Create a pentagonal pattern
1: the pentagon
First turn on Smart guides.
Then start with a vertical line (press Shift while drawing with the line tool).
Take the Rotate tool and Alt-click on the bottom point of the line. In the dialog enter 360/5 to let Illustrator calculate the rotation angle. Then press Copy.
Press Cmd/Ctrl+D three more times.
Take the Polygon tool, click and drag from the centre of the lines to the end of the top right line. You'll mabe have to use the arrow keys to turn the number of sides to 5.
With the live paint bucket fill the 5 parts of the pentagon. Then select the live paint object with the selection tool and set the stroke to None.
When you're done, expand the object.
3: Prepare the pattern
Grab the pentagon by its top point with the selection tool, press Alt and move to the rightmost point.
Take the rotate tool and click on the rightmost point. Then drag by one of the other points to get a little gap.
Then grab by the bottom left point and snap to the left polygon.
4: Make copies
You can't make a pattern directly, because the pattern mode can't provide the needed offset. So we need some copies.
Select both pentagons at their bottom joint with the selection tool, press (+ hold) Alt to make a copy and move to the top of the left pentagon.
Then grab the leftmost point, press Alt and move to the right.
5: Make pattern
Select all the pentagons and select Object > Pattern > Create
Take the pattern tile bound tool and scale the pattern tile like so:
There's still a gap in the pattern. This will be solved by applying two fills to the object: apply the pattern to the top fill and a solid color to the bottom one.
See other English tutorials:
EPS – the zombie of file formats
Saving legacy files
Long Shadows with Illustrator
Problems with align to pixel grid
Proposals for better better workflow in Illustrator
Freeform-artbrush with a gradient
Pathfinder doesn‘t react
Stylish guilloches with gradients
Outline object and outline stroke