Random Pretzel Hand Theory

This page has a semi-random selection of bits of theory on Pretzel Hand. It was too geeky to put on the main page, and that page was pretty geeky in the first place. Enjoy.

– Time Reversing a Pattern
– Pattern Identification
– Outside Mills/Flos Transformations
– Have we found all the 3:3 Messes?
– How many Rubenstein’s Revenges are there?

– Time Reversing a Pattern

If we have a three object outside pattern that is a distorted reverse cascade (not a siteswap) and we want to find the time reverse it’s a simple process. If we say the mess pattern is:

3 o| Af Ba Cb : Dc Ed Fe

Then the time reverse pattern is going to take the six throws and play them backwards changing catches into throws. “f” is the last catch so the pattern starts with the reverse of that pair: “Fa”. Then we go backwards so the next pair is taken from the end of the pattern and we turn “Fe” into “Ef”. Then continue back through the pattern to get:

3 i| Fa Ef De : Cd Bc Ab

– Pattern Identification

Let’s say we want to know whether a 3:3 pattern is outside or inside. If we generalise 3:3 mills mess and flos we get have:

o\ RUL:LUR – regular mills mess
o\ LUR:RUL – flipped mills mess
o/ URU:ULU – regular flos mess
o/ ULU:URU – flipped flos mess

o\ Xu Ux Xx : Xu Ux Xx – mills mess [Ru = regular, Lu = flipped]
o/ Ux Xu Uu : Ux Xu Uu – flos mess [Ur = regular, Ul = flipped]

Now we can work out what the time reverses are depending on the last digit of the siteswap and when it gets caught (because that becomes the first throw of the time reverse pattern). I’m not going to bother with throws higher than 8. The square brackets refer to the first letter pair and can be used to tell if the pattern is regular (the listed letters) or flipped.

TR 3:3 Mills Mess
[ _0_ , _3_ or _6_ ] i/ Ux Xx Xu : Ux Xx Xu [for 3 = Ur & 0 or 6 = Ul]
[ __1 , __4 or __7 ] i\ Xu Ux Xx : Xu Ux Xx [for 4 = Lu & 1 or 7 = Ru]
[2__ , 5__ or 8__ ] i\ Xx Xu Ux : Xx Xu Ux [for 5 = Ll & 2 or 8 = Rr]

TR 3:3 Flos Mess
[ _0_ , _3_ or _6_ ] i\ Xu Uu Ux : Xu Uu Ux [for 3 = Ru & 0 or 6 = Lu]
[ __1 , __4 or __7 ] i/ Ux Xu Uu : Ux Xu Uu [for 4 = Ur & 1 or 7 = Ul]
[2__ , 5__ or 8__ ] i/ Uu Ux Xu : Uu Ux Xu [for 5 = Uu Ur & 2 or 8 = Uu Ul]

So let’s say we knew the siteswap of a pattern but didn’t know if it was outside or inside (this can happen when trying to identify patterns belonging to different fmilies). If the Pretzel Hand was: Xu Ux Xx : Xu Ux Xx we’d know tthat it was a mills and outside if the siteswap didn’t end in 1, 4 or 7. If it did end in one of those numbers then we’d know it could be done as an outside regular mills or outside flipped mills AND an inside regular mills or inside flipped mills. If it starts with Ru it can be done as a flipped inside pattern and as a regular outside pattern. If it starts with Lu it can be done as a regular inside or flipped outside.

Then just for funsies here are the 5:5 versions:

o\ RULUL:LURUR – regular RLL mills mess
o\ LURUR:RULUL – flipped RLL mills mess
o\ RURUL:LULUR – regular RRL mills mess
o\ LULUR:RURUL – flipped RRL mills mess
o/ URULU:ULURU – regular RL flos mess
o/ ULURU:URULU – flipped RL flos mess
o/ URURU:ULULU – regular RR flos mess
o/ ULULU:URURU – flipped RR flos mess

o\ Xu Ux Xu Ux Xx : Xu Ux Xu Ux Xx – mills mess [Ru = regular]
o/ Ux Xu Ux Xu Uu : Ux Xu Ux Xu Uu – flos mess [Ur = regular]

TR 5:5 Mills Mess
___0_ i/ Ux Xu Ux Xx Xu : Ux Xu Ux Xx Xu [RLL = Ur, RRL =Ul]
____1 i\ Xu Ux Xu Ux Xx : Xu Ux Xu Ux Xx [Ru]
2____ i\ Xx Xu Ux Xu Ux : Xx Xu Ux Xu Ux [Rr]
_3___ i/ Ux Xx Xu Ux Xu : Ux Xx Xu Ux Xu [Ur]
__4__ i\ Xu Ux Xx Xu Ux : Xu Ux Xx Xu Ux [RLL = Lu, RRL = Ru]
___5_ i/ Ux Xu Ux Xx Xu : Ux Xu Ux Xx Xu [RLL = Lu, RRL = Ru]

TR 5:5 Flos Mess

___0_ i\ Xu Ux Xu Uu Ux : Xu Ux Xu Uu Ux [RL = Ru, RR = Lu]
____1 i/ Ux Xu Ux Xu Uu : Ux Xu Ux Xu Uu [RL = Ur, RR = Ul]
2____ i/ Uu Ux Xu Ux Xu : Uu Ux Xu Ux Xu [RL = Uu Ur, RR = Uu Ul]
_3___ i\ Xu Uu Ux Xu Ux : Xu Uu Ux Xu Ux [Ru]
__4__ i/ Ux Xu Uu Ux Xu : Ux Xu Uu Ux Xu [Ur]
___5_ i\ Xu Ux Xu Uu Ux : Xu Ux Xu Uu Ux [RL = Lu, RR = Ru]

– Outside Mills/Flos Transformations

For outside mills and flos messes we can generalize them as follows

o\ Xu Ux Xx : Xu Ux Xx – mills mess
o/ Ux Xu Uu: Ux Xu Uu – flos mess

To convert one to the other we take the first two letters and put them at the other end of the pattern, then change the highlighted red letters so if they were crossed they are now uncrossed or vice versa to match the letters around them that are already in the pattern. Then move the colon into the center of the pattern.

If they are in a specific siteswap then you have to modify the siteswap in the same way which could affect the aesthetic of the pattern.

– Have we found all the 3:3 Messes?

We can show we have all the possible 3:3 messes by looking at the throwing patterns. Rotating RRULLU gives us all mills messes and flipped messes inside and outside: RRU:LLU, RUL:LUR ULL:URR, LLU:RRU, LUR:RUL and URR:ULL.

While rotating RUULUU give us flos messes and flipped flos messes inside and outside: RUU:LUU, UUL:UUR, ULU:URU, LUU:RUU, UUR:UUL and URU:ULU.

RRULLU and RUULUU are the only two ways of always having a U between the R and the L. Then there’s RURLUL which gives us the hectic messes: RUR:LUL, URL:ULR, ULR:URL, LRU:RLU, RLU:LRU and LUL:RUR.

RRULLU, RUULUU and RURLUL are the only three ways of having the letters so right and left are mirrored in the middle. So unfortunately there are no more base messes to find.

– How many Rubenstein’s Revenges are there?

There are four 5:5 regular outside patterns for us to find Rubens in: RULUL:LURUR, RURUL:LULUR, URULU:ULURU and URURU:ULULU. Rubens is a 22335 pattern and the main characteristic of the Rubens pattern is the double loop from the 223 part of the pattern. To complete this loop we’ll need a RUL starting with the right hand or a LUR starting with the left. That rules out URURU:ULULU. And then there were four.

Now we need to see if we can fit the other patterns to the siteswap. We want the 223 on the RUL so that gives us: 22335 o\ RULUL:LURUR, 35223 o/ RURUL:LULUR and 52233 o/ URULU:ULURU. We can take those add in the catches and work out the time reverse of the patterns to get these six Rubenstein’s Revenges:

22335 o\ Ru. Ul. Lu Ul Ll : Lu. Ur. Ru Ur Rr – Rubenstein’s Revenge
52233 i/ Ul Lu. Ur. Rr Ru : Ur Ru. Ul. Ll Lu – TR Rubenstein’s Revenge
35223 o\ Ru Ur Ru. Ul. Ll : Lu Ul Lu. Ur. Rr – Alt Rubenstein’s Revenge
35223 i/ Ur Rr Ru. Ul. Lu : Ul Ll Lu. Ur. Ru – TR Alt Rubenstein’s Revenge
52233 o/ Ul Lu. Ur. Ru Uu : Ur Ru. Ul. Lu Uu – Flos Rubenstein’s Revenge
33522 i\ Lu Uu Ul Lu. Ur. : Ru Uu Ur Ru. Ul. – TR Flos Rubenstein’s Revenge

We could also work out the flipped versions to get six more. However, when we flip them the arms won’t be able to uncross in the same manner as the arc, we’d have to windmill them or loop in the opposite direction. The opposite direction loop is clearly the superior option and that would make it a Butterfly pattern so let’s call them Butterfly Loop Rubenstein’s.

22335 o\ Lu. Ur. Ru Ur Rr : Ru. Ul. Lu Ul Ll – Butterfly Loop Rubenstein’s
52233 i/ Ur Ru. Ul. Ll Lu : Ul Lu. Ur. Rr Ru – TR Butterfly Loop Rubenstein’s
35223 o\ Lu Ul Lu. Ur. Rr : Ru Ur Ru. Ul. Ll – Alt Butterfly Loop Rubenstein’s
35223 i/ Ul Ll Lu. Ur. Ru : Ur Rr Ru. Ul. Lu – TR Alt Butterfly Loop Rubenstein’s
52233 o/ Ur Ru. Ul. Lu Uu : Ul Lu. Ur. Ru Uu – Flos Butterfly Loop Rubenstein’s
33522 i\ Ru Uu Ur Ru. Ul. : Lu Uu Ul Lu. Ur. – TR Flos Butterfly Loop Rubenstein’s

So there are six pure Rubens, six butterfly loop Rubens and many, many more hectic and butterfly versions.

– Top of This Page
– Return to Main Pretzel Hand Page
– Appendix A: List of Patterns
– Appendix B: Building Ingenious Messes From Hulking Windmills