Wheels can be fastened on a wheel axle[?], which when fitted to a third object such as a vehicle body, will efficiently impart the energy of the converted linear motion to that body. Since the strongest practical motions available in nature to pre-industrial humans (exertion of their own muscle, exertion of animal muscle, flowing water, etc.) were linear and horizontal, the wheel/axle/object combination enabled them to leverage these sources in any direction just by tilting the wheel (vertically in a horse driven mill wheel, for instance). Expended linear force, instead of being pushed ahead and scattered in unlimited forward space, was now cyclically captured along the compact 360 degree surface of the turning wheel. Compared to a vehicle without wheels, such as a travois[?], barrow[?] or sled, this compaction of momentum, along with a related reduction in surface friction, allows wheeled vehicles to travel farther even while less push or pull force is applied.
With its cycling/compaction of momentum, reduction of friction and resulting lower amount of work necessary to move items, the wheel is essential in transport. In order for wheeled vehicles to work well, one must employ them on roads as a part of road transport, or fit the vehicle with superior suspension. A development of wheels to allow them to travel through rough terrain is the caterpillar track found on some tractors and armored vehicles. On many wheeled vehicles (for instance, automobiles and motorcycles), a rigid wheel does not come directly in contact with the surface, but instead serves as a mounting point for a flexible tire, usually filled with compressed air. For rail vehicles this is not the case. Land vehicles without wheels are the hovercraft, the electro-magnetic monorail, and relying on surface conditions the sled, the snowboard and skis.
Since a wheel is a rigid object, it will only be non-rotating when all the torques on it are balanced. Since forces produce larger torques when they are closer to the axis, a wheel can be used to transform between large and small forces applied by friction with belts or other wheels. Other variations on wheels produce the pulley and the windlass. The combination of the wheel with the wedge produced the toothed gear, which was fundamental to the advent of industrial class complex machines.
Most authorities credit the ancient Mesopotamians (Sumerians) with the invention of the wheel at about 4000 BC, with an independent invention in China at around 2800 BC. The Incans and certain other western hemisphere cultures seem to have approached the concept, as wheel-like worked stones have been found on objects identified as children's toys dating to about 1500 BC. The wheel was unknown in Sub-saharan Africa and Australia until relatively recent contacts with Eurasians. While the 4000 BC date for the invention of the wheel may seem ancient, it is curious to note in this context that most anthropologists date the emergence of fully modern human beings (Homo sapiens sapiens) to at least 98,000 BC. That people with capacities fully equal to our own walked the earth for 94,000 years before hitting upon the wheel is surprising at first blush. But it becomes more understandable when one realizes that populations were extremely small through most of this time and that the wheel, which requires an axle and socket to be truly useful, is not so simple a device as it may seem. Perhaps nothing testifies more to how non-obvious the wheel is than the fact that biological evolution itself, at least on the macro level, did not "invent" it (unless one counts certain species of lizards and spiders that sometimes curl themselves into a rough ball when attacked, attempting to roll to safety).
In July 2001, the wheel was the object of an innovative, but non-inventive, patent as a "circular transportation facilitation device"  (http://news.bbc.co.uk/hi/english/world/asia-pacific/newsid_1418000/1418165.stm). The patent was obtained by John Keogh[?], a lawyer from Melbourne, Australia, with the declared intention of demonstrating the unfairness and inaccuracy of the modern patenting system.
In Buddhism, the wheel is a central conceptual metaphor. Life is said to be an endless cycle of births and rebirths, which one can only escape by achieving enlightenment. Accordingly the wheel to a Buddhist has a meaning akin to that of a cross for a Christian or a crescent for a Muslim, although it is not a symbol of a deity as such. The mandala is a common representation which is based on the wheel's image.