Ribbon
A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, velvet, cotton, and jute and of synthetic materials, such as polyester, nylon and polyproylene. Ribbon is used for innumerable useful, ornamental and symbolic purposes. Cultures around the world use ribbon in their hair, around the body, and as ornamentation on animals, buildings, and packaging. Some popular fabrics used to make ribbons are satin, organza, sheer, silk, velvet and grosgrain.
Etymology
The word ribbon comes from Middle English ribban or riban from Old French ruban, which is probably of Germanic origin.
Cloth ribbon
Ribbon (mathematics)
In mathematics (differential geometry) by a ribbon (or strip) 
is meant a smooth space curve 
given by a three-dimensional vector 
, depending continuously on the curve arc-length 
(
), together with a smoothly varying unit vector 
perpendicular to at each point (Blaschke 1950).
The ribbon is called simple and closed if is simple (i.e. without self-intersections) and closed and if 
and all its derivatives agree at 
and 
.
For any simple closed ribbon the curves 
given parametrically by 
are, for all sufficiently small positive 
, simple closed curves disjoint from .
The ribbon concept plays an important role in the Cǎlugǎreǎnu-White-Fuller formula (Fuller 1971), that states that
where 
is the asymptotic (Gauss) linking number (a topological quantity), 
denotes the total writhing number (or simply writhe) and 
is the total twist number (or simply twist).
Ribbon theory investigates geometric and topological aspects of a mathematical reference ribbon associated with physical and biological properties, such as those arising in topological fluid dynamics, DNA modeling and in material science.
Ribbon (rhythmic gymnastics)
Ribbon is a component of rhythmic gymnastics composed of a handle (called "stick"), a ribbon and attachment.
Stick
This can be of any colour and made of wood, bamboo or plastic with a maximum diameter of 1 cm (2/5") at its widest, a cylindrical or conical shape and should have a length of 50–60cm (20–24") (including the fastening ring). The bottom end of the stick may be covered by an adhesive, anti-slip tape or may have a rubber handle for a maximum length of 10 cm (4") at the level of the grip. The top of the stick, where the ribbon will be attached, may consist of:
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