[Catalyst 2018]

opposite side of an odd-sided polygon, where
the ratio should be 0. In this case, we have
that k=(n–1)/2, so that we get the following
result: R n,k
= (1–cos 2 π)/(1–2cosπ cosθ+cos 2 θ)
= 0/(1–2cosθ +cos 2 θ) = 0, as expected. In
addition, using n=6 and k=2, we get that
R 6,2
=(1–cos 2 5π/6)/(1–2cos 5π/6 cos π/6 +cos 2
π/6) = (1–(–√3/2) 2 )/(1–2(–√3/2)(–√3/2)+(–√3/2) 2 )
= (1– 3/4)/(1+2(3/4)+3/4) = (1/4)/(1+3/2+3/4)
= (1/4)/(13/4) = 1/13, which is exactly what
Fig. 3: Octagon crosscuts using k=1,2,3,4,5,6. Comparing the
2 lines shows that the octagons using k=1,2,3 are the same as
those using k=4,5,6.
was found in earlier papers. The other known
results can also be checked using a similar
method.
RATIONALITY
Now we try to find which values of n and k
will produce a rational value for R n,k
. We can
easily check different values of n and k and
find that the following are rational: R n,0
, R 4,1
,
R 6,2
, R 6,1
, R 8,2
, R n(odd),(n-1)/2
. However, there are an
infinite number of combinations for n and k,
and there is no straightforward way to prove
that these are or are not the only rational
values of R n,k
. One approach is to look at
different values for the possible parts of the
ratio that could be irrational. For example,
when both cosθ and cosψ are rational, then
R n,k
is rational. This approach works for some
combinations, but a problem arises when
both cos 2 θ and cos 2 ψ are irrational, as there
is no way to tell what will happen with R n,k
.
This is because both the sum and product
of irrational numbers can be rational, thus
even though all components of R n,k
are
irrational, the final output could still be a
rational number. Because this is the majority
of the values that will occur, an approach that
directly applies to this case is needed.
One possible approach is to look at the
polynomial representation for R n,k
and the
minimal polynomials for cosθ. The minimal
polynomial of the value cosθ is the smallest
nonzero, monic polynomial with rational
coefficients that has cosθ as a solution. For
example, the minimal polynomial of √2 is
x 2 –2=0. Note that this polynomial cannot
be reduced, and thus it is the smallest
polynomial with √2 as a solution. All algebraic
numbers have a minimal polynomial, by
definition, and thus for each value of n, cosθ
has a minimal polynomial. 2
First, we use R n,k
to create a polynomial
expression of one variable. Using the multiple
angle formula, we know that cosψ=cos((2k+1)
θ)=cos(sθ)=T s
(cosθ) where T s
(cosθ) is the
s th Chebyshev Polynomial. For example,
cos(4θ)=8cos 4 (θ)–8cos 2 (θ)+1because T 4
(x)=8x 4 –
8x 2 +1. 7 This allows us to express R n,k
as an
expression of θ: R n,k
=(1–T s2
(cosθ))/(1–2T s
(cosθ)
cos θ +cos 2 θ, where s=2k+1. Setting x=cosθ,
and R n,k
=r, where r is a rational value, we get
the polynomial expression P(R n,k
)=1-T s
2(
x) –
r+2rxT s
(x)–rx 2 =0.
It is known that if a certain value solves a
polynomial, then the minimal polynomial
(minpoly) of that value has to be able to divide
the larger polynomial, with a remainder of
0. 4 Thus we want to see what happens when
we use different combinations of n and k,
and see if there is any way to find a pattern
in the remainders of P(R n,k
)/minpoly(cosθ).
In this case, a random collection of values
from within the bounds n≤100 and k1
–1) ∏ ψ2 (x) for n even.
d
d|n, d>2
where ψ n
(x) is the minimal polynomial of
cos(2π/n) multiplied by a constant and d|n
means that d is a divisor of n, so that the
product runs through all the divisors of n. 3
Exploring this relationship in more detail is
an important and rapidly expanding area
of theoretical mathematics, and could be
very helpful in creating a final proof of the
rationality of Rn,k. In our exploration, we
came across a need to understand more
about the relationship between these two
concepts and have seen how a simple
question concerning area ratios can produce
complicated mathematical questions that
are still being explored purely for theory.
This is just one application of the possible
knowledge that can be gained by the abstract
investigation of these two concepts and the
inner workings of their natural connection.
A special thanks to Dr. Zsolt Lengvarszky of
the Louisiana State University, Shreveport,
Mathematics Department for his help and
mentorship in the project.
WORKS CITED
[1] Ash, J.M., et al., Constructing a Quadrilateral Inside
Another One, Mathematical Gazette, 2009, 528, 522–532.
[2] Calcut, J.S., Rationality and the Tangent Function, http://
www2.oberlin.edu/faculty/jcalcut/tanpap.pdf (accessed Feb.
8, 2018).
[3] Gürtaş, Yusuf Z., Chebyshev Polynomials and the Minimal
Polynomial of Cos(2/n), The American Mathematical Monthly,
2017, 124, 74-78.
[4] Leinster, T., Minimal Polynomial and Jordan Form, http://
www.maths.ed.ac.uk/~tl/minimal.pdf (accessed Feb. 8, 2018).
[5] Mabry, R., Crosscut Convex Quadrilaterals, Math Mag,
2011, 84, 16–25.
[6] Mabry, R., One-Thirteenth of a Hexagon, n.d. 1-11.
[7] Mason, J.C., Handscom, D.C., Chebyshev Polynomials;
Chapman and Hall: Boca Raton, 2003; sec 1.2.1.
DESIGN BY Kaitlyn Xiong
EDITED BY Olivia Zhang
CATALYST | 37