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Each model has a group of isometries that is a subgroup of the Mobius group: the isometry group for the disk model is SU(1, 1) where the linear fractional transformations are "special unitary", and for the upper half-plane the isometry group is PSL(2, R), a projective linear group of linear fractional transformations with real entries and ...
The Ford circle associated with the fraction / is denoted by [/] or [,]. There is a Ford circle associated with every rational number . In addition, the line y = 1 {\displaystyle y=1} is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity , which is the case p = 1 , q = 0. {\displaystyle p=1,q=0.}
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest ... solutions for n = 2, 3, 4, 7, 19, ...
The lattice has vertex types (1 ⁄ 2)(3 3,4 2) + (1 ⁄ 2)(3,4,6,4), while the dual lattice has vertex types (1 ⁄ 15)(4 6)+(6 ⁄ 15)(4 2,5 2)+(2 ⁄ 15)(5 3)+(6 ⁄ 15)(5 2,4). The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond ...
The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere. The counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one (in a one-dimensional universe, the circle analogue is just two points). That is ...
Thus the first term to appear between 1 / 3 and 2 / 5 is 3 / 8 , which appears in F 8. The total number of Farey neighbour pairs in F n is 2| F n | − 3. The Stern–Brocot tree is a data structure showing how the sequence is built up from 0 (= 0 / 1 ) and 1 (= 1 / 1 ), by taking successive mediants.
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Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container.