traced after
n
full rotations around the fixed wheel. If
R/r
is not a fraction—if it is irrational—the curve will never close completely, although it will nearly close after many rotations.
For some special values of
R/r
the ensuing curves can be something of a surprise. For example, when
R/r
= 2, the hypocycloid becomes a straight-line segment: Each point on the rim of the rolling wheel will move back and forth along the diameter of the fixed wheel (
see below, at left
). Thus, two circles with radii in the ratio 2:1 can be used to draw a straight-line segment! In the 19th century this type of curve provided a potential solution to a vexing problem: how to convert the to-and-fro motion of the piston of a steam engine into a rotational motion of the wheels. It was one of many solutions proposed, but in the end it turned out to be impractical.
When
R/r
= 4, the hypocycloid becomes the star-shaped astroid (from the Greek astron, a star)
.
This curve has some interesting properties of its own. Its perimeter is 6
R
(as with the cycloid, this value is independent of π), and the area enclosed by it is 3π
R
^{
2
}
/8, that is, three-eighths the area of the fixed circle. Imagine a line segment of fixed length with its endpoints resting on the
x
- and
y
-axes, like a ladder leaning against a wall. When the ladder is allowed to assume all possible positions, it describes a region bound by one-quarter of an astroid. This shows that a curve can be formed not only by a set of points lying on it, but also by a set of lines tangent to it.
Turning now to the epicycloid, the case in which the fixed and the moving wheels have the same radius (
R/r
= 1) is of particular interest: It results in a cardioid, so called because of its heart-shaped form. This romantic curve has a perimeter of 16
R
and its area is 6π
R
^{
2
}
.
The Greek astronomer Claudius Ptolemaeus, or Ptolemy (ca. 85–165 C.E.), invoked epicyclic motion in an attempt to explain the occasional retrograde motion of the planets—a movement from east to west in the sky, instead of the usual west to east. He ascribed to them a complex path in which each planet moved along a small circle whose center moved around Earth in a much larger circle. The resulting epicycle has the shape of a coil wound around a circle. When this model still failed to account for the positions of the planets accurately, more epicycles were added on top of the existing ones, making the system increasingly cumbersome. Finally, in 1609, Johannes Kepler discovered that planets move around the Sun in ellipses, and the epicycles were laid to rest.
The illustration at right shows a five-looped epicycloid (
blue
) and a prolate epicycloid (
red
) similar to Ptolemy’s planetary epicycles. This latter curve closely resembles the apparent path of Venus against the backdrop of the fixed stars. Earth and Venus follow an eight-year cycle during which the two planets and the Sun will be aligned almost perfectly five times. Surprisingly, eight Earth years also coincide with 13 Venusian years, locking the two planets in an 8:13 celestial resonance and giving Fibonacci aficionados one more reason to celebrate!
Notes