 Its statement needs some background . A tiling <formula> of a ring <formula> ( i.e. , a closed annulus ) gives two invariants , <formula> and <formula> , called approximate moduli . These are similar to the classical modulus of a ring . They are defined by the use of weight functions . A weight function <formula> assigns a non @-@ negative number called a weight to each tile of <formula> . Every path in <formula> can be given a length , defined to be the sum of the weights of all tiles in the path . Define the height <formula> of <formula> under <formula> to be the infimum of the length of all possible paths connecting the inner boundary of <formula> to the outer boundary . The circumference <formula> of <formula> under <formula> is the infimum of the length of all possible paths circling the ring ( i.e. not <unk> in R ) . The area <formula> of <formula> under <formula> is defined to be the sum of the squares of all weights in <formula> . Then define 
