Explore rotational and reflective symmetry groups. Draw a motif and see it replicated under dihedral and cyclic group operations.
Every image this tool shows you is a copy of one thing: the motif you draw. Pick a symmetry group such as C_6 or D_4 and copies appear under the group's operations: rotations through multiples of 360/n degrees, plus, for dihedral groups, reflections across axes spaced 180/n degrees apart. The result is a rosette showing the orbit of your drawing under the group. The surprise: the visible count is not always the group order. A motif that is itself mirror-symmetric, or lies on a reflection axis, is fixed by some operations, so overlapping copies merge and the count drops to |G| divided by the stabilizer size.
A concrete input and expected output from the current implementation.
Input
Choose the dihedral group D_4 and draw a single dot one unit to the right of the center, at (1, 0).
Expected output
Four copies, not eight: dots at (1, 0), (0, 1), (-1, 0), and (0, -1).
The dot lies on the 0 degree mirror axis of D_4 (the other axes are at 45, 90, 135 degrees), so the reflection in that axis fixes it and the other three reflections only send it to positions the rotations already produce. Eight group elements divided by a stabilizer of size 2 gives 4 distinct images. A dot at (1, 0.5), off every axis, gives 8.
The group selector fixes the operations applied to your motif. C_n applies the n rotations through multiples of 360/n degrees. D_n applies those same n rotations plus n reflections, one across each mirror axis at a multiple of 180/n degrees, giving 2n operations in total. Reflecting and then rotating by one step lands on the next mirror axis, which sits halfway between two rotation angles.
The number of distinct images equals the group order divided by the stabilizer size: the subgroup of operations that leave your motif unchanged. A motif with mirror symmetry, or one placed on a reflection axis, has a stabilizer larger than the identity alone, so several operations draw identical copies on top of one another. The group still acts fully; only the visible count shrinks, to at most n copies under D_n.
Because the motif is fixed by some group element. The stabilizer is the set of operations that leave your drawing exactly where it is, and it always contains at least the identity. The distinct copies number |G| divided by the stabilizer size, so for a dot on a reflection axis of D_4 the stabilizer has two elements and eight divided by two gives four.
Yes, when the motif is not mirror-symmetric. Reflections change chirality, so an asymmetric drawing such as a small letter 'R' appears in left-handed and right-handed versions under D_n, while rotations alone under C_n can never reverse handedness. If your drawing has mirror symmetry of its own, the two groups produce identical pictures, so the chirality difference becomes invisible.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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