Parabella
adapted from a letter to the January 1985 Mathematics Teacher
by Alfinio Flores, Guanajuato, Mexico
Instructions:
Replace each of the dashes below with a letter so that it
makes a math word and completes the story.
Mother Q-------- was very happy with her new daughter, Parabella.
The doctor
was happy, too. He had feared a d--------- case,
but fortunately Parabella
was a r---- and healthy child. "She is
beautiful, " said Mother Q--------.
But what daughter is not
beautiful in her mother's eyes? Parabella's elder
sisters, Hyp and
Elli, thought otherwise.
"Look how ugly she is. She has only one a--- of s-------," Hyp
exclaimed.
(see figure 1a)
"She has only one f----," Elli added.
"She certainly is not like us!" (figures 1b,c)
When Elli and Hyp wore their C-------- dresses, everybody could
tell they were
sisters:
But Parabella looked different:
y2 = 4px.
Parabella was bright, and soon discovered that her t-- sisters had
other s-----------.
With Elli, the s-- of the d-------- from each
p---- to the foci was constant:
dl + d2 = c
(figure 2).
For Hyp it was the d--------- of the
d-------- that was constant:
dl - d2 = c
(see figure 3).
Parabella was sad because she could not find a similar p-------
in herself.
She only had one f----.
0-- day, when playing with a
l---, she made an amazing
discovery. If she put the line p------------ to her a---, so that her vertex was
the same
d------- from the line as it was from her f---- (see figure 4), the
same would be true for all her p----- (see figure 5)!
Parabella ran to tell her sisters the news.
"Look, with my friend d-------- I have a property similar to yours!"
The two elder sisters responded coldly, "Who needs a stupid
l---?
We don't need anything but our
two f---."
Parabella was a little hurt. She wanted so much to convince
her
sisters that she was like them! The next day Parabella
discovered
that she could look at very distant objects. She was
able to do this
because she could concentrate p------- rays in
her f---- (see figure 6).
She had observed that Elli had a
similar ability: Elli could concentrate
the r--- emitted from one
f---- in the other focus after reflecting
them (see figure 7).
This time, however, Parabella did not tell her sister about her
discovery. Parabella also discovered that this property of how
the r--- were reflected enabled her to remember how to get a
t------ line,
which she always forgot (see figure 8). The same
was true for
Elli (see figure 9).
In the afternoon, Parabella read the book that her friend,
the h------, had given her. It related the strange adventures of
her friend's grandfather, a s-----. In the evening, Parabella ate
lots of s------ of pie and ice cream. She went to sleep and had a
weird dream.
Parabella's dream
Parabella and her sisters were prisoners in Dandelin's
castle.
It was like a double ice cream Cone (see figure 10).
The castle was guarded by Dandelin's s------. Parabella
could hear her sisters, but she knew they were not in the same p----
because the voices seemed to come from everywhere.
0-- of the
spheres was holding her at the f----. Two s------ held Elli tight
(see figure 11), and t-- more held Hyp. Parabella managed to
escape by slicing the c--- with a p----. Another slice freed
Elli, and Soon Hyp was free also (see figure 12).
Parabella woke up very excited. She realized that she was
more like her sisters than she thought. They were all c---- sections.
In one sense, she was between her sisters. The slice
that set Parabella free was between the slice that freed Elli and
the o-- that freed Hyp.
She ran to tell her sisters the dream. But as soon as she
started to describe the castle, that c--- in the third d--------,
her sisters laughed at her. "That is what you get for eating so
much at night and reading nonsense like Flatland."
This time Parabella was very upset: "You and your p----- thinking!"
she cried and broke into tears. While she was weeping,
the P---- Fairy appeared.
"Do not cry, Parabella. I will get you some polar clothes
and you will
look like your sisters. You will also see that you
are between them, as in your dream. I will even get a d-------- for
everyone."
The P---- Fairy did what she promised, and soon the t-- sisters were
dressed
in their p---- clothes (see figure 13). The
only difference
among them was that
for Elli: e < 1
for Parabella: e = 1
for Hyp e > 1
The elder sisters at last recognized that they all shared s------
characteristics. Parabella was glad, and she lived happily ever
after.
