How can I tile normal map in Cycles?
I tried to use Texture Coordinates and Mapping nodes with different settings, it works with the texture, but doesn't work with the normal map.
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Normal maps, like any other texture it's automatically tiled in cycles unless you are specifically clamping the tiling.
Scaling the incoming texture coordinates, should be enough to show up the repetition of the pettern regardless of the map type.
The problems from your setup comes from the usage of an Environment Image texture with Color data options, while the usual setup requires an Image Texture with Non Color data for normal maps.