My question: Is there a generalization of the result below, either involving more digits or a fraction other than $\cfrac{1}{7}$?
The Futility Closet has this surprising result:
The One-Seventh Ellipse
The decimal expansion of $\cfrac{1}{7}$ is $0.142857142857\dots$, a repeating decimal. Arrange the six repeating digits into overlapping ordered pairs, like so: $$ (1, 4), (4, 2), (2, 8), (8, 5), (5, 7), (7, 1), $$ and, remarkably, all six lie on an ellipse: $$ 19x^2 + 36yx + 41y^2 – 333x – 531y + 1638 = 0. $$
Even more remarkably, if we take the digits two at a time: $$ (14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42), $$ these points also lie on an ellipse: $$ -165104x^2 + 160804yx + 8385498x – 41651y^2 – 3836349y – 7999600 = 0 $$
That’s from David Wells, The Penguin Dictionary of Curious and Interesting Numbers (1986). Victor Hugo wrote, “Mankind is not a circle with a single center but an ellipse with two focal points, of which facts are one and ideas the other.”