Symmetric patterns

Symmetric patterns are patterns in which every throw made on one side of the pattern is repeated on the other side of the pattern. These patterns are good for strengthening both hands. Patterns can exhibit symmetry in different ways:

Complete symmetry[]

Complete symmetry is the strongest form of symmetry in juggling patterns and exists when a sequence of throws made by the right and left hand is repeated in the same order and starting with the opposite hand. This category of the wiki is for patterns with complete symmetry.

Any vanilla siteswap with an odd period (odd length, e.g. 561 but not 5551) always exhibits symmetry.

Synchronous siteswaps that can be written with an asterisk (*) are symmetric because each throw made with the left hand is also made by the right hand, e.g. (4,2x)*. This type of pattern always exhibits symmetry. This type of synchronous symmetry is a subset of rotational symmetry, where this is a special case when the offset between the left and right hands is equal to zero or half the period of the siteswap. E.g. (4x,2x)(4,2x)*, also written as (4x,2x)(4,2x)(2x,4x)(2x,4), has a period of 4 and has its right side throws offset from its left by two beats, which is half of the period.

Another type of synchronous siteswap that is always symmetric is a pattern where both throws in each pair of throws are identical, e.g. (8,8)(4,4).

Rotational symmetry[]

Vanilla siteswaps in which the left hand throws are repeated on the right side at a later time do not necessarily exhibit complete symmetry. Nonetheless, vanilla siteswaps can still have rotational symmetry if they have even periods and have left hand and right hand sequences of throws that are rotations of one another. For example, in 6622 the right hand does everything that the left hand does, and in the same order, but the pattern 6622 always starts with the same hand, meaning it does not have complete symmetry. Another example is e2ae6a26; one hand throws "e a 6 2" and the other "2 e a 6", which means that each hand throws the same sequence in the same order (when the sequence is extended by juggling it multiple times), but at different times in the sequence.

Synchronous siteswaps can also have rotational symmetry. Such patterns like (6x,4x)(2x,6x)(4x,2x) have the same sequence of throws on the left side as on the right side, by offsetting the right throws from the left throws by one beat-pair (i.e. left hand: 6x 2x 4x, and right hand: 4x 6x 2x), but do not exhibit reflectional symmetry because the right-left throw pairs are not expressed reversed elsewhere in the pattern.