
Superegg
Written March 2, 2019
Students,
I've trained my camera on this object, which is barely an inch tall
and weighs four ounces. It's standing on end, and even if I tip
it a little sideways, it will return to an upright position.
No bird
lays an egg like this. Ordinary eggs won't stand on end without
some help.
And no,
this object isn't weighted on the bottom, like a “bop bag”
that bounces right back from a punch. My shiny egg is solid
metal, all the way through.
So what's
the secret? The explanation begins with some geometry. 

(x/3)^{2}
+ (y/4)^{2}
= 1


Inside the
upper diagram, the red line traces an egglike ellipse.
It's the graph of the function shown below it.
The raised ^{2}
is an exponent, meaning that the quantity inside the parentheses
should be raised to the second power, or “squared.” 


But what
if we used a larger exponent than 2? Then we'd obtain a fatter
curve, a super ellipse.
The green
shape uses an exponent of 3. It's sort of partway between a
rectangle and an ellipse, because if we were to increase the exponent
even further, the corners would fill out and the shape would approach
a rectangle. 
This
fountain in the center of Stockholm is a reallife example.
“Sergels torg” used to be a commonplace square.
When it was redesigned in the 1950s, it could have become a
commonplace traffic circle. But Danish mathematician Piet Hein
suggested a pleasing compromise between the two shapes, with the
curvature of the wall continuously varying. 


The north,
south, east, and west curbs are almostbutnotquite straight.
Between them, there are no sharp corners; the curve gradually becomes
tighter, then gradually eases.
Hein
called his shape a superellipse. He also extrapolated it into a threedimensional
superellipsoid, the Superegg. I read about all of this in Scientific
American in the 1970s.

Back then,
I was directing television shows on Cable TV3 in Washington, PA.
I noticed
that the superellipse somewhat resembled the shape of the rounded
picture tubes of that era, so when it came time to construct a
backdrop for one of our programs, I plotted the function on poster
board and used the shape for a logo. 
(The
actual program concept, I must admit, lacked “focus.”
What was it? If some random local people asked for a onetime
shot on TV, this show is where we'd put them.)
So back to
my shiny object. How does it stay upright?
First
consider a child's block. Tip it a little so it's up on its
right edge.
As long as
the center of gravity stays to the left of the edge that's in contact
with the table, shown by the point of the yellow arrow, the
preponderance of the block's weight will pull it back to the left. 

The
superellipse, because it approximates a rectangle, acts the same
way. If we tilt the Supergg ten degrees to the right, the point
of contact with the table (the point of the yellow arrow) rolls
farther to the right than the center of gravity does. Compare
it to the black plumbbob line to see that the center of gravity
remains to the left of the arrow. Just as in the child's
block, the slightly greater mass to the left of the arrow outpulls
the mass to the right, and the object falls back to the left.

On the
other hand, if we stand an ordinary elliptical egg on its end and
tilt it to the same angle, the point of contact rolls to the right
more quickly, moving the center of gravity to the right of
that point of contact. The egg topples even further to the right.
Mystery solved! 
