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In Digital Geometry, a gap is a location of a digital object through which a discrete ray can penetrate with no intersection. More specifically, for a 3D digital object we distinguish between 0-and 1-gaps depending on the relative position of such a ray. Although in some applications it is important to know how many gaps has a set of voxels, it is quite complicated to find an efficient algorithm to directly count them. In this paper, we provide a formula that states the number of 1-gaps of a generic 3D object using the notion of free cell of dimension 1 and 2.

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We provide a formula that expresses the number of (n − 2)-gaps of a generic digital n-object. Such a formula has the advantage to involve only a few simple intrinsic parameters of the object and it is obtained by using a combinatorial technic based on incidence structure and on the notion of free cells. This approach seems suitable as a model for an automatic computation, and also allow us to find some expressions for the maximum number of i-cells that bound or are bounded by a fixed j-cell.By the above expression and the definition of dimension of a cell, we have that the dimension of the dual e ′ of a cell e = (x, θ) coincides with the number of non-null components of the direction θ, that is dim(e ′ ) = n j=1 [θ j 0]. Consequently, the dual e ′ of an i-cell e is an (n − i)-cell.Definition 3. Let D be a digital object. The dual D ′ of D is the set of all dual cells e ′ , with e ∈ D.We say that two n-cells v 1 , v 2 are i-adjacent (i = 0, 1, . . . , n − 1) if v 1 v 2 and there exists at least an i-cell e such that e ⊆ v 1 ∩ v 2 , that is if they are distinct and share at least an i-cell. Two n-cells v 1 , v 2 are strictly i-adjacent, if they are i-adjacent but not j-adjacent, for any jn . The set of all n-cells that are i-adjacent to a given n-voxel v is denoted by A i (v) and called the i-adjacent neighborhoods of v. Two cells v 1 , v 2 ∈ C n are incident each other, and we write e 1 Ie 2 , if e 1 ⊆ e 2 or e 2 ⊆ e 1 . Definition 4. Let e 1 , e 2 ∈ C n . We say that e 1 bounds e 2 (or that e 2 is bounded by e 1 ), and we write e 1 < e 2 , if e 1 Ie 2 and dim(e 1 ) < dim(e 2 ). The relation < is called bounding relation.Definition 5. Let e be an i-cell of a digital n-object D (with i = 0, . . . n − 1). We say that e is simple if e bounds one and only one n-cell.