A "spiral of digits" will help us illustrate an appealing "upward iteration." This spiral S is built from a familiar sequence D of distinct numbers (the order of its terms will be explained later):
D = 0, 1, 3, 2, 5, 7, 9, 4, 10, 6, 8, 21, 11, 20, 22, 23, 13, 24, 26, 28, 40, 42, 44, 25, 15, 17, 46, 48, 60, 62, 19, 31, 12, 33, 27, 35, 29, 14, 64, 66, 16, 37, 68, 41, 80, 18, 82, 84, 86, 43, 30, 88, 32, 200, 45, 34, 202, 47, 49, 201, 204, 36, 61, 206, 208, 220, 222, 63, 39, 65, 51, 67, 53, 224, 226, 228, 240, 69, 55, 81, 57, 203, 205, 59, 83, 71, 73, 85, 75, 38, 50, 242, 87, 89, 244, 77, 79, 210, 91, 52, 54, 93, 212, 56, 58, 70, 214, 216, 207, 218, 72, 74, 76, 78, 95, 211…
The sequence of digits in D forms the spiral of digits S:
S = 0, 1, 3, 2, 5, 7, 9, 4, 1, 0, 6, 8, 2, 1, 1, 1, 2, 0, 2, 2, 2, 3, 1, 3, 2, 4, 2, 6, 2, 8, 4, 0, 4, 2, 4, 4, 2, 5, 1, 5, 1, 7, 4, 6, 4, 8, 6, 0, 6, 2, 1, 9, 3, 1, 1, 2, 3, 3, 2, 7, 3, 5, 2, 9, 1, 4, 6, 4, 6, 6, 1, 6, 3, 7, 6, 8, 4, 1, 8, 0, 1, 8, 8, 2, 8, 4, 8, 6, 4, 3, 3, 0, 8, 8, 3, 2, 2, 0, 0, 4, 5, 3, 4, 2, 0, 2, 4, 7, 4, 9, 2, 0, 1, 2, 0, 4, 3, 6, 6, 1, 2, 0, 6, 2, 0, 8, 2, 2, 0, 2, 2, 2, 6, 3, 3, 9, 6, 5, 5, 1, 6, 7, 5, 3, 2, 2, 4, 2, 2, 6, 2, 2, 8, 2, 4, 0, 6, 9, 5, 5, 8, 1, 5, 7, 2, 0, 3, 2, 0, 5, 5, 9, 8, 3, 7, 1, 7, 3, 8, 5, 7, 5, 3, 8, 5, 0, 2, 4, 2, 8, 7, 8, 9, 2, 4, 4, 7, 7, 7, 9, 2, 1, 0, 9, 1, 5, 2, 5, 4, 9, 3, 2, 1, 2, 5, 6, 5, 8, 7, 0, 2, 1, 4, 2, 1, 6, 2, 0, 7, 2, 1, 8, 7, 2, 7, 4, 7, 6, 7, 8, 9, 5, 2, 1, 1…
These digits will be drawn in a particular way. The very simple—and vaguely "Brutalist"—typeface used here is the one chosen by mathematician Dean Hickerson. The digits from 0 to 9 are drawn as follows within a vertical 3-by-5 rectangle.
