“Math is beautiful. It’s like art,” Professor Gregory Conner says.

According to Dr. Conner, math is beautiful for what it is, in and of itself. Its beauty is not diminished even when the math cannot be directly applied in some way to our lives.

Mathematicians often discover theorems and develop proofs that may not immediately find an application in science or other fields, and Professors Gregory Conner and Chris Grant from the Department of Mathematics are no exception.

In 1964, the Ukrainian mathematician Oleksandr Mikolaiovich Sharkovsky discovered a beautiful theorem about periodic points of functions. Say F is a continuous function, and x and F(x) are real numbers. Let F2(x) denote F(F(x)), and Fn+1(x) = F(Fn(x)). Then if Fn(x) = x, x is called a periodic point of F, with period n. Sharkovsy proved the amazing fact that any function F that has a point of period 3 also must have periodic points with all other periods!

With little communication between the United States and the Ukraine during the cold war, two American mathematicians, Li and Yorke, independently discovered this theorem in 1975 and published it in their landmark paper, “Period Three Implies Chaos.” Today, however, Sharkovsky is credited as the originator of the theorem that now bears his name.

One day after discussing Sharkovskii’s Theorem with Dr. Grant, Dr. Conner mentioned the theorem to grad student Mark Meilstrup. Meilstrup guessed that Sharkovskii’s Theorem could apply to more complicated spaces, ones with names like the Topologist’s Sign Curve and the Warsaw Circle.

Finding how Sharkovskii’s Theorem applied in more complicated spaces launched Conner, Grant, and Meilstrup on a journey—one with a complex destination which is challenging to explain to non-mathematicians.

Grant said, “I can remember very clearly that I’d given what I thought was a simple explanation of this theorem, and my friend’s reaction was, ‘Whoa, hold on there Einstein.’”

Non-mathematicians may wonder if such mathematical results apply in any way to real life. “It could have applications down the road, perhaps in fifty years, or maybe one hundred [years],” Conner said.

Meanwhile, the sheer beauty of mathematical insights such as Sharkovskii’s Theorem holds great intrinsic value for those who have eyes to see it.