For fully edible artichokes: Cut the stem flush with the bottom of the artichoke so the cooked artichoke will rest steadily on the plate. Cut a lemon in half, squeeze the juice into a large bowl of cool water, and keep the halves close by so you can rub them over the cut surfaces  of the artichokes to prevent them from darkening.  With a paring knife, trim the base of the leaves flush with the artichoke bottom. Snip or pluck off the outer layers of leaves to expose the leaves that are pale yellow almost all the way to the tip. Cut off the top third of the artichoke leaves with a sturdy serrated knife. With a vegetable peeler or paring knife, trim the dark outer layer of the artichoke stem. Dip the artichoke in the lemon water as you go to prevent it from darkening. Cut the artichoke in half through the stem. Scrape out the fuzzy choke with a small teaspoon or espresso spoon. Pull out the inedible purple-tipped leaves, if there are any, that surround the choke. Place the trimmed artichoke halves in the lemon water as you go. These completely edible artichoke halves may be cooked as is or sliced or cut into wedges.