Submit manuscript...
eISSN: 2574-8092

International Robotics & Automation Journal

Research Article Volume 9 Issue 3

Shortest paths of Rubik’s snake composite knots up to 8 crossings

Songming Hou,1 Jianning Su,2 Ramon Mufutau1

1Program of Mathematics and Statistics and Center of Applied Physics, Louisiana Tech University, USA
2Department of Mathematics, Computer Science, and Engineering, Georgia State University, Perimeter College, USA

Correspondence: Songming Hou, Program of Mathematics and Statistics and Center of Applied Physics Louisiana Tech University Ruston, Louisiana, 71272, USA, Tel 3182573270

Received: October 15, 2023 | Published: October 26, 2023

Citation: Hou S, Su J, Mufutau R. Shortest paths of Rubik’s snake composite knots up to 8 crossings. Int Rob Auto J. 2023;9(3):109-113 DOI: 10.15406/iratj.2023.09.00272

Download PDF

Abstract

A Rubik’s Snake is a toy that has been around for 40 years. It can be twisted to many interesting shapes. In particular a Rubik’s snake can be twisted to form a knot. In this paper we study how many blocks are needed to form a composite knot with up to 8 crossings. Also, we improved a couple of previous results for Rubik’s snake prime knots.

Keywords: Rubik’s snake, composite knot

Abbreviations

MB, Mitochondrial diseases; mtDNA, mitochondrial DNA; MRI, magnetic resonance imaging; GTCS, generalized tonic-clonic seizure; HSV, herpes simplex virus; ENMG, electroneuromyography

Introduction

Researchers and amateurs alike have been enthralled by Prof. Rubik’s adaptable puzzle toy, “Rubik’s Snake”,1–3 for over 40 years. The right isosceles triangle prisms that make up Rubik’s Snake are joined together to form blocks, and this system allows for a variety of configurations that let users build complex structures and shapes. 24 blocks were initially included in the toy, but more blocks are now needed because of the possibility of creating designs that are more intricate. The Rubik’s Snake has been used as a tool for the study of protein folding4,5 and for the construction of reconfigurable modular robots.6–8 Some ideas in the study of Rubik’s Snake such as the use of rotation matrix is also used in rigid Origami folding.9,10 In previous papers that the first author collaborated with others, strategies have been given for the design of a Rubik’s Snake,11 and some mathematical problems concerning a Rubik’s Snake have been studied.12 Rotations that are not 0, 90, 180, or 270are mentioned in11 but not much theoretical work is presented. On the other hand,12 has quite some theoretical work but is only concerned with integer multiple of 90 degree rotations. In,13 Rubik’s snakes with general rotation angles were studied with theoretical work presented. In,14 theorems about palindromic, periodic and Möbius Rubik’s snakes were proved. In,15 a general strategy for designing box shapes using a Rubik’s snake was presented and a counting formula was derived. Knot theory16 is an interesting research area in mathematics that drew a lot of attention. It also has interesting applications in architectural design, for example, some work by Zaha.17 Knot classification tool is available.18 In,19–21 shortest path problems for Rubik’s Snake prime knots are studied. These are the pioneering work that combines Rubik’s snake with knots. Rubik’s snake composite knot paths have not been studied in the literature. In this paper, we study the shortest path for a Rubik’s snake composite knot with up to 8 crossings. The organization is as follows: in Section 2, we summarize our earlier work on the Rubik’s Snake prime knots. Some results are needed for our discussion here. In section 3, we proved a theorem needed for some of our constructions. In Section 4 to 6, we present results for Rubik’s Snake composite knots up to 8 crossings. We conclude in Section 7.

Prime knots

The Rubik’s Snake prime knots with shortest path up to 6 crossings have been studied. We just briefly summarize the results. The 31 (trefoil) knot was studied in our previous paper19 and 41, 51 and 52 studied in our previous work.20 The prime knots with 6 crossings 61, 62 and 63 were studied in our recent work.21 In this paper, in addition to the study on composite knots, we will make a couple of improvements for 41 and 52.

A Rubik’s snake prime knot we frequently use in this paper is the following palindromic 31 sequence from19: θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@38E4@ = [1, 0, 1, 2, 1, 0, 3, 0 ,0, 1, 3 ,0, 3, 3, 0, 3, 1, 1, 1, 3, 0, 3, 3, 0, 3, 1, 0, 0, 3, 0, 1, 2, 1 ,0], where 0, 1, 2, 3 mean starting with a straight ruler and rotating a joint by 0, 90, 180, or 270.

A theorem about [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ sequence

 We found a convenient tool to generate short path for composite knots based on our previous research on prime knots. The main idea is in this theorem and we will apply it to construct composite knots in the next section:

Theorem:

If the 1st and the k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4AaiabgUcaRiaaigdaaaa@39BB@ th blocks of a Rubik’s Snake has opposite entrance direction and opposite exit direction, and t is a sequence connecting the first blocks k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4AaiabgUcaRiaaigdaaaa@39BB@ , then [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ is a closed loop.

Proof. If a rotation index is 0 or 2, and we change both the entrance and exit direction to negative of the original, then the exit direction of the next block is also changed to negative of the original. If a rotation index is 1 or 3, and we change both the entrance and exit direction to negative of the original, then the exit direction of the next block is unchanged. However, when we use t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeyOeI0IaamiDaaaa@3914@ instead of t sequence, those 0 and 2 are unchanged but 1 and 3 interchanged. This will make the case of 1 or 3 also have the exit direction of the next block changed to negative of the original. It follows by induction that we will have k pairs of opposite vectors. Therefore the sum of 2k vectors has to be zero, forming a closed loop. Further, since the entrance and exit directions of the first and k+1th blocks are opposite respectively, the entrance and exit directions of the first and 2k+1 th blocks must be the same respectively. Therefore the 2k blocks form a closed loop.

We are going to use this [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ pattern to construct some composite knots. This is very convenient, as the shift from the first block to the k+1 th block does not matter. What matters is to find a pair of blocks with opposite entrance directions and opposite exit directions. Note that the converse is not true. For example, t = [2, 1, 0, 2, 1, 1, 2]. Then [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ is a closed loop (though collision occurs). However, the exit directions of the 1st and 8th blocks are the same.

31#31 knot and 41#41 knot

Our general strategy for constructing composite knot paths is to remove a part of a prime knot that is not knotted. We start with the simplest composite knot 31#31. The original sequence θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@38E4@ for a trefoil path is changed to [0, 3 ,0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 3 ,0, 3, 1, 1, 1, 3, 0, 3, 3, 0, 3, 1, 0] after a rotation redefining which one is the first block. Now we only use the first 28 of the 34 elements to remove a part that is not knotted. Now if we repeat the first 28 elements twice, it does not quite work. However, the condition in the theorem is met and by using where is the sequence with the first 28 elements, we get the 31#31 knot without crashing, which has 28∗2=56 blocks: [0, 3, 0,1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 3, 0, 3, 1, 1, 1, 3, 0, 0, 1, 0, 3, 2, 3, 0, 3, 0, 3, 2, 3, 0, 1, 0, 0,3,1,0,1,1, 0, 1, 3, 3, 3, 1, 0]

Note that instead of doing [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ we interchange 1 and 3. The reason is that in mod 4 sense, −2 is 2 and −1 is 3.

Figure 1 shows the 31#31 knot with 56 blocks.

Figure 1 31#31 knot with 56 blocks.

Figure 2 shows the line representation corresponding to it that clearly reveals it is a composition of two trefoil knots.

Figure 2 Line representation of 31#31 knot.

Note that if we are too greedy to use t = [3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 3, 0, 3, 1, 1, 1, 3], even though we still have the condition in the theorem to ensure [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ is a closed loop, we have collisions. This also serves as a counter example that the theorem does not ensure no collision even if t part alone has no collision.

In,20 the shortest 41 path found at that time had 46 blocks. Motivated by the symmetry associated with 41 knot, we searched for period 2 sequence [ t,t,t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshacaGGSaGaaGPaVlaa dshacaGGSaGaeyOeI0IaamiDaaWdaiaawUfacaGLDbaaaaa@4288@  and found that the shortest path is improved to only 44 blocks. For example, t = [0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 1].

Figure 3 shows the improved 41 knot with 44 blocks.

Figure 3 41 knot with 44 blocks that improved the previously published result.

Figure 4 shows the line representation corresponding to it that clearly reveals it is a 41 knot. The knot symmetry of 41 is also displayed.

Now by using the [ t,t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacaWG0bGaaiilaiabgkHiTiaadshaa8aacaGLBbGaayzx aaaaaa@3CBE@ idea again, we found that 76 blocks are enough to make 41#41 knot: [0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0, 0, 3, 3, 0]

Figure 4 Line representation of 41 knot.

Figure 5 shows the 41#41 knot with 76 blocks.

Figure 5 41#41 knot with 76 blocks.

Figure 6 shows the line representation corresponding to it that clearly reveals it is a composition of two 41 knots. We could also constructed a period two version of 41#41 with 76 blocks:

Figure 6 Line representation of 41#41 knot.

[0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0, 3, 3, 0, 2, 3, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 3, 0] repeated twice.

31#41

The trick to get 31#41 is to reuse most of the first part of 31#31 and most of the reverse order of first part of 41#41: [0, 1 ,2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 3, 0 ,3, 1, 1, 1, 3, 0, 3, 3, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 1, 0, 0, 0, 0, 3, 2, 0, 3, 3, 0, 3, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0]

Figure 7 shows the 31#41 knot with 68 blocks.

Figure 7 31#41 knot with 68 blocks.

Figure 8 shows the line representation corresponding to it that clearly reveals it is a composition of 31 and 41 knots.

Figure 8 Line representation of 31#41 knot.

31#51 and 31#52

These knots with 31 component have a similar construction. Recall that we had a sequence for palindromic 31 with 34 blocks in Section 2 cited from.19 First, let  be the sequence for the 51 knot with 52 blocks: [1, 1, 3, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 0, 0, 1, 3, 0, 0, 0, 3, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2]

Note that this is not exactly the same 51 construction in.20 As mentioned there, the solution is not unique and we choose this one for our purpose to construct the shortest composite knot. We construct [ θ 27,... θ 34, θ 1,... θ 21 ,1, α 48,... α 52 , α 1,... α 42 ,0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacqaH4oqCdaWgaaWcbaGaaGOmaiaaiEdacaGGSaGaaiOl aiaac6cacaGGUaaabeaakiabeI7aXnaaBaaaleaacaaIZaGaaGinai aacYcacaaMc8oabeaakiabeI7aXnaaBaaaleaacaaIXaGaaiilaiaa c6cacaGGUaGaaiOlaaqabaGccqaH4oqCdaWgaaWcbaGaaGOmaiaaig daaeqaaOGaaiilaiaaykW7caaIXaGaaiilaiaaykW7cqaHXoqydaWg aaWcbaGaaGinaiaaiIdacaGGSaGaaiOlaiaac6cacaGGUaaabeaaki abeg7aHnaaBaaaleaacaaI1aGaaGOmaaqabaGccaGGSaGaaGPaVlab eg7aHnaaBaaaleaacaaIXaGaaiilaiaac6cacaGGUaGaaiOlaaqaba GccqaHXoqydaWgaaWcbaGaaGinaiaaikdaaeqaaOGaaiilaiaaicda a8aacaGLBbGaayzxaaaaaa@68B6@ . This has 78 blocks. Then we could save 4 blocks by a shift to get [ θ 29,... θ 34 , θ 1,... θ 19 ,0, α 48,... α 52 , α 1,... α 42 ,0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaaeaa aaaaaaa8qacqaH4oqCdaWgaaWcbaGaaGOmaiaaiMdacaGGSaGaaiOl aiaac6cacaGGUaaabeaakiabeI7aXnaaBaaaleaacaaIZaGaaGinaa qabaGccaGGSaGaaGPaVlabeI7aXnaaBaaaleaacaaIXaGaaiilaiaa c6cacaGGUaGaaiOlaaqabaGccqaH4oqCdaWgaaWcbaGaaGymaiaaiM daaeqaaOGaaiilaiaaicdacaGGSaGaeqySde2aaSbaaSqaaiaaisda caaI4aGaaiilaiaac6cacaGGUaGaaiOlaaqabaGccqaHXoqydaWgaa WcbaGaaGynaiaaikdaaeqaaOGaaiilaiabeg7aHnaaBaaaleaacaaI XaGaaGPaVlaacYcacaGGUaGaaiOlaiaac6caaeqaaOGaeqySde2aaS baaSqaaiaaisdacaaIYaaabeaakiaacYcacaaIWaaapaGaay5waiaa w2faaaaa@65A8@ . This has only 74 blocks and is the shortest 31#51 we found.

We now take a different shortest 31 from [19]. θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@38E4@ = [1, 0, 1, 2, 1, 0, 3, 0, 3, 2, 3, 3, 2, 3, 0, 3, 1, 1, 1, 3, 0, 3, 2, 3, 3, 2, 3, 0, 3, 0, 1, 2, 1, 0]

The shortest 52 sequence found in20 had 56 blocks. Although it was verified to be optimal locally, we found that a non-local change could improve it to only 54 blocks below: α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdegaaa@38CD@ = [0, 3, 0, 0, 2, 1, 0, 3, 0, 0, 0, 3, 2, 0, 3, 3, 0, 0, 1, 2, 1, 1, 3, 0, 3, 3, 0, 3, 1, 0, 0, 3, 0, 0, 3, 2, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 1, 3, 3, 0, 3]

Figure 9 shows the improved 52 knot with 54 blocks.

Figure 9 52 knot with 54 blocks that improved the previously published result.

Figure 10 shows the line representation corresponding to it that clearly reveals it is a 52 knot.

Figure 10 Line representation of 52 knot.

We construct. This 31#52 has 76 blocks.

Figure 11, 12, 13 and 14 show the composite knots and their line representations constructed in this section.

Figure 11 31#51 knot with 74 blocks.

Figure 12 Line representation of 31#51 knot.

Figure 13 31#52 knot with 76 blocks.

Figure 14 Line representation of 31#52 knot.

Conclusion

Finding the shortest path for certain Rubik’s snake nontrivial knot is a challenging problem. With each joint having 4 choices, the total number of attempts is too many to have an exhaustive search. Based on our previous results for prime knots, we found possible shortest composite knots as follows: 31#31 with 34 + 34 − 12 = 56 blocks, 41#41 with 44 + 44 − 12 = 76 blocks, 31#41 with 34 + 44 − 10 = 68 blocks, 31#51 with 34+52−12 = 74 blocks, and 31#52 with 34 + 54 − 12 = 76 blocks. We also improved the 41 knot from 46 to 44 blocks and 52 knot from 56 to 54 blocks. It appears all our results are of the form, where and are the smallest number of blocks we found for the corresponding prime knots, except 31#41. Table 1 summarized the above results. As before, we verified that no local improvement could be made. Although there is no rigorous proof, the results look reasonable that they might be the shortest paths.

Shortest A

 Shortest B

 Shortest A#B

31:34

 31: 34

 31#31: 34+34-12=56

41:44

 41: 44

 41#41: 44+44-12=76

31:34

 41: 44

 31#41: 34+44-10=68

31:34

 51: 52

 31#51: 34+52-12=74

31:34

 52: 54

 31#52: 34+54-12=76

Table 1 A list of length of shortest paths found for composite knots based on prime knots

Acknowledgments

We thank Zhiyan Liu for some discussions about Rubik’s snake composite knots and thank Yunjie Chen for some discussions about architecture application and references. Songming Hou’s research is partially supported by the Walter Koss Endowed Professorship. The title of the professorship is made available through the State of Louisiana Board of Regents Support Funds.

Conflicts of interest

Authors declare that there is no conflict of interest.

References

  1. Fiore A. Shaping Rubik’s Snake. Penguin Books, Harmondsworth, Middlesex, England; 1981.
  2. Fenyvesi C. Rubik’s snake of infinite possibilities. The Washington Post; 1981.
  3. Jensen G. Now meet rubik’s snake –bigger than rubik’s cube!. United Press Interntational; 1981.
  4. Iguchi K. 1998. A toy model for understanding the conceptual framework of protein folding: Rubik’s magic snake model. Mod Phys Lett B. 1981;12(13):499–506.
  5. Iguchi K. Exactly solvable model of protein folding: Rubik’s magic snake model. Int J Mod Phys B. 1999;13(4):325–361.
  6. Ding X, Lu S, Yang Y. Configuration transformation theory from a chain-type reconfigurable modular mechanism-rubik’s snake. The 13th World Congress in Mechanism and Machine Science; 2011.
  7. Zhang X, Liu J. Prototype design of a rubik snake robot. Mechanisms and Machine Science. 2016;36:581–591.
  8. Liu J, Zhang X, Zhang K, et al. Configuration analysis of a reconfigurable rubik’s snake robot. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2019;233(9):3137–3154.
  9. Hull TC, Belcastro SM. Modelling the folding of paper into three dimensions using affine transformations. Linear Algebra and its applications. 2002;348(1-3):273–282.
  10. Tachi T. Simulation of rigid origami. Origami. 2009;4(08):175–187.
  11. Li Z, Hou S, Bishop T. Computational design and analysis of a magic snake. J Mech Rob. 2020;12(5):054501.
  12. Hou S, Chen Y, Li Z. Some mathematical problems related to the rubik’s snake. J Mech Rob. 2021;13(1):014502.
  13. Hou S, Atkins S, Chen Y. A rubik’s snake with general rotation angles. SCIREA Journal of Information Science and Systems Science. 2021;5(6):123–135.
  14. Hou S, Su J, Chen, Y. Palindromic, periodic and mobius rubik’s snakes. Int Rob Auto J. 2021;7(3):84–88.
  15. Hou S. Designing paths for box shapes using a rubik’s snake. Int Rob Auto J. 2022;8(2):66–68.
  16. Adams C. The Knot Book: An elementary introduction to the mathematical theory of knots. American Mathematical Society, ISBN 978-0-8218-3678-1; 2004.
  17. Serrazanetti F, SchubertM. Zaha Hadid: Inspiration and Process in Architecture. Matteo Schubert Print Book.; 2011.
  18. Livingston C, Moore AH. Knotinfo: Table of knot invariants.notinfo.math.indiana.edu, October; 2023.
  19. Hou, S, Su J. Shortest paths of trefoil knot designs using rubik’s snakes. Int Rob Auto J. 2022;8(1):18–20.
  20. Hou, S, Su J. Shortest paths of rubik’s snake prime knots up to 5 crossings. Int Rob Auto J. 2022;8(2)47–50.
  21. Hou, S, Su J, Mufutau R. Shortest paths of rubik’s snake prime knots with up to 6 crossings and application to roller coaster design. Int Rob Auto J. 2023;9(1):30–33.
Creative Commons Attribution License

©2023 Hou, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.