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Möbius Cellular Automata Scarves
| Content Provider | Semantic Scholar |
|---|---|
| Author | Matsumoto, Elisabetta A. Segerman, Henry Serrière, Fabienne |
| Copyright Year | 2018 |
| Abstract | In 2015, the third author launched a successful Kickstarter campaign to fund the purchase of an industrial knitting machine. The Kickstarter rewards were scarves, each procedurally knitted in a unique two-colour pattern: the output of a elementary cellular automaton. The scarves are double knit (double bed jacquard in machine knitting parlance), meaning that the scarf has two layers of different colours, that swap positions back and forth to produce the pattern on the scarf. This implies that the pattern on the back side of the scarf is the colour reverse of the pattern on the front side. A corresponding “inverse” elementary cellular automaton produces the pattern on the back. We wondered if it would be possible to produce a Möbius strip scarf, in which the front becomes the back whilst seamlessly continuing the development of a single elementary cellular automaton. This paper describes our discoveries. Elementary Cellular Automata An elementary cellular automaton [1] is a one-dimensional cellular automaton with two states, for which the state cn+1 i of a cell in position i at generation n + 1 is determined by c n i−1, c n i and c n i+1 in the previous generation n. There are eight possible states of the three consecutive cells that determine the state of the middle cell in the next generation, so there are 28 = 256 possible elementary cellular automata (“rules”). These are numbered by reading the eight output bits as a binary number. See Figure 1. The rules act on binary strings, which in this paper are assumed to be cyclic, avoiding special cases at the ends of a string. Figure 1: The eight possible states of three consecutive cells determine the state of the middle cell in the next generation. Two rules are shown here. The first is numbered as 01101001 in base 2, which is 105 in base 10. The second is numbered as 10010110 in base 2, which is 150 in base 10. The Scarf Inverse of a Cellular Automaton Figure 2: Front and back of double knit. In double knitting, two colours of yarn are used, each colour making a planar knitted surface, one on the front of the piece (eg. black) and one on the back (eg. white). Colour work in double knitting exchanges stitches between the back surface and the front surface, swapping the two colours. Therefore, if we see a binary string in a row of double knitting, the opposite side of the work will show that binary string both reversed and with colours interchanged. See Figure 2. This motivates the following definitions: Bridges 2018 Conference Proceedings |
| Starting Page | 523 |
| Ending Page | 526 |
| Page Count | 4 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://archive.bridgesmathart.org/2018/bridges2018-523.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
