In the History of the wheel, the wheel pre-dates driven wheels by about 6000 years. The wheel alone is not a machine, but when attached to an axle in conjunction with bearing, it forms the wheel and axle, one of the simple machines. Bearings are used to help reduce friction at the interface. In the simplest and oldest case the bearing is just a round hole through which the axle passes.
The English word wheel comes from the Proto-Indo-European *kwekwlo-, which was an extended form of the root *kwel- meaning "to revolve, move around". This is also the root of the Greek κυκλος kuklos, the Sanskrit chakra, and Persian charkh, all meaning "circle" or "wheel", and also in Lithuanian, sukti means "to rotate". The Latin word rota is from the Proto-Indo-European *rotā-, the extended o-grade form of the root *ret- meaning "to roll, revolve".
Most authorities regard the wheel as one of the oldest and most important inventions, which originated in ancient Mesopotamia (present-day Iraq) in the 5th millennium BC (Ubaid period), originally in the function of potter's wheels. The fast wheel, a completely mobile, carefully balanced apparatus of stone, appeared the Sumerian city of Ur in southern Mespotamia around 3500 BC. After being used for various industrial or manufacturing applications, the first wheeled transportation appeared in the form Mesopotamian chariots around 3200 BC. The fast wheel soon appeared in ancient Egypt around 3000 BC, and by 2000 BC there were Egyptian chariots that used spoked wheels. 
The wheel spread through Western Asia in the 4th millennium BC, and reached the Indus Valley of the Indian subcontinent by the 3rd millennium BC. In China, the wheel is certainly present with the adoption of the chariot in ca. 1200 BC, although Barbieri-Low (2000) argues for earlier Chinese wheeled vehicles, circa 2000 BC. The Greeks later adopted wheeled chariots from the Egyptians. 
Although they did not develop the wheel properly, the Olmec and certain other western hemisphere cultures seem to have approached it, as wheel-like worked stones have been found on objects identified as children's toys dating to about 1500 BC. Early antiquity Nubians used wheels for spinning pottery and waterwheels. It is thought that Nubian waterwheels may have been ox-driven It is also known that Nubians used horse-driven chariots imported from Egypt.
The invention of the wheel thus falls in the late Neolithic, and may be seen in conjunction with the other technological advances that gave rise to the early Bronze Age. Note that this implies the passage of several wheel-less millennia even after the invention of agriculture. Looking back even further, it is of some interest that although paleoanthropologists now date the emergence of anatomically modern humans to ca.150,000 years ago, 143,000 of those years were "wheel-less". That people with capacities fully equal to our own walked the earth for so long before conceiving of the wheel may be initially surprising, but populations were extremely small through most of this period and the wheel, which requires an axle and socket, and must be made round to actually be useful, is not as simple a device as it may seem. Only with the domestication of the horse did the wheel show its true potential in Eurasia. Larger or heavier wheels benefit greatly from balancing, which requires a skilled wheelwright, a profession limited to organized societies.
Wide usage of the wheel was probably delayed because smooth roads were needed for wheels to be effective. Carrying goods on the back would have been the preferred method of transportation over surfaces that contained many obstacles. The lack of developed roads prevented wide adoption of the wheel for transportation until well into the 20th century in less developed areas.
Early wheels were simple wooden disks with a hole for the axle. Because of the structure of wood a horizontal slice of a trunk is not suitable, as it does not have the structural strength to support weight without collapsing; rounded pieces of longitudinal boards are required. The spoked wheel was invented more recently, and allowed the construction of lighter and swifter vehicles. The earliest known examples are in the context of the Andronovo culture, dating to ca 2000 BC. Soon after this, horse cultures of the Caucasus region used horse-drawn spoked-wheel war chariots for the greater part of three centuries. They moved deep into the Greek peninsula where they joined with the existing Mediterranean peoples to give rise, eventually, to classical Greece after the breaking of Minoan dominance and consolidations led by pre-classical Sparta and Athens. Celtic chariots introduced an iron rim around the wheel in the 1st millennium BC. The spoked wheel had been in continued use without major modification until the 1870s, when wire wheels and pneumatic tires were invented.
The invention of the wheel has also been important for technology in general, important applications including the water wheel, the cogwheel (see also antikythera mechanism), the spinning wheel, and the astrolabe or torquetum. More modern descendants of the wheel include the propeller, the jet engine, the flywheel (gyroscope) and the turbine.
The critical angle can be found by solving the equation:
- is the radius of the wheels;
- is the horizontal distance of the center of blah blah blah from the rear axle; and
- is the vertical distance of the center of mass from the axles.
For big wheels, this formula can be simplified to:
The maximum height of an obstacle can be found by the equation:
where is the wheelbase.
The wheel has also become a strong cultural and spiritual metaphor for a cycle or regular repetition (see chakra, reincarnation, Yin and Yang among others). As such and because of the difficult terrain, wheeled vehicles were forbidden in old Tibet.
The winged wheel is a symbol of progress, seen in many contexts including the coat of arms of Panama and the logo of the Ohio State Highway Patrol.
The introduction of spoked (chariot) wheels in the Middle Bronze Age appear to have carried somewhat of a prestige. The solar wheel appears to have a significance in Bronze Age religion, replacing the earlier concept of a Solar barge with the more "modern" and technologically advanced solar chariot.
The wheel is also the prominent figure on the flag of India. The wheel in this case represents law (dharma). It also appears in the flag of the Romani people, hinting to their nomadic history and their Indian origins. The wheel can also appears in the flag of Mahl Kshatiyas Kings (kattiri buvana maha radun).
In recent times, the custom aftermarket car/automobile roadwheel has become a status symbol. These wheels are often incorrectly referred to as "rims". The term "rim" is incorrect because the rim is only the outer portion of a wheel (where the tire is mounted), just as with a coffee cup or meteor crater. These "rims" have a great deal of variation, and are often highly polished and very shiny. Some custom "rims" include a bearing-mounted, free-spinning disc which continues to rotate by inertia after the automobile is stopped. In slang, these are referred to as "Spinners".
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- "wheel". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=wheel.
- kwel-1. The American Heritage Dictionary of the English Language: Fourth Edition. 2000
- ret- The American Heritage Dictionary of the English Language: Fourth Edition. 2000
- Dyer, Gwynne, War: the new edition, p. 159: Vintage Canada Edition, Randomhouse of Canada, Toronto, ON
- Ekholm, Gordon F (1945). "Wheeled Toys in Mexico". American Antiquity 11.
- CRAFTS; Uncovering Treasures of Ancient Nubia; New York Times
- Ancient Sudan: (aka Kush & Nubia) City of Meroe (4th B.C. to 325 A.D.)
- What the Nubians Ate
- The Cambridge History of Africa
- bookrags.com – Wheel and axle