118.

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Figure 5.11 Binary multiplication process for a box of unit length containing P points. Points division ratio u₀=1/3, and u₁=2/3.

(5.17)

For example, as shown in Figure 5.11, at iteration n=2, there are three different point distributions, i.e. ρ₀= ρ₁=, and ρ₂=4/9. Similarly, for n=3, there are four different ρ_k, namely ρ₀=1/27, ρ₁=2/27, ρ2= 4/27, and ρ₄=8/27. , and ρ₂=4/9. Similarly, for n=3, there are four different ρ_k, namely ρ₀=1/27, ρ₁=2/27, ρ2= 4/27, and ρ₄=8/27.

Let N(k) denote the number of sub-boxes containing the same ρ_k. N(k) can be expressed as:

(5.18)

For example, in the case n=3 and k=1 (i.e. p₁=2/27, see Figure 5.11) the value N(1) is calculated as 3, which indicates that the number of sub-

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