Translational Symmetry: Periodic Patterns via Vectors and Sequences

① This GeoGebra example demonstrates translational symmetry by repeating a base figure along a fixed vector, creating a periodic pattern. ② The core mathematical concepts include translation transformations, symmetry groups, and the use of modular arithmetic and sequences to generate repeated structures. ③ Construction highlights: Points (A, B, C, D, E, F) define the base shape, a vector u specifies the translation direction and magnitude, and commands like Mod and Sequence generate lists (list1, list2) of translated objects, with text1 providing annotations. ④ Suitable for middle school geometry or introductory abstract algebra, this example helps visualize translational symmetry groups and periodic tiling, and can be used in math art or pattern design lessons.