Greenery pruned into geometric and abstract forms are called topiary. So, yes, your box hedges are a type of topiary. Topiary has been around since the Romans planted their first gardens. Originally as a way to keep animals out of gardens it transformed into an art form. Topiary is most closely associated with the geometric designs of parterres in formal European gardens.
Any woody shrub, small tree or herb that is evergreen, with small leaves and compact shape can be used for topiary. The most popular shrubs for outdoor, and occasionally indoor, topiary are yews and boxwood. Their popularity rests on their sturdy single trunk, their ability to thrive with hard pruning and their compact, relatively fast, growth. Hollies, viburnums, laurels, hornbeam and privets can also be successfully used. For indoor topiary, woody herbs like rosemary, lavender and thyme are most popular. Ivy and other vines are used for "formed" topiary. Formed topiary are wire shapes stuffed with sphagnum moss where several ivy plants or vines are actually wrapped around the form, cloaking the wire with greenery.
Small Hedges and Intricate Topiaries
The most recognizable topiary form is the hedge. Hedges are popular because they are easy to maintain and even an inexperienced gardener can successfully maintain one. Plants used for hedges should grow from a sturdy trunk, tolerate crowding, have compact growth with many closely growing branches, small, preferably evergreen, leaves and can withstand heavy pruning. Yews, boxwood and privet have all of these characteristics and are the most popular hedge plants. Some dwarf hollies and barberries will also make handsome hedges, however, both shrubs produce berries and will drop leaves in the fall, giving you naked hedges. Intricate and geometric shapes are typically evergreen, stand-alone specimen plantings. Again you need shrubs that can tolerate heavy pruning, have close growing branches and compact growth. Yews, boxwood and dwarf spruces should be used exclusively. Their compact, quick growth hides pruning cuts, their size is easily managed and they look good all year round. For hedges choose the same plant throughout the hedge for a uniform look. Shrubs with medium to dark green leaves look better as hedges and intricate shapes although you can use variegated and yellow-leaved varieties.
Pyramidal, Large Hedge and Container Topiaries
Any plant, evergreen or deciduous, can be pruned into a pyramid, as long as it grows from a single, strong trunk. Yews are the go-to shrub for large hedges (hedges over 4 or 5 feet tall). But, if plant trees closely together and trim the tops into a "box" shape and you have an unusual hedge. For these types of tree hedges, choose hornbeam, laurels, hollies and beeches. Lollipops are trees that have their tops trimmed into a mophead or umbrella shape. Again look to hornbeam, laurel, and beeches for these shapes. Any slow-growing shrub or woody herb can be grown as a container topiary. Boxwood is especially good for an evergreen asymmetrical shape. Woody herbs like rosemary, thyme and lavender all make excellent container topiary and are usually trimmed into a ball, pyramidal or lollipop shapes. Any type of ivy or evergreen vine can be used for formed topiary. You must plant three to four vines around the form and wrap the vines around the form to make the topiary. You can choose shrubs and plants with variegated or yellow foliage, although pruning cuts are easier to see on light colored leaves than dark green ones.