Carefully turn the white flesh into strips. Once again, use your paring knife to create thin strips starting from the cut edge of the cucumber. This time, you will be forming "inner petals" of the flower, from the hard, white portion of the cucumber.
This may be more difficult than the peel, as you need to create strips thin enough to bend, but thick enough that they won't break. Go slowly, and take a break if your eyes or hands feel strained.