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  "body": "\n## Definition\n\nLet $H = \\{\\, x \\in \\mathbb{R}^n : x \\cdot a = c \\,\\}$ be a hyperplane, with $a \\neq 0$, and let $\\sigma_H$ be the [[reflection|06BRDF3F0E]] in $H$. Let $w \\in \\mathbb{R}^n$ be a nonzero vector with $w \\cdot a = 0$, and let $T_w$ be the [[translation|087VJM5Q77]] by $w$.\nThe glide reflection with the axis $H$ and the glide vector $w$ is the map $T_w \\circ \\sigma_H$.\n\n## Remarks\n\nBecause $w$ is parallel to $H$, the two maps commute: $T_w \\circ \\sigma_H = \\sigma_H \\circ T_w$.\nA glide reflection is an indirect isometry with no fixed point. Its square is the translation by $2w$.\nIf the translation vector is perpendicular to $H$ in place of parallel, then $T_w \\circ \\sigma_H$ is the reflection in a hyperplane parallel to $H$.\nIn the plane, the axis is the glide line. Glide reflections are symmetries of many frieze and wallpaper patterns.\n\n## Explanations\n\n### L1 human\n\nFirst reflect the plane in a line, then slide it along that line. Neither step alone gives the result. The map $(x, y) \\mapsto (x + 1, -y)$ is the glide reflection with the axis $y = 0$ and the glide vector $(1, 0)$.\n",
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