Here is a simple and fun way to create a pattern and wrap it on an egg using the 3D feature in Illustrator. Faster than decorating the real thing and the design possibilities are endless.
Create a pattern that you want to wrap around the egg. This one starts with triangles.
Add more colors and shapes.
Add some circles, you can do fills and strokes and try different combinations.
Try different colors.
Add some lines and use the Effects to make them wavy.
Select all the shapes and make sure the are expanded. This will not work if there are lines or effects. Choose Object > Expand Appearance (if there are effects) the choose Object > Expand to outline all the paths.
Open the Symbols panel from the Window Menu. Select the expanded shapes and drag them into the Symbol Panel. Name the symbol and click OK.
The graphic will now show up in the Symbols Panel, ready to be mapped to the egg.
Create an egg by starting with a circle.
Select the top point with the Direct Selection Tool and move the top point up to create the egg.
Select the left point with the Direct Selection Tool and delete it, leaving half the egg. Set the stroke color to the color you want the egg to be.
Choose Effect > 3D > Revolve. This will create a 3D egg. Click on the Map Art… button.
From the Map Art dialog box, choose the Symbol from the menu that you want to map to the egg. Move the symbol the the light gray area on the mapped area. Scale it to fit the egg. Click OK.
The end result will be the artwork mapped to the egg.
If you need to edit the 3D artwork, open the Appearance panel and click on the 3D Revolve link to open the 3D panel again.