Prime Numbats

by Simon Whitechapel

To see how the shape is formed, examine the digits of 1/19₁₀, which also form a nycteridomorph (a bat-like shape or similar). Each digit d of the reciprocal represents a movement of distance l on a bearing θ of d x (2&pi / 9) radians:

Animated nycteridic of 1/19 in base 10

The period of 1/19 is at a maximum in base 10: that is, each of the numbers 1-18 is represented in the remainders during the division, so the sequence of digits repeats in blocks of 18:

If the first and second halves of the 18-digit sequence are examined, it will be seen that d_n + d_n+9 = 9: 0 + 9 = 9; 5 + 4 = 9; 2 + 7 = 9... Because the line corresponding to a digit d is the mirror image of the line corresponding to the digit 9-d, the nycteridic of 1/19 consists of two halves of 9 lines each. Here are more nycteridomorphs created by reciprocals having maximum period in a particular base:

Nycteridic of 1/37₃₉

Nycteridic of 1/59₆₁

Nycteridic of 1/67₆₉

Nycteridic of 1/67₇₈

Nycteridics can be formed using other formulae. Suppose each remainder r of 1/n represents a movement of distance l on a bearing θ of r x (2&pi / (n-1)) radians. If this formula is applied to 1/17, a repeating series of nycteridics occurs in various bases:

Animated nycteridic of 1/17_{3, 5, 6, 7, 10, 11, 12, 14...}

Now suppose the digits d of 1/n in base b represent successive movements of distance l on a cumulative bearing θ of (d x 2&pi / (b-1) radians + θ₀), where θ₀ is the previous bearing.

Nycteridic of 1/29₁₅ using cumulative θ of 2π

When the radian multiplier 2π is adjusted, the shapes become rotationally rather than mirror symmetrical.

Nycteridic of 1/89₄ using cumulative θ of 1·375π

Nycteridic of 1/37₁₆ using cumulative θ of 1·875π

Nycteridic of 1/37₇ using cumulative θ of 2·125π

Nycteridic of 1/97₆ using cumulative θ of 1·75π

Nycteridic of 1/97₆ using cumulative θ of 1·875π

Nycteridic of 1/97₆ using cumulative θ of 2·125π

Nycteridic of 1/97₆ using cumulative θ of 3·125π