Site swap

I may point to the progression written according to the notation (ジャグリングノーテーション) by the non-negative integer line for the site swap (Site swap) to refer to the pattern of the toss jag ring or the notation. I was formulated independently in the 1980s by three researchers [¹]. I can explain it if I explain the scale intuitively when each numerical value expresses height (time distance before catching it after throwing it closely) to throw a tool. Because I do not have other information, it is simple and is easy to treat it, but on the other hand cannot define the pattern to make special how to throw. Because it is progression, it is possible to create a judgment of the jag ring possibility and new patterns by mathematical processing.

In addition, I am used for heights and a description of the passing to bandy a tool in higher than pattern of the windmill and two to perform with more than 2 diabolos. In addition, even the tool (cigar boxes) which cannot apply site swap may be used by analogy of how to throw closely.

Table of contents

Vanilla site swap

 

The site swap called the simplest vanilla site swap is expressed as an integer line of the limited head such as 3 and 7531. I put the following premises by the vanilla site swap.

• I ignore all the state of the tool or a kind, the physical state [^{note 1}].
• I throw plural objects with one hand at the same time and cannot catch it.
• I throw a tool with both hands in turn every a given period of time [^{note 2}].
• The numerical value expresses distance at time before being caught after the tool was thrown.

For example, the right animation is 3 ball cascades, but all balls are thrown every three. Therefore the site swap is referred to 3. Because there is a premise throwing with both hands in turn, I catch it by the hand which is reverse to the hand which I threw with the odd number and will receive it in the hand which I threw by the even number adversely. Three special how to throw is defined then: I mean what I continue having each without 2 throwing an object in what I hand a reverse hand a state to hold nothing in a hand at the moment as for 1 or throw 0 linearly and hand in that time. In addition, when it is necessary to write the numerical value that is bigger than 9; using the alphabet 10 a, 11 b,…It is common to express と.

Each pattern is repeated every some beats. I call the shortest length of this repeated beat a period of the site swap and usually write only the shortest period. It is 1 in the period of 3 ball cascades which I nominated for an example earlier. In addition, because both hands repeat the same pattern in this way when a period is an odd number, I call a pattern with the site swap of the odd number period when it is symmetry (symmetrical) and call a thing having an even period like a shower [^{note 3}] with the pattern that is asymmetry (asymmetrical).

Jag ring possibility

A certain several lines of site swap having period n+1${\displaystyle {\begin{Bmatrix}a_{n}\end{Bmatrix}}=a_{0}a_{1}\cdots a_{n}}$ But, to be jag ring possibility (jugglable),
What kind of; is different${\displaystyle 0\leq i,j\leq n}$ に vs. even if do it${\displaystyle (a_{i}+i)\not \equiv (a_{j}+j){\pmod {n+1}}}$ であることである.

It may be said that this formulated, "balls more than two do not fall into one hand at the same time at the moment of which beat either" intuitively. In other words, this is because it may be said that I do not fall at the same time if the numerical value that chopped the sum in a period of the beat of the flight duration expresses it when the ball falls in n+1 periods, and this is all different from the turn of the beat given up if a period of a certain site swap is n+1.

From a definition of this jag ring possibility, the cyclic shift (cyclic shift)^{[ note 4}) of the site swap that a jag ring is available for understands that a jag ring is possible promptly. However, it is common to usually write a particularly unreasonable limit, a maximum number first.

In addition, the mean of several lines of site swap becomes the number of the object to throw then. For example, as for 531, three, 91 are five.

State

I can throw a pattern of 441 from 3 ball cascade "3" immediately, but cannot throw 3 ball shower 51. It is a concept called state (state) of the site swap to explain this.

The correction of this knob is expected.

Synchronized eggplant site swap

When express a synchronized pattern to throw with both hands at the same time; numerical value of each right and left (bind it in), and divide で. In addition, the numerical value of the synchronized pattern is limited to an even number and transcribes the cross throw by adding x after numerical value. When I repeat a symmetric pattern, I may substitute the one side in *. For example, (6x, 4) (4, 6x) (can describe it with 6x, 4)*.

マルチプレクシング site swap

When I express plural multi-throws to give up at the same time in one hand, I bind the numerical value of the object to throw at the same time and transcribe it. For example, [97] 121 is 5 balls multi-shower throwing two balls from one hand at the same time.

Multi-hand site swap

The normal site swap is made for a lot of general people having two arms, but can think about site swap when I have the arm of other numbers. In the case of one arm, the pattern of the heights of the diabolo is used to express a pattern of the passing that two people bandy a tool in the case of four arms.

Characteristic

• When I throw an object to the top, I am proportional to the value that the height to throw it, and to raise pulls 1 from numerical value and squared.

  　In the case of 7531, the ratio of each height is, for example, 9:4: It is 1:0.

  　(it goes down and squares 1 6420 from 7531 and becomes 36, 16, 4, 0, but I do it reduction to a common denominator (/4), and it is 9410)

• There is it in the pattern that there is much number to throw one when I add the value that is equal to length of the site swap to a certain value.

  　For example, 7535 that added 4 of several lines of length to 7531-1 that is four patterns is five patterns, and 7131 that pulled 4 from 7531-5 is three patterns.

• Even if I add n toward the left after replacing n unit remote numerical value and pull n from right one, as for the pattern, a jag ring is possible.

  For example, I replace 3 with 7531-5 and assume it 7351, and 7441 that is the progression that adds 1 to 3, and pulled 1 from 5 is jag ring possibility.

Explanatory note

1. They use ^, e.g., the ball or use the club, or information such as the number of revolutions of the club and the information such as the body throw are not expressed by the site swap.
2. I call distance beat (beat) in ^ this time.
3. The jag ring pattern that a tool is thrown like a traditional bean bag of ^ Japan only by one hand.
4. 315,135 for ^ 531

References

1. ^ "The Mathematics of Juggling". December 18, 2009 reading.

Allied

Program

I can use it in a freelance, and there is the software which can simulate the site swap.

• JuggleMasterJAVA version

I can simulate a jag ring pattern on a browser. I can let you display the original skill by renewing a pattern file.

• Juggling Lab animator

The jag ring simulator which was written by JAVA. It is the software of the open source distributed in GPL. I can let you display ラダーノーテーション. In addition, the applet version exists, too.

• Site swap analysis onWEB

Of the site swap to meet small analysis and condition of the site swap listing it is possible.

Bookm

A Japanese book commenting on site swap includes the following things.

"The ball jag ring guide second edition" (ナランハ, written by Junichiro Nakajima, ISBN 493157100X)

"Ball jag ring all sorts of subject" (game company, Charlie Dan sea, written by Keisuke Inoue, ISBN 4896595181)

If it is a foreign book, the following has a lot

"The Mathematics of Juggling" (Springer, Burkard Polster work, ISBN 0387955135)

Allied item

• ジャグリングノーテーション
• Jag ring

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