Bengt Magnusson was one of the pioneers of siteswaps. He made numerous posts on the subject to the original juggling mailing list -- Peter Mark saved some of the original posts for historical interest, and they are reproduced below. Although these posts may refer to programs that may not exist anymore or use now dated terms, generally the descriptions are clear and informative, and in several places they give insight into how the folks who developed the notation and programs thought about it and came up with it. (All Bengt's posts are available via the JIS news service; search for "bengt" in the "From" line.) Bengt, a widely regarded juggler and a student at CalTech, died in 1991.
From @voodoo.physics.ucsb.edu:bengt@pcs1.physics.ucsb.edu Mon Jan 21 23:48:29 1991
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From: bengt@voodoo.physics.ucsb.edu (Bengt Magnusson)
Message-Id: <910121191618.20a02399@pcs1.physics.ucsb.edu>
Subject: Text file, explaining our notation.
To: pdm@cs.uoregon.edu
X-St-Vmsmail-To: ST%"pdm@cs.uoregon.edu"
Status: RO
A few explanations about the program and the notation. First, the notation. A
trick is written as a string of numbers. Each number indicates how high and
where a ball should be thrown. Generally, it should be thrown as if you were
juggling that many balls, with the requirement that odd numbers are always
juggled in a cascade, and even numbers in a fountain. THE HANDSPEED ALWAYS STAYS
THE SAME! Thus, if you are juggling five balls, making about five throws per
second, you always continue to throw five times per second, no matter how high
or low any individual ball is being thrown. The throw heights have to be related
by our formula
Hn = Hm * ( (n-1)/(m-1) )^2,
where n and m are the number of balls, and Hn and Hm the throw heights needed.
(We have for simplicity assumed theta=0.5 .)
The two hands are always supposed to be staggered, as in a normal cascade
pattern. As an example, suppose you are juggling five balls, making each throw
about 1 m high. Then a "6" is a 1.6 m high throw to the SAME hand. A "7" is a
2.25 m high throw over to the OTHER hand. An "8" is a 3 m high throw to the
SAME hand, while a "9" is a 4 m high throw to the OTHER hand. A "4" is a 0.6 m
high throw to the SAME hand, and a "3" is a 0.25 m high throw to the OTHER hand.
"0" - "2"s deserve special mention. In principle, you could make a "2" be a
0.06 m high throw to the same hand, but it's usually done by just holding on to
the ball for one beat. A "1" is a quick hand-off to the other hand, like the
short throw in a shower. A "0" simply means the hand is empty for that moment.
This still satisfies the notion of throwing a ball as if you were juggling that
number: 2 could be "juggled" by just holding the balls, one in each hand. 1 is
"juggled" by just constantly passing it back and forth between the hands. 0 is
"juggled" by not doing anything at all with two empty hands.
As an example, let's give a detailed description of the five ball trick 7562.
I let the clock tick from 0 to 2 s, at each instant telling you how each hand is
throwing.
Time s Left hand Right hand .
| .
| .
| 0.0 1 m across "5"
v
0.2 1 m across "5"
0.4 1 m across "5"
0.6 2.2 m across "7" start trick
0.8 1 m across "5"
1.0 1.6 m same "6"
1.2 hold "2" end trick
1.6 1 m across "5"
1.8 1 m across "5"
2.0 1 m across "5"
.
.
.
If you were actually attempting this trick, you wouldn't be thinking "2.2 m
across," etc. Most of you have a pretty good idea of how high a seven ball
pattern would have to be, so you'd probably think of the trick as "high across,
normal, not-so-high same, hold," or something like that.
To further help clarify the notation, I'll give the notation for some tricks
that most of you already know. The first example is the shower. A three ball
shower is called "51," a four ball shower "71," a five ball shower "91," etc.
The "5" is the high throw, while the "1" is the quick throw underneath. Imagine
the numbers repeated endlessly: "...5151515151..." I will always only write out
the shortest string possible, that can then be repeated forever.
A four ball half shower thrown with right (R) and left (L) hands alternating
as in ...R L R L R L... (as opposed to ...RL RL RL...) is called "53."
(Note that no five ball (or any odd number) half shower is written as "64;" the
balls never switch hands for this pattern. "64" is three balls done in one hand
and two in the other. As an exercise, visualize the difference between "64" and
"6662" and "666660". Both involve three in one hand and two in the other.)
Three balls in one hand (with the other hand empty) is "60." Two in one hand,
with a third held in the other hand, is "42." Two in one hand, with the third
thrown every other time, is "4440." "Five with a hole," i.e., four balls all
thrown in the five ball pattern, is "55550." Many people practice this when they
try to learn five balls. Other five ball practice patterns used with four balls
are "552," and, less commonly, "5551."
A fairly common four ball pattern is where one ball is thrown high, and the
other three are briefly cascaded like three balls. There are two ways of doing
this, the more common one being "633." This one is also popular for bouncing,
the "6" being bounced on the floor instead of thrown high. (The other way of
doing the pattern is "7333." Of course you can also do "83333," etc., but this
is getting silly.)
As you can see, you already know many of our tricks, which we call "site-
swaps." There are literally thousands of these tricks. Once you start learning
the basic throws of a trick, you can then figure out ways to modify them to
your own taste and skill. As I mentioned above, you can bounce some of the
throws. You can perhaps do Mill's mess with some of these patterns. One nice
application of "2"s occurring in patterns is to do something with the hand that's
just holding a ball. You can scratch your nose, wipe your forehead, pretend to
look at your watch, swat a bug (or your partner), etc. Limitless opportunities!
(Of course you can do the same whenever "0"s occur, too, but tricks with "0"s
always involve higher (=harder) throws than do similar tricks with "2"s.)
The tricks can of course be done with clubs or rings also. Here, "3"s may be
of interest. Any single throw three club trick can be done then. For example,
with five clubs, throw a "663." When you do the "3," don't just throw it across.
Do a chin roll, a head roll, a helicopter roll, a Nizer, an under the leg, a
back cross, an ego, or an oge! Perhaps more humanly, do this during a "53" or
"633" with four clubs. "6"s with clubs are typically triples, while "5"s are
more lofted doubles.
Now for the program. It's in badly written fortran (it has goto's in it!), but
it's fairly well commented, and it works. On a VAX it takes about 15 CPU-minutes
to generate all tricks up to word length seven. (There are many thousand of
them.) The program will ask you for the number of balls you intend the tricks to
be. It will also ask you for maximum word length, and for the highest throw you
allow. (Throwing 15's isn't easy; typically, 9's are the most people can deal
with.) I suggest you don't try to generate tricks of word length longer than 9.
The program will generate all true site-swaps, as well as all excited ones. A
true site-swap is one you can just do from a regular pattern, while an excited
one needs a special start-up sequence. For example, you can't just start doing
a shower from a three ball cascade; you need to prepare for the shower by
throwing a "4" first. Similarly, you can't just stop the shower either. You have
to do a "2" before resuming the cascade. Therefore, the "51" is said to be an
excited site-swap. The "64" with five balls, on the other hand, is a true site-
swap. The program will also generate a start-up and a finish sequence for the
excited tricks. (I say A sequence; there are many possible ones. My program
tries to form the one with the lowest throws. Another selection criterion could
be to select the one creating the lowest number of throws, or the one with the
most 0s or 2s or whatever. For example, some finish sequences will be 111, which
is dumb. 200 would be better. You can usually figure those out on your own, or
modify the program.)
Rather than working with the actual numbers in the trick sequences, the
program works with the numbers minus the number of balls. Thus, instead of
thinking about the 7562, it ponders 201-3. The reason is that these lower
numbers are more general; add 6 to them, and you get a valid 6 ball trick
(8673); add 3 and you get a 3 ball trick (5340), etc. It starts by generating
(for a trick length of four throws) 1 -5 -5 (for five balls; you can't throw
less than a 0). It then tries to fill in the last digit so the string sums to
zero. This requires a 10 in the last position, or 6 0 0 15. If you allow 15's,
it will consider this a legal trick candidate, and perform further tests. (It
has to check for collisions.) If you don't, it will increment the second to
last digit (now 1 -5 -4), and try again, until a legal trick is found. After a
legal trick is found, it will again increase the penultimate digit, until it
reaches the maximum throw height you allow. It will then increase the digit
before that, reset the penultimate one to -n, and start over. It keeps doing
this until the first digit reaches the maximum throw height.
All the time, the program tries to make the string sum to zero. This is the
fundamental criterion for a legal trick. No strings that do not sum to zero are
legal tricks. However, it is not a sufficient criterion. Many zero-sum strings
are not legal tricks, either. However, as a consequence of the zero-sum rule,
the average of the numbers in a string like 7562 is the number of balls the
trick involves. You can therefore tell at a glance how many balls a given trick
involves, even if you've never tried to perform it (or seen it) before. The
second test is for collisions. If any two throws ever aim to the same future
site, a collision will occur, and the trick candidate is rejected.
The program avoids repetitions. Thus, you will find "64" in the table, but you
won't find "6464." Also, "744" is in the table, but "64744" or "74464" aren't.
You can just string together tricks in any way you want. Moreover, the table is
generated so that the highest throw is always the first one. This means that
some tricks you'd expect to find aren't in the table directly. For example,
"771" is an excited trick, with start up sequence "66" and finish "3." This is
tabulated under word length three as 66 771 3, and it means you can do
...55566771771771...7717713555... You may notice that 667713 is then also a
valid trick, that could be repeated: ...555667713667713...667713555..., and
therefore expect to find 667713 in the table for word length six. You won't. You
will, however, find 771366, which is the same trick shifted over two throws. If
you do ...667713667713... over and over again, you can't tell whether it's
667713 or 771366. The only difference is in the start-up and finish sequences.
667713 won't need any, while 771366 will, but that's only a minor issue that you
can adjust as you learn the tricks. I'm just avoiding duplications; there are
enough tricks as there are.
Some suggestions for worthwhile tricks to start out with: Three balls 441,
51414. Four balls 661515. Five balls 7562, 75751, 771, 97531 (the first ball you
throw is the last you catch, the last you throw is the first you catch), 95551.
You can do a lot of sifting while running the program. For example, sequences
like "11" are pretty dumb. Tricks with "0"s always have similar tricks with "2"s
instead, and the latter will always be easier as they involve lower throws. "3"s
are really hard to throw in most patterns; they must be higher than "1"s, but
much lower than any other throws, so you end up doing most "3"s blind. Just get
rid of those tricks during execution of the programs, and the table will be
somewhat shorter and easier to read. The sample program I'm sending has such a
rejection scheme (in the commented loop to address 122). Just delete that part
if you want, or add your own.
So why do we call the tricks site-swaps? The name comes from the way we first
thought about the tricks. Suppose you are juggling imaginary numbered boxes,
each one containing a ball. The boxes are always moving in the same basic
pattern (cascade or fountain). Now suppose you can throw the balls in such a
way that they switch boxes, f.ex. let the balls in box numbers 2 and 4 switch
places. Then you have performed a site-swap: the balls have had their imaginary
boxes, or sites, swapped.
This obviously ties together strongly with the permutations of numbers. If you
juggle five balls, you have five boxes, and so there should be 5! = 120 site-
swaps. There are, and in a sense the set of all site-swaps of length n can be
made isomorphic to Sym(n). However, many of these site-swaps would be dumb.
64555, 56455, 55645, and 55564 would all permute the imaginary boxes differently
while looking absolutely identical. The generated tables will therefore have
fewer elements than n!. To this must then be added the excited tricks, so the
result is that there is no simple formula predicting the number of tricks at a
given word length. Our best guess is that is goes roughly like n!; maybe someone
can derive an exact formula. The simple interpretation of imaginary boxes
becomes less obvious when you have five balls and do a six (or seven or...)
throw trick. (Number of throws = word length.) However, the name had already
stuck in our minds, and we kept it. The excited tricks were discovered even
later, but clearly belong to the same general category of tricks, so we call
them site-swaps too.
Bengt Magnusson and Bruce Tiemann
448 Pitzer Court
Goleta, CA 93117
(805)685-3360
Bengt@Voodoo.Ucsb.Edu
Bengt@Voodoo.Bitnet
Voodoo::Bengt
Available node numbers: 44.124, 44.125,
44.126, 44.128, 44.129, 44.130, 44.133
(i.e., 44124::Bengt, etc.)
From JUGGLING@INDYCMS.BITNET Sat Jan 26 19:45:21 1991
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Date: Sat, 26 Jan 91 19:35:57 PST
From: Bengt Magnusson <bengt@VOODOO.BITNET>
Subject: Explanation of new site swap program
Sender: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
To: "Peter D. Mark" <pdm@CS.UOREGON.EDU>
Reply-To: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
Message-Id: <CB315BDF291F200B1C@oregon.uoregon.edu>
X-Envelope-To: pdm@CS.UOREGON.EDU
Comments: Warning -- original Sender: tag was
Status: RO
I have finally created a better version of the site swap program. It is
written in a real language (ANSI C), and it uses a more efficient and
elegant algorithm. Moreover, there are no goto's in the code! Note, however,
that this program only generates ground state site swaps. Another program
dealing with the excited state tricks will follow later (maybe much later).
The following description will assume you know something about site swaps
already.
A site swap is a string of numbers that tell you how high each throw must
be. If the trick contains n numbers, you are to imagine you are juggling n
imaginary sites. They happen to be inhabited by the real objects you are
juggling, but there may be more or fewer of them than you have objects!
After the trick, the sites land in some permutation of their initial order.
If you subtract off m, the number of real objects you have, from the string
of numbers, you get a string whose sum is zero. F.ex., the 7562 is a five
object trick. Subtract off five, and you have 2 0 1 -3; 2+0+1-3=0. If you
count forward two positions from the '2' (which is in pos. 1), you end up by
the '1' (which is in pos. 3); if you count forward zero positions from the
'0' (in pos. 2), you stay there; if you count one position forward from the
'1' in pos. 3, you end up by the '-3' in pos. 4; finally, if you count
minus three positions forward, i.e., three positions backwards, from the '-3'
you end up by the 2 in pos. 1 .
Notice how in all of this, we always ended up on a position inside the
trick itself. Nothing ever pointed outside the string of numbers we have.
This is what is meant by a ground state trick. To illustrate what an excited
state looks like, consider the 771, again with five objects. It's a 2 2 -4
with five subtracted off. Notice how, if you count four positions backwards
from the -4 you end up two positions before the first position. This is a
common feature of all excited state tricks. This new program only generates
the ground state tricks! Another, more practical difference between ground
and excited states, is that you can throw a ground state trick directly from
a regular (no trick) pattern. You can also stop it after a single cycle (or
however many cycles you want) and resume the normal juggling. (This also
means that you can string any number of different ground state tricks
together, side by side.) You can not do this with an excited trick. They all
require a particular sequence of throws to get started from the regular
pattern. Once started, the excited trick can be repeated however many times
you want. When you want to stop, you again need to do a particular set of
throws to be able to get into the regular pattern. You are also not free to
combine excited tricks any way you want; that typically require their own
coupling sequences.
So if my old program generated all tricks, why write a new program that
generates fewer tricks? The reason is that the old program was badly written,
it was slow, and it had bugs in it. It did not generate all the tricks it
was supposed to, and it kept on generating tricks that shouldn't be there
(as you may have noticed from my repeated postings of patched up versions).
In particular, the old program demanded the first throw be the highest one.
The reason for that was that once you are running the trick indefinitely,
you can cyclically the starting point to any throw you want. However,
if you only want to do the trick a single time, you can't do that, and the
program therefore missed many tricks.
I have also used more intelligent choices for candidate throw heights,
thereby reducing the execution time. Instead of generating all strings of
zero-sum tricks, I use the restrictions imposed by the ground state. Since
no throws are allowed to refer to positions outside the trick itself, I can
place very strict limits on the range for throw heights. F.ex., the first
throw can be at most n-1 high, or it will point beyond the last throw. It
must also be greater than 1. (A 0 = a normal throw, so the trick is really
a site swap one unit shorter (word length n-1 instead of n), and you can't
start a trick with a smaller than normal throw: you have to start by buying
time.) A throw in the last position must be negative (again, a zero means
the trick is one unit shorter, while a positive number points beyond the end
of the trick), but not smaller than -n+1, or it points before the first
throw. Throws in position i, 1<i<n, must be smaller than n-i, but greater
than 1-i. Of course all throws must also be greater than -number of real
objects juggled.
The algorithm starts by calling a routine that generates the possible
throw heights for pos. 1. It then calls itself to generate the throws for
pos. 2, etc. The routine keeps calling itself to generate the throw height
for pos. i, assuming pos. 1 to i-1 have already been assigned heights. It
also keeps track of running sums from pos. 1 to i: if it ever reaches zero
before the last position, we are just juxtaposing two shorter site swaps,
and since we already know they can be juxtaposed any way we want, it will be
unnecessary to tabulate those tricks over again. The program also checks for
impossible conditions: if pos. 1 to i have been assigned heights that are too
big so we can never generate a zero sum trick, no matter how small the
remaining throws are made, it backtracks, rather than trying all the possible
attempts at the (doomed to fail) zero sum. The variables summax and summin in
the code check this condition. The sum from i+1 to n of the lowest possible
throws is -0.5*n*(n-1) + 0.5*i*(i-1), while the sum of the highest possible
throws is -1 + 0.5*n*(n-1) + 0.5*i*(i+1-2n). Add the running sum to this,
and you get the highest and lowest possible sums. Then must bracket zero if
a zero sum trick is to be possible. If they don't, the algorithm again
backtracks one position, and tries new throws.
Collisions are checked for as we go along, rather than in the finished
trick. In this way, we can stop right away when two throws are in conflict,
rather than trying to generate all other tricks with those two throws first,
and then throwing them out. A throw of height j in pos. i will collide (land
in the same hand at the same time) with a throw of height m in pos. k if
m-j = k-i.
Before a trick is output, you can eliminate unwanted ones in the last
routine, called 'nuke'. Say you don't want to tabulate any tricks containing
'3's. Just make nuke return 1 whenever a trick contains a 3. (Since the
tricks are generated with the number of objects subtracted off, you have to
make the test for 3 - # of objects, rather than just 3.) See the comments in
the program for more information. I'm sure the algorithm can be improved
upon further, and I invite you to do so. However, I hope the improvement of
this algorithm compared to the old one will be apparent.
Bengt@Voodoo.Ucsb.Edu
From bengt%pcs0.physics.ucsb.edu@nobbs.physics.ucsb.edu Mon Jan 28 13:41:42 1991
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Date: Mon, 28 Jan 91 13:38:42 PST
From: bengt@pcs0.physics.ucsb.edu (Bengt Magnusson)
Message-Id: <910128133842.20601c85@pcs0.physics.ucsb.edu>
Subject: RE: more about siteswaps
To: pdm@cs.uoregon.edu
X-St-Vmsmail-To: ST%"pdm@cs.uoregon.edu@spencer.cs.uoregon.edu"
Status: RO
I knew it, I screwed up the address. I just made someone else confused. You
see, after I replied to you about the exc./gnd., I realized one thing, and
added that in a note I sent to what I remembered as your address. Well, it
wasn't. This is the comment that was meant for you: I said that the 4514142
will appear in the three ball, seven word length ground state site swap
table. Well, it won't, and this is why. 45141 is a ground state site swap of
word length five, and 42 is another ground state site swap (of word length
length two). 4514142 is then just the juxtaposition of two ground state site
swaps, and will therefore not appear in the table, since we know that we can
string any site swaps together back to back in any way we want. Therefore, we
have yet another reason to tabulate the excited state tricks separately.
Bengt@Voodoo.Ucsb.Edu
From @voodoo.physics.ucsb.edu:bengt@pcs1.physics.ucsb.edu Mon Jan 28 10:56:06 1991
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Date: Mon, 28 Jan 91 10:52:55 PST
From: bengt@voodoo.physics.ucsb.edu (Bengt Magnusson)
Message-Id: <910128105255.20a03b86@pcs1.physics.ucsb.edu>
Subject: RE: more about siteswaps
To: pdm@cs.uoregon.edu
X-St-Vmsmail-To: ST%"pdm@cs.uoregon.edu@spencer.cs.uoregon.edu"
Status: RO
First, please notice the trivial bug the posted version had: replace 'infp'
(in routine 'trick') by 'outfp'.
About graphics: read my new posting.
Ground state vs. excited: Bruce and I are writing articles for JW that will
hopefully explain all this. However, since the excited state article will not
appear for at least two issues, you might want to know right away. Basically,
a ground state trick is one you can throw right away out of a normal pattern,
and return to the normal pattern as soon as you've finished the trick. Take
the 441 (3 balls). You can merrily juggle along in your 3 ball cascade, and at
any time decide to do a 441, and then return to the cascade: ...3333441333...
Excited state tricks don't work that way. Take the 51414 (3 balls). If you try
to do that straight out of the cascade, you run into trouble: ...33351 is as
far as you'll get. The last 3 you threw will collide with the 1 (land in the
same hand at the same time). You have to throw a special start up sequence to
create a configuration that allows you to begin the trick. In this case it's
simple: it's a single throw, a 4. Therefore, if you throw ...3333 4 you can
then start the 51414: ...3333 4 51414 . You can now run the 51414 as many
times as you like: ...333 4 51414 51414 ... 51414 (the spaces are for clarifi-
cation only, they don't mean you have any spaces in the juggling). Now, if you
want to return to the cascade, you run into problems again. You have to throw a
special finishing sequence as well. In this case, it's a 2: ...51414 2 333...,
so the full excited trick looks like ...333 4 51414 51414 ... 51414 2 333...
The tricks are called excited because they require these special coupling
sequences, and live their own life only within these couplants. You may
realize that the sequence 4514142 can be done directly out of a cascade (this
is the excited trick plus the coupling sequences); therefore it should be a
ground state site swap. Indeed, the 4514142 is found in the seven word length
ground state table. One might therefore ask why we should bother tabulating
excited state tricks separately, as they are always included in the ground
state tables, although at a higher word length. The reason is that from looking
at the 4514142 in the table, it is not at all clear that the 51414 piece can
be repeated indefinitely within the trick. Repeating the inner trick often
yields neat tricks on their own, and therefore merit their own table. (Notice
that not all site swaps hide these 'inside' tricks, only some do. However,
enough of them do that there are, at a given word length (not counting
connecting sequences), more excited state tricks than there are ground state
tricks.)
Yes, a future program will do the excited state tricks, as well as their
coupling sequences.
Bengt@Voodoo.Ucsb.Edu
From JUGGLING@INDYCMS.BITNET Sun Jan 6 20:41:33 1991
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Date: Sat, 5 Jan 91 15:16:26 PST
From: Bengt Magnusson <bengt@ucsb.ucsb.edu>
Subject: Site-swap Program
Sender: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
To: "Peter D. Mark" <pdm@CS.UOREGON.EDU>
Reply-To: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
Message-Id: <DAE0E37D955F6012CD@oregon.uoregon.edu>
X-Envelope-To: pdm@CS.UOREGON.EDU
Comments: Warning -- original Sender: tag was
Status: RO
The version of the program I posted has an unnecessary limitation in it.
Fairly early on there is a line like "If ( Sum .Eq. 0 ) Goto 300". What is
does is remove needless duplication of tricks. F.ex., 6464 is not really
different from 64, and it's therefore unnecessary to clutter up the four
word length list with this duplicate. That program line does just that.
However, it also removes some useful tricks, like the 51414. If you just
comment out that line, you will get all the tricks printed out, but there will
be a lot of duplicates. A better way of doing it is to not just check for
Sum=0, but to also check if the first part of the trick (the zero-sum part
that made sum=0) is a genuine site-swap. If it is, then the trick will indeed
be a useless duplicate, but if it isn't, the trick will probably be useful.
This would mean tricks like 6464, 64744, 7575195551, etc., would all be
removed from the table (being just two regular site-swaps in succession),
while tricks like 51414 will be retained. The 51 (three shower) is an
excited site swap, requiring a start-up sequence (a 4) and a finish sequence
(a 2). If it is immediately followed by another trick, something new and
interesting is likely to result. The 51414 is really just a three shower (51)
and a 441 in succession, but the 441 has been cyclically shifted to a 414,
and there is no finish throw after the 51, so the juxtaposition of those two
site-swaps is sufficiently different to merit its own spot in the table.
I'll post an ugly version of the program that does just this. Later on in
the program is a piece that checks for genuine site-swaps, and I've just
duplicated that part, and do it separately for Sum=0 situations. Of course
that piece should really be in a subroutine. If fortran only lent itself to
useful subroutining... Anyway, try this new version, and you'll get more
site-swaps to practice (as if there werent' enough already!).
Bengt
From JUGGLING@INDYCMS.BITNET Fri Feb 8 17:03:51 1991
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Date: Fri, 8 Feb 91 16:29:23 PST
From: Bengt Magnusson <bengt@VOODOO.BITNET>
Subject: Excited State Site Swaps, Part II
Sender: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
To: "Peter D. Mark" <pdm@CS.UOREGON.EDU>
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Status: RO
Here is the promised program for generating excited state site swap
tricks. Well, the program is actually in a different file. This file
will attempt to explain what the program does, as well as describe a
few general facts about excited state tricks.
First, let's recall what is meant by an excited state trick, as
opposed to ground state tricks. All site swaps are described by a
string of numbers, where each number tells you how high to throw the
object. The average of the numbers in the string equals the number
of objects juggled. If you subtract off the number of objects from
each number in the string, you end up with a string of numbers whose
sum is zero. Each number in the string now tells you how many sites
each object is advanced or delayed. F.ex., the 4 4 1 trick with three
becomes 1 1 -2 after this operation. It tells us the first throw is
delayed one position, as is the second one, while the third one is
advanced two positions.
In a ground state trick, no throw is ever advanced or delayed beyond
the string of the trick. In the above example, the first 1 is delayed
one position, to the second position. The second 1 is also delayed one
position, from the second to the third position. The -2 is advanced two
positions, from pos. 3 to pos. 1. No throw ever points outside the
trick. If you draw little arrows from one number to the position it
points to, you'll end up with a closed loop. There are no open ends
pointing away from the trick.
Consider now a simple excited state trick, the three object shower.
It's called a 5 1 in our notation, or 2 -2 after the subtraction. Here,
the 2 points from pos. 1 to pos. 3, but there is no position three: the
trick is only two throws long. Similarly, the -2 points from pos. 2 to
pos. 0, which again doesn't exist. This is a common mark of all excited
tricks. They all point to positions beyond the trick. Sure, if you
repeat the trick: 2 -2 2 -2 2 -2..., you'll always end up on an existing
position, but remember, a trick is always written in its shortest form,
which is 2 -2 in this case.
Another way of stating the difference between a ground state and an
excited state trick is that you can always do a ground state trick
directly from the normal cascade or fountain pattern, and you can string
any number of ground state tricks together back to back. You can not do
that with excited state tricks. In order to get it going, you will have
to do some funny throws to couple the cascade (or fountain) to the
excited trick. Once in the excited trick, you can run it as long as you
like, but when you want to stop, you again have to throw a coupling
sequence to return to the normal pattern. Consider the three object
shower. If you're juggling along in a cascade, and decide you want to
start showering all of a sudden, you have to adjust a few throws before
you get into the shower. Many people do this by abruptly changing their
juggling speed, but you don't have to do this. In fact, when you're
throwing site swaps, the notation always assumes that your hand speed
remains EXACTLY THE SAME. In the case of the shower, you can throw a '4'
to get started, and a '2' to finish: ...3333 4 5151...5151 2 3333...
Excited state tricks can also in general not be juxtaposed the way
ground state tricks can. Any random pair of excited tricks is likely to
require a set of special connection throws. The only exception is for
tricks with the same start up sequence.
We have to be a bit careful about what we call an excited trick. Let's
look at the 4 4 1 again (ground state). Suppose we rewrite it 4 1 4; this
is just a cyclic permutation of the same trick, and when you run it,
there's no way to telling them apart: 441441441... or 414414414.. look
the same, you just your starting position one throw. However, the
4 1 4 becomes 1 -2 1 after subtraction, and here the -2 points from pos. 2
to pos. 0, and the second 1 points from pos. 3 to pos. 4. How can a
ground state trick point outside the string? Well, you can either say that
it doesn't: 414 is excited, not ground state (unlike 441), but this is
unsatisfactory. The best solution is to say that a trick is ground state
as long as there is at least one cyclic permutation of the trick that is
ground state. A trick is then, equivalently, said to be excited only if
there is NO cyclic permutation of the trick which is ground state.
Consider again 5 1 --> 2 -2. The only cyclic permutation of this trick
is -2 2, which is also excited. There is therefore no cyclic permutation
of the three object shower which is ground state, so it is a genuinely
excited trick. 5 1 4 1 4 was long our favorite example of a three object
excited trick. Written as zero-sum, it's 2 -2 1 -2 1, and the first -2 and
the trailing 1 both point outside the trick. However, we recently
discovered that the cyclic 1 2 -2 1 -2 is no longer excited! Older
computer codes had overlooked this possibility. The currently posted
version takes this into account.
Another issue in tabulating excited state tricks is that of repetition.
For the ground state tricks, it's obvious: since any two ground state
tricks fit together back to back, there is no reason to include pairs
(or triples or...) of tricks in the table. However, for excited state
tricks that is not as clear, since they do in general not fit together
randomly. Typically, tricks with the same start up sequence do fit
together. I decided to let possible combinations stay in the table,
except for repetitions of the same trick. Therefore, 5 1 5 1 will not
appear in the table, while 6 0 3 5 0 4 will. The first one just repeats
the same trick twice, while the second one shows you that the 6 0 3
trick can in fact be followed by the 5 0 4 trick, both of which are
legal excited state tricks on their own.
Ground state tricks are never allowed to begin or end on a throw of
normal height, as such a trick is the same as a trick of shorter
length. I relax this restriction for excited state tricks, as they
normally require a finish sequence to return to the normal pattern. If
it would so happen that the trick can be ended on a single normal
throw, that is something I consider non-obvious, and I also tabulate
those. F.ex., three in one hand is 6 0. You can follow that with a
single normal throw: 6 0 3 is a legal trick. Note, however, that 6 0 3 3
is not legal! There seems to be significant merit in allowing 6 0 3
to appear in the table.
Once a perfectly legal and genuinely excited trick has been found,
it has to undergo some more processing. First, we don't want both 5 1
and 1 5 to appear in the table. The program therefore generates the
tricks under the condition that the first throw is the highest one.
This works fine for tricks with just one throw of that height. But
consider the five ball trick 777171. It has no less than four 7's.
Clearly, the program must prevent all four possible orderings of this
(777171, 771717, 717177, and 717771) from appearing. It does that by
keeping only the cyclic that collects all the highest throws at
the beginning. 777171 gathers the most number of high throws at the
beginning, and is the one listed.
The algorithm basically loops through all values that throws could
possibly have in any position. You get to specify a highest throw, as
excited state tricks rapidly accommodate ridiculously high throws.
Already at word length four, three ball tricks can contain "12"s, while
six ball tricks can contain "24"s! Clearly you don't want to waste the
computer's time (or disk space) with all these utterly useless tricks.
Generally, if you have n objects, and word length m, all but one of the
throws can be a -n (in zero-sum notation); to compensate, the remaining
throw must then be n(m-1). The program loops from 1 to n(m-1) for the
first position (as it has to be the highest throw). Once that has been
assigned, the algorithm calls itself, and loops from -n to pos[1] for
the throws in the next position, where pos[1] is the throw height in the
first position.
This recursion proceeds until the second to last position is filled.
At that point, the last throw height is fixed by the zero sum condition.
The program checks along the way to make sure impossible conditions
aren't created along the way (such as collisions from throw heights
already assigned, to non-zero sums).
There is then the issue of start up and finish sequences. These are
computed for each trick that passes all the filters above. These sequences
aren't unique, and it'd be painful to generate all the possible ones. I
chose to limit the search for the one involving the lowest throws
possible. Start up sequences can be at most n-1 throws long for n object
patterns. This is seen by the fact that the lowest throw you could have
is a -n, and it could appear in the second position at the earliest.
(Remember: the first throw has to be the highest one!) It can therefore
point at most n-1 positions in front of the first throw. I basically
walk through the trick, and set 1's in a work array at the positions
referred to by the trick. I then assign each location with a 1 a throw
that carries it to the next unoccupied (non-1'd). The landing position is
given a 1 to mark that it is now being referred to, and I continue with
the next 1 after the original one, until I reach the beginning of the
trick. This method is a bit hard to explain, so if you want to know how
that's done, just stare at the code until you figure it out. Finishing
sequences are generated similarly.
As mentioned above, start up sequences aren't unique. I welcome you to
find your own. I selected the ones with the lowest throws; another
selection rule would be to generate those with the fewest number of
throws deviating from the regular pattern. F.ex., the five object trick
7 7 1 has the start up sequence 6 6 (listed), but you can also start it
with a 7 5. The 7 5 involves a 7, which is higher than the 6's (and
therefore harder), but on the other hand, it only has one abnormal
throw, while the 6 6 has two.
Some finishing sequences will perhaps be a bit dumb, like 1 1 2 2, etc.
However, once you get interested in a trick enough to learn it and want
to stop it after along run, you should be able to find your own
variations of the stop sequences. F.ex., a 1 1 is equivalent to a 2 0,
which may be easier to do.
I explicitly opted against trying to find connection sequences between
different excited tricks, as that would produce far too many numbers. If
1000 tricks are generated, each trick would need 999 coupling sequences
listed next to it. Clearly that's ridiculous. If you really want to
string excited tricks together, you can either do the finish sequence
listed, and then the listed start up sequence, or you can work out your
own connecting sequence for your particular combination of tricks.
It's great fun going down the lists of site swaps, and try to figure
out what some tricks are doing. Take the 8 8 5 1 7 1, a hard five object
trick. First you throw two up high, which you follow by a 5 1 and a 7 1.
5 1 is a three shower, while 7 1 is a four shower. You therefore park two
up high, shower three for one cycle, then shower four for one cycle, and
then start over. An easier trick, also with five, is the above mentioned
7 7 7 1 7 1. It's mostly a four shower (7 1 7 1), but with one extra that
gets showered 'the wrong way'.) I don't think you could ever think of tricks
like those on your own. Similar examples exist all over the tables, and at
all number levels. Enjoy the tricks, and be sure to let the rest of the
juggling community know of especially neat tricks you find in the tables!
Bengt
From JUGGLING@INDYCMS.BITNET Wed Feb 6 01:18:54 1991
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Date: Wed, 6 Feb 91 01:11:57 PST
From: Bengt Magnusson <bengt@VOODOO.BITNET>
Subject: Excited State Tricks
Sender: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
To: "Peter D. Mark" <pdm@CS.UOREGON.EDU>
Reply-To: "Juggling - For Jugglers of all abilities." <JUGGLING@INDYCMS.BITNET>
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Comments: Warning -- original Sender: tag was
Status: RO
First, another bug in the posted version of the ground state trick
program. It's fairly likely to not have caused much trouble, but the
potential is there, and it should be fixed. I simply fell into a common
C trap, and thought my array was numbered from 1 to wlen, when it's
really numbered from 0 to wlen-1. The program will therefore ask the
array pointer to point one position beyond the array limit. For most
storage allocators, this will probably be OK, but you never know. Even
though my version runs fine with that bug, it might not always do so.
The simplest fix I can think of is to insert the line "pos--;" right after
the storage allocation, which is in line 59. In this way, the pointer will
always be within the array limits.
Also, remember to change the two occurrences of 'infp' to 'outfp,' or
the program will not compile at all!
I just finished writing a program that deals with the excited state
tricks. The old old program that claimed to do those really didn't, at
least not very well. In particular, many of the start up and finish
sequences were given incorrectly by that program. (Recall: a ground state
trick can be thrown directly from the normal pattern, while an excited
state trick has to be 'gotten into' by throwing a start up sequence. Once
that is done, the trick can be done indefinitely. When you want to return
to the normal pattern, you must likewise throw a special finish sequence.)
Hopefully this new version will be correct. It's written in ANSI C, as is
the ground state trick program. I will post it in a little while, when I'm
sure there are no bugs in it.
Excited state tricks are difficult to categorize. The incorrect start up
and finish sequences, together with the many tabulated tricks that aren't
really excited, show some of the difficulties. F.ex., 441 is a well known
ground state trick with three objects. However, 414 and 144 are both excited
in the sense that you can't just throw a 414 from a cascade. Still, it seems
like 414 and 144 do not merit their own place in an excited table, as they
can just be cyclically shifted to a ground state trick. The algorithm must
look for these things. Further evidence of how easy it can be to misidentify
excited tricks is given by the fact that our long time favorite example of
a three object excited trick, 51414, is not excited at all! The cyclic
45151 is a ground state trick. Oops. It's still not clear to me whether
excited tricks that are just the juxtaposition of two shorter excited tricks
should be tabulated. In the case of ground state tricks, it's fairly clear
to me that they shouldn't, as we know any two ground state tricks can be
strung together. However, we don't know right off hand which excited state
tricks can follow each other, and therefore maybe excited combinations should
be included in the table. (It seems like any excited tricks with the same
start up sequences should be able to follow each other. Maybe each excited
word length should be further subdivided into tricks with identical start up
sequences? This would define an equivalence class of tricks, by the way.
(The last comment for the mathematically interested.))
Finally, it should be noticed that the number of excited tricks is much
larger, for a given word length, than the number of ground state tricks.
However, many of these require ridiculously high throws (try throwing 15's),
and the program will ask you for the highest throws you'll allow in order to
cut down the size of the tables.
Once again, the program for excited tricks will appear in a few days.
Bengt@Voodoo.Ucsb.Edu
From bengt@voodoo.physics.ucsb.edu Wed Jan 30 12:01:46 1991
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Date: Wed, 30 Jan 91 11:44:53 PST
From: bengt@voodoo.physics.ucsb.edu (Bengt Magnusson)
Message-Id: <910130114454.20a04206@pcs1.physics.ucsb.edu>
Subject: RE: site swaps
To: pdm@cs.uoregon.edu
X-St-Vmsmail-To: ST%"pdm@cs.uoregon.edu@spencer.cs.uoregon.edu"
Status: RO
Two in one hand, with the other on empty, would be denoted 4 0. Two in one,
and one in the other, with the throws staggered, is 4 4 4 0. The published
notation requires the hands to be staggered. However, it is easily adapted to
parallel throws, and even crossing throws with even numbers (and half
showers with odd numbers). F.ex., an odd number half shower requires the hands
to throw simultaneously, and we'd call those throws 6_c 4_c, where the _c is
to be read as a subscript, and 'c' stands for 'crossing.' Similarly, you can
add a _p subscript for 'parallel,' and you'd call the 2 in one, 1 in the
other a 4_p 4_p 4_p 0_p. BM