What is the formula for a superellipse with integer Cartesian points?

Question

The September, 1965 Scientific American introduced me to the superellipse, specifically the one designed by Pete Hein that was used as a roundabout in Stockholm. (https://www.piethein.com/page/superellipse-24/)

I discovered that placing integer points in a fixed pattern on a Cartesian plane generated a superellipse that exactly superimposed over the graphic in the original article.

The problem was that I did not (and still don’t) have the necessary mathematical acumen to find the equation that allows for the integer coordinates.

The method for plotting the points is as follows: Let N be the total number of moves between each point. In this example, N=4.

Starting at (0,0), count up 4 units to (0,3). Next count up 3 units and over 1 (3+1=N) to (1,5). Then up two and over two, then up one and over three, then over four, ending up at (6,6). Continue the process to complete the points on the superellipse. The arbitrary placement of this example places the center of the superellipse at (1.5,4.5). When N is odd, the center is also an integer Cartesian point.

Over the years I have casually presented this problem to several mathematicians, none of whom found a solution, although to be fair, I’m not sure just how much effort they might have put into solving it!

Answer 1

For your particular example, which seems to have $N=3$ rather than $N=4$, it seems the points satisfy (to rounding)

$$|x-4.5|^{2.51446672707687}+|y-1.5|^{2.51446672707687} = 46.6735667414061$$

where $k=2.51446672707687$ is the non-zero solution to $9^k+3^k=7^k+7^k$ and $46.6735667414061 = 4.5^k+1.5^k$

For $N=4$, I would have thought your points would start $(0,0)$, $(0,4)$, $(1,7)$, $(3,9)$, $(6,10)$, $(10,10)$, etc. and these would satisfy

$$|x-8|^{2.46836241138465}+|y-2|^{2.46836241138465} = 175.027781385891$$

where $k=2.46836241138465$ is the non-zero solution to $8^k+2^k=7^k+5^k$ and $175.027781385891 = 8^k+2^k$

For $N=1$ and $N=2$ there is not a unique solution. For $N = 5$ (or larger $N$) there is no exact solution since there is no non-zero solution for $25^k+5^k=23^k+13^k=19^k+19^k$ though something like $k=2.448$ is not too bad

Answer 2

Despite a somewhat confusing presentation, I think I understand the intended construction: the first coordinate increases by successively larger increments while the second coordinate increases by correspondingly smaller increments. The coordinates are given by $x = \frac12 n(n-1)$, $y = \frac12n(2N-n-1)$ for $0 \le n \le N$, the "center" by $(\frac12 (N-1)^2, \frac12 (N-1))$.

For a continuous parameter $t$, upper left portion of the curve is, after shifting and scaling, given parametrically by $x = t^2, y = 1 - (1 - t)^2$ for $0 \le t \le 1$. The "center" is at $(1,0)$; shifting it to $(0,0)$ and reflecting in the $y$-axis to move the arc to the first quadrant, the curve becomes $x = 1 - t^2$, $y = 2t - t^2$. After eliminating $t$, the equation of the curve is $(x-y)^2 + 2(x + y) = 3$. This describes a parabola symmetric about the line $y=x$, opening to the lower left. To produce the desired closed, symmetric curve, it is only necessary to introduce absolute values: $$ (|x| - |y|)^2 + 2(|x| + |y|) = 3. $$ Here is an image as rendered by Wolfram|Alpha:

It is indeed very similar to the "standard" superellipse, $|x|^{2.5} + |y|^{2.5} = 1$:

I suspect an exponent of $2.4$ would produce an even better match, but I have not succeeded in persuading Wolfram|Alpha to produce a plot with the same scale.