exhausting siteswaps, global shifts, Bacon, and feasts

[Daniel S.]

Hey All,

I was looking over my siteswap notes and noticed I didn't have much for
when you add 1 or subtract 1 to every number in a siteswap. Miika calls
this a global shift. Sounds like a good term.

So I started doing some tinkering...

if you always subtract the period whenever it shows up and keep making
positive global shifts then for example in period 6 you get a list like
this:

001203
112314
223425
334530
445041
550152
001203 .. hey we already had that one!

if you keep going as I did then it starts back at the beginning.
I've been referring to these siteswap families as "feast families" because
they remind me of feasts in passing and can be used to generate
interesting feast patterns. I can explain if anyone's interested.

And well it's not a family unless they have something in common so what is
it?
One thing is that the difference between numbers are all the same.

Using 001203 as an example
0 + 0 = 0
0 + 1 = 1
1 + 1 = 2
2 + 4 = 0
0 + 3 = 3
3 + 3 = 0

now there is a new set of numbers in the middle column 011433
This number can generate any siteswap in it's feast family by starting
with 0-5
I'll refer to this number as a Bacon for the time being (I've never seen
it, but I guess Kevin Bacon is in the movie 'six degrees of separation' ).

Next we find a way to exhaust the Bacons in a given period. I came up
with a method and wrote out my results. This really feels like something
that would be a lot faster for someone who knew how to program.

Here's the link:
http://siteswapgeneration.com/images/6feast.pdf

At the end of the paper I translate all of the Bacons back into siteswaps.

This got me to thinking that there must be a bunch of ways to exhaust
siteswaps given certain restraints. I'm curious to know of other ways if
anyone's up to share.

Also I'm always curious to know how to write something in math. If
there's some nice way to write out the algorithm I've used at the end of
the link above using math symbols and such I'd be happy to expand my math
grammar and vocab. It's also interesting that one can contain such a long
string of things in an equation that takes up such little space.

cheers,

Daniel

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[Miika]

Daniel S. wrote:
> Hey All,
>
> I was looking over my siteswap notes and noticed I didn't have much for
> when you add 1 or subtract 1 to every number in a siteswap. Miika calls
> this a global shift. Sounds like a good term.
>

I didn't originate it, but it does pair up nicely with a 'local shift'
which is adding/subtracting a multiple of the period to just one throw.
And of course we are all familiar with the cyclic shift as well :-)

> So I started doing some tinkering...

[snip]

>
> Also I'm always curious to know how to write something in math. If
> there's some nice way to write out the algorithm I've used at the end of
> the link above using math symbols and such I'd be happy to expand my math
> grammar and vocab. It's also interesting that one can contain such a long
> string of things in an equation that takes up such little space.
>

I'll have to look at it with more time, but to speed that up, would you
mind explaining a bit more why you exclude some of the 'bacon numbers' and
how do you know which ones to exclude? Do you have to work backwards from
the end with the siteswaps or what?

-Miika

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Siteswaps of the day: 123 , 456 , 789

[Daniel Shultz]

Miika wrote:
> I didn't originate it, but it does pair up nicely with a 'local shift'
> which is adding/subtracting a multiple of the period to just one throw.
> And of course we are all familiar with the cyclic shift as well :-)

these are much better terms than I was using ^_^
is this a cyclic shift?
531
315
153

> I'll have to look at it with more time, but to speed that up, would you
> mind explaining a bit more why you exclude some of the 'bacon numbers' and
> how do you know which ones to exclude?

I had to exclude certain bacons in order to have a valid siteswap.
I realized that the difference between numbers next to each other would never
be 1 less than the period (mod p). An example:
*by convention I'll always start the associated siteswap with 0, but this
could be any number.

period 7
bacon: 6666666
yields...
siteswap: 0654321

bacon: 6 n n n n n n
yields...
siteswap: 0 6 n n n n n

it doesn't matter what numbers are after the bacon 6 in period 7. It's
already invalid. You can't have a difference of 1 less than the period between
2 numbers or they'll collide.

This means that in period 7 all single digit bacons which add up to 6 (mod p)
will yield invalid siteswaps and must be excluded.

thus results omitted using the _ symbol as a bacon place holder:
_ = 6
...or in other words any one number that adds up to 6

If any 2 numbers add up to 5 (or 2 less than the period) then it is also not
valid.
I've shown this as _ + _ = 5

examples of bacons and their associated siteswaps:
bacon: 2311000
siteswap: 0256000
the 5 in the above siteswap lands at the first position because 2 + 3 = 5

bacon: 1411000
siteswap: 0156000
the 5 in the above siteswap lands at the first position because 1 + 4 = 5
4 + 1 also equals 5 so the 6 and the 1 will also collide.
You can try it with other examples, but if _ + _ = 5 then the pattern is
invalid.

Results omitted for period 7 would be:
_ = 6 (mod 7)
_ + _ = 5 (mod 7)
_ + _ + _ = 4 (mod 7)
_ + _ + _ + _ = 3 (mod 7)
_ + _ + _ + _ + _ = 2 (mod 7)
_ + _ + _ + _ + _ + _ = 1 (mod 7)
and in order for the pattern to start where it left off then
_ + _ + _ + _ + _ + _ + _ = 0 (mod 7) must be true

I'm not sure if I used the "mod p" or "mod 7" correctly... what I mean by it
is that, for example, in "mod 7" 5,12,19,... are all the same.

___________

The tables I've written out in the link I gave before sequentially exclude
bacons for every period until I reach the length of the siteswap.

To make the tables for period 6 I started with a single digit and excluded the
number 5 from being used in any following tables. Then I went to double
digits which didn't add up to 4. The double digits that passed the test got
put into the triple digit table where invalid bacons were crossed out and the
valid ones got taken into the 4 digit bacon contenders. I crossed out any 4
digit bacons that were omitted by the rule and the remaining valid ones got
transferred into the 5 digit table. The invalid 5 digit bacons got crossed
out. Now the next step was to add one number to each 5 digit bacon so that it
added up to 0 mod 6.

I then made cyclic shifts until each bacon was it's in it's lowest numerical
form so for example if I had 014100 i wrote it as 000141. This was because
there were multiples of the same answer and it made it easier to delete the
extras if they were all aligned in order.

Then I translated all of the remaining bacons into siteswaps starting with 0
which gave me all of the 'feast families' for period 6. To get all siteswaps
for period 6 (mod p) I made global shifts to every feast family until they
repeated themselves. All feast families except for 2 required 5 global shifts
to be complete. The 2 families that only required 1 shift were 042042 and
002244.

Also worth noting... you can't get from one feast family to another by making
local shifts alone. ...at least I'm pretty sure :P

Daniel


posted via JuggleHub
...where there's a siteswap tab!
btw, Niall it would be nice if the thread links were still visible somewhere
while posting so that referencing posts in the thread is easier to navigate.

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[Miika]

Yes, that's a cyclic shift. Not too many of these basic operations have
standard names, but siteswapping and sitesliding come to mind.

Ok, I had time to take a closer look at what you've done. So the bacon
numbers
are dependent on the length of the siteswap you are trying to construct,
and
you build the bacon numbers one digit at a time. To avoid confusion, you
might
try to include that number that they depend on in the name; like calling
the
numbers you used here as bacon-6 numbers (of lengths 1-6) and so on. In
any
case I confirm that the method does generate all siteswaps in the desired
range
:-)

In your definition of a feast family you only allow for throws mod p, so
local
shifts (+p) will in fact always leave the family, though of course this
won't
result in entering some other feast. Perhaps we can extend the definition
to
include patterns with higher throws, so this might be considered a feast
family
as well:

630
741
852

or

86420
97531
58642
69703
70814

And again you can construct all siteswaps by starting from the basic
feasts and
using local shifts methodically.

I find the ideas you've come up with as interesting, probably worth
exploring
more, and perhaps even not entirely useless. For example, it was
interesting
that some feasts had fewer members, when the original pattern repeated
sooner
than after p iterations. (Like for 002244 , 042042 , 01234 ,
c9630c9630c9630 )
If someone would take these into account and find some way to count the
different bacon numbers, they might be able to derive the formula for
siteswaps
of this type.

When can we expect a game around the concept? ;-)

-Miika

--

btw, tämäkin on ensimäinen viestini tältä sivustolta

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