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How can I make a tileable polygon background?

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Step 1: Modeling:

Start with a flat plane. Next, add some modifiers, first a sub surf (set the division to 6 or 8 & set the mode to simple), then a displace with a default cloud texture (set the strength to taste - you may want to adjust it after the next step), and finally, a decimate modifier. Adjust the ratio until the poly-gon pattern looks right:

enter image description here

Next it is time to define the edge that will be tiled.

First, apply all the modifiers, and then make any adjustments to the polygons (make ones that are too small and thin etc. the right shape).

Finally, grab all the vertices along the edges you wish to tile, and delete them:

enter image description here

Now we will need to cut an edge. Any spot works, I pick one near the left edge. Go into top view Numpad7, Then use the kinfe tool K, locked on an axis C. Click and drag up, and then hit enter.

enter image description here

Now we need to split the mesh along this edge. To do this, hit V and drag to the side, so that the edge comes away. Use RMB to put it back into place. Del-select everything with A, and hover over one of the halfs and hit L to select it.

Drag it over to the other side, and fill in the seem. The two opposing edges will now be contiguous when lined up.

enter image description here

Step 2: Shading

See this answer for details. This answer other may also be found useful.

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