Before we're going to analyze the structure of Calcite crystals in order to find the Complex Motif (that lies at the base of the crystal's tectological structural aspect), something must be said with respect to chemical bonding and ionic radii, both controlling the internal structure of crystals.
In crystals we can encounter several distinct  b o n d i n g  t y p e s , namely covalent bonding, van der Waals bonding, hydrogen bonding and ionic bonding.

Covalent bonds controlling structure.
Covalent bonds are typically quite strong, so covalent crystals are hard and have high melting points. The strong  d i r e c t i o n a l  nature of covalent bonds requires that the atoms be placed in specific  o r i e n t a t i o n s  so that orbitals can overlap (See the first part of this website -- accessible via   back to homepage -- The Chemical Bond (Non-classical Series)). This often prevents the atoms from acquiring a close-packed arrangement.

Molecular bonds controlling structure.
Molecular crystals consist of discrete molecules packed together in a systematic way. Each molecule in the structure is typically bonded internally with covalent bonds and lacks a residual charge or valence with which to bond with other atoms. The forces that hold the molecules to each other are limited to relatively weak van der Waals and hydrogen bonds. Molecular crystals are therefore quite soft. The geometry of the structure is controlled by the shape and charge distribution of the individual molecules and the pattern by which they can be systematically packed together. For example in Ice (H₂O) hydrogen bonding holds water molecules in a hexagonal structure that is reflected in the hexagonal shape of snow flakes.

Ionic bonds controlling structure.
Ionic bonding involves the electrostatic force obtaining between positively charged ions and negatively charged ions (an ion is an electrically charged atom or atomic group).
Because ionic bonds are not directional, we can, as a first approximation, look to the manner in which cations (positively charged ions) and anions (negatively charged ions) bond together in purely geometrical terms, unencumbered by the complexities of aligning orbitals in the specific orientations required for covalent bonds. In doing so we can imagine a cation -- that is normally relatively small -- being in contact with several anions (that are normally larger). In what way (i.e. in what kind of arrangement) those anions group themselves around a cation is expressed by the  c o o r d i n a t i o n  p r i n c i p l e.  The coordination principle states that how anions and cations pack together in a crystal structure depends on their relative sizes (radii). A cation's coordination number (CN) is the number of anions with which it is in contact. When the bonding is dominantly ionic, each cation will attract or coordinate with as many anions as it can pack around itself. The  c o o r d i n a t i o n  p o l y h e d r o n  is the shape defined by anions coordinating with a cation. Twelve-fold coordination is the largest common coordination number. The coordination number is dependent on the  r a d i u s  r a t i o  between cation and anion, where the radius ratio is the ratio of the radius of the cation and that of anion.
The most important coordination numbers are listed below.
As has been said, a certain coordination of anions around a cation defines a polyhedron, a  c o o r d i n a t i o n  p o l y h e d r o n.  We find such a polyhedron when we interpret the centers of the anions (that are grouped around one central cation) as the corners of a polyhedron. The particular coordination is called after the coordination polyhedron that it involves. As has been promised, some important coordinations will be listed :

• Coordination number = 12. The geometric shape defined by the 12 anions is not a regular polyhedron. Each cation can be surrounded by no more than 12 anions, such that each anion touches the cation. The two different ways in which 12-fold coordination can be achieved are based on cubic closest packing (ABC-structure) and hexagonal closest packing (AB-structure).
• Coordination number = 8. The eight anions, surrounding a central cation, can be thought to lie at the corners of a cube :   Cubic Coordination.
• Coordination number = 6. The six anions, surrounding a central cation, lie at the corners of a regular octahedron :   Octahedral Coordination.
• Coordination number = 4. The four anions, surrounding a central cation, lie at the corners of a regular tetrahedron :   Tetrahedral Coordination.
• Coordination number = 3. The three anions, surrounding a central cation, lie at the angles of an equilateral triangle :   Triangular Coordination.
• Coordination number = 2. The two anions, lying at either side of a central cation, define a straigt line connecting the three centers of the three ions :   Linear Coordination.

The Electrostatic Valence Principle
(NESSE, W., 2000, Introduction to Mineralogy, p. 62).
For an ionic bond, the  b o n d i n g  c a p a c i t y  of an ion is proportional to its oxidation state (charge). The  s t r e n g t h  of the electrostatic valence bonds (EVB) reaching an ion is therefore equal to the charge of the ion divided by the number of oppositely charged ions with which it coordinates, i.e., the coordination number (CN) :

EVB = ion charge/CN

Although this relationship is straightforward, it leads to some interesting consequences.

UNIFORM BOND STRENGTH
First consider the case in which all ionic bonds have the same strength. We call this  i s o d e s m i c.  This will be the case for all ionic minerals and other ionic compounds with a single anion and single cation, as well as some minerals (and other compounds) with multiple cations and/or anions. If all the cation-anion bonds are the same strength, it is impossible for any bond to occupy more than half of the anion's charge because the smallest possible coordination number is two. Because the anion-cation bonds are all uniform, the anions tend to pack together in a highly symmetrical arrangement and the cations occupy polyhedra formed by the interstices among the anions. Some of the common structures are based on cubic or hexagonal closest packing of the anions. It should be no surprise that most of these substances are higly symmetrical and crystallize in the Isometric, Tetragonal, or Hexagonal Crystal Systems.
Substances with uniform bond strengths are conveniently classified as Oxides, Fluorides, Chlorides, etc., based on the identity of the anion.

NON-UNIFORM BOND STRENGTH
In some structures certain anion-cation bonds are stronger than others. The cations forming stronger bonds are small and possess a high charge, for example C⁴⁺, S⁶⁺, P ⁵⁺, and Si⁴⁺. Because these cations strongly bond to the anions (usually O^2-) the result is the formation of CO₃^2-, SO₄^2-, PO₄^3-, and SiO₄^4- anionic groups that serve as structural elements in their crystal's structures.
Crystals with non-uniform bond strengths typically have lower symmetry than those in which bond strengths are uniform because of the geometric complexity involved with packing the anionic groups together and coordinating with other cations. Coordination polyhedra formed between these anionic groups often are irregular. The two types of non-uniform bonding are  a n i s o d e s m i c  and  m e s o d e s m i c.
With anisodesmic bonding, some anion-cation bonds take more than half of the anion charge.
With mesodesmic bonding, some anion-cation bonds take exactly half of the anion charge.
A n i s o d e s m i c  b o n d i n g  is found in crystals containing small, highly charged cations. Concider the mineral Calcite (CaCO₃), with which we are going to occupy ourselves below :   Carbon (C) is a small cation with +4 charge that coordinates with three O^2-. The electrostatic valence bond (EVB) strength of the C--O bonds is therefore 4/3 = 1.33. The Ca²⁺ is in 6-fold coordination with these same O^2- anions with an EVB of 2/6 = 1/3. Each O^2- bonds with two Ca²⁺ cations so the total EVB reaching each Oxygen is 2 (See Figure 1). We see that the two-valency of Oxygen is in this case unevenly distributed around the Oxygen atom (4/3 goes to Carbon, 1/3 to a Ca atom and another 1/3 to a second Ca atom). The Oxygen anions are more strongly bonded to the Carbon than to the Calcium. As a consequence, distinct CO₃^2- structural anionic groups are formed.

Figure 1.   Anisodesmic carbonate (CO₃^2-) group. Each O--C bond occupies one and one third of the available - 2 charge on Oxygen. Only two-thirds of a charge on the Oxygen is available to bond with other cations such as Ca²⁺.
Black disc = Carbon. Red disc = Calcium. Light blue disc = Oxygen.
( Redrawn from NESSE, W., 2000, Introduction to Mineralogy )

The mineral  C a l c i t e  is one of the two crystalline varieties in which the substance CaCO₃, Calcium carbonate, can occur. The other one is the mineral Aragonite.
The point symmetry of Calcite crystals is 3* 2/m, which means that they belong to the Ditrigonal-scalenohedric Crystal Class of the Hexagonal Crystal System (Rhombohedric Division). The Space Group is R 3* 2/c, which means that there is a c-glide present in the Calcite structure, i.e. three glide planes containing the c-axis.
The essential structural element in all carbonate minerals (and thus also in Calcite) is the CO₃^2- anionic group. It consists of one Carbon cation that coordinates with three Oxygen anions. Carbon, with its +4 charge, commands two-thirds of the net -6 charge of the three Oxygen anions with which it coordinates. These Oxygen anions are therefore more strongly bonded to the central C⁴⁺ cation than to any other cations in the mineral structure. Further, an Oxygen anion in one CO₃^2- group cannot also be shared with another CO₃^2- group, because that would imply the configuration C-O-C in which one bond strength is 4/3, leaving only 2/3 for the other bond strenght, while the latter should also be 4/3. The structure of all carbonates must therefore be based on CO₃^2- groups that are bonded laterally through various cations (in the case of Calcite this is Calcium).
The CO₃ groups lie in planes at right angles to the 3-fold (c) axis, and the Ca ions, in alternate planes, are in 6 coordination with Oxygens of the CO₃ groups. Each Oxygen is coordinated to a Carbon ion within its own layer, i.e. to a Carbon ion at the center of its CO₃ group, and to two Ca ions, one above and one below it. The next Figure illustrates the alternation of Ca cations with CO₃ anions in the Calcite structure.

Although the  p o s i t i o n  of the C atoms is repeated in every third CO₃ layer, the corners of the triangular CO₃ groups alternate by a mirror plane every layer, i.e. the  o r i e n t a t i o n  alternates every layer. This is because of the mentioned c-glides :

Figure 4.   Unit cell of the structure of Calcite, CaCO₃.
The unit cell (which is an elongated rhombohedron) contains two differently oriented CO₃ groups, one situated above the central Calcium ion and one below it. The central Calcium ion corresponds to the Calcium ion at the lower end of the rhombohedron of Figure 2. This unit cell contains 8x1/8+1=2 Ca atoms and 2 CO₃ groups, and thus this unit cell contains two formula units of Calcium Carbonate (CaCO₃).
( Adapted from STRÜBEL, G., Mineralogie, 1977 )

Figure 5.   Unit cell of the structure of Calcite, CaCO₃, in relation to the cleavage rhombohedron. We can imagine two such rhombohedrons placed on top of each other (with one Ca in common). Here a lower cleavage rhombohedron is depicted. Above we had described the upper cleavage rhombohedron in relation to the unit cell.
( After STRÜBEL, G., Mineralogie, 1977 )

Figure 6.   Two unit cell choices (elongated rhombohedron and hexagonal prism) of the structure of Calcite, CaCO₃, in relation to the cleavage rhombohedron. The extra elements of the cleavage rhombohedron falling within the hexagonal prism should be repeated within the outline of a second cleavage rhombohedron on top of the first one (with a Calcium ion in common).
( After HURLBUT, C. & KLEIN, C., Manual of Mineralogy, 1977 )

Figure 6a.   Alternation of Ca-layers and CO₃-layers, and within each layer an alternation in turn. The unit cell thickness is indicated. Only two CO₃ groups, namely B' and B, fall within the confines of the unit cell. Their position is identical within the stacking sequence -- Both occupy B-positions -- but their orientation is different ( indicated by the absence and presence of a ' sign ). Compare with Figure 4 :   The lower X and upper X in the present Figure correspond to the lower and upper Ca in Figure 4.
X' in the present Figure corresponds to the central Ca ion in Figure 4.
B' and B in the present Figure correspond to the two CO₃'s in Figure 4.

In order to find the Complex Motif of Calcite, i.e. the translation-free motif of the Calcite crystal structure, we first eliminate all simple translations (which are responsible for the periodic repetition of the unit cell). What is left is evidently one unit cell as depicted in Figure 4. In this unit cell there is still a translational element present (which must also be eliminated) : We can detect this element as the translational element of the c-glide evident in the different orientation of the two CO₃ triangles which fall within the confines of the unit cell.
The next Figure depicts an extention of the Calcite structure such that two unit cells (with one Ca atom in common) are visible.

Figure 7.   Two unit cells of the structure of Calcite, CaCO₃. They share one Calcium atom. The CO₃ triangles that fall within the confines of the two unit cells are numbered. We can see that their orientation alternates, indicating the presence of a c-glide.

Figure 8.   Two unit cells of the structure of Calcite, CaCO₃, and a copy of these unit cells shifted with respect to the original as indicated. The length of the shift is equal to 1/2 c, i.e. equal to half the length of the unit cell measured along the c-axis.

Figure 9.   Concentrating on the area involved by one unit cell.

Figure 10.   The unit cell of Calcite (left) and what must be added (right) to its content to obtain the Complex Motif.