Superposable objects

C symmetry rotation of an object about this axis by an angle of 360/n gives a superposable object. 

The shapes of similar molecules can be compared using the idea of the symmetric difference. The symmetric difference of two superposed objects is their union minus their intersection (Figure 2). By superposing molecules so that their overlap is maximal and then measuring the volume of the symmetric difference, it is possible to define a shape metric. A shape metric is a measure of dissimilarity that obeys the triangle inequality the difference of A and B plus the difference of B... 

The requirement for the existence of enantiomers is a chiral structure. Chirality is solely a symmetry property a rigid object is chiral if it is not superposable by pure rotation or translation on its image formed by inversion. Such an object contains no rotoinversion axis (or rotoreflection axis cf. Section 3.1). Since the reflection plane and the inversion center are special cases of rotoinversion axes (2 and 1), they are excluded. 

Objects (and molecules) that are superposable on their mirror images are achiral. 

Figure 5.5 A demonstration of chirality of a generalized molecule containing one tetrahedral stereocenter, (a) The four different groups around the carbon atom in III and IV are arbitrary, (b) III is rotated and placed in front of a mirror. Ill and IV are found to be related as an object and its mirror image, (c) III and IV are not superposable therefore, the molecules that they represent are chiral and are enantiomers.

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

Chirality (handedness, from Greek cheir = hand) is the term used for objects, including molecules, which are not superposable with their mirror images. Molecules which display chirality, such as (S)-(+)-lactic acid (/, Fig. 1) are called chiral. Chirality is often associated with a chiral center (formerly called an asymmetric atom ), such as the starred carbon atom in lactic acid (Fig. 1) but there are other elements that give rise to chirality the chiral axis as in allenes (see below) or the chiral plane, as in certain substituted paracyclophanes.1,2)... 

Chirality is the geometric property of a rigid object (or spatial arrangement of points or atoms), which is nonsuperposable on its mirror image such an object has no symmetry elements of the second kind (a mirror plane, a center of inversion, a rotation-reflection axis,. ..). If the object is superposable on its mirror image, the object is described as being achiral. 

Superposability is the ability to bring two particular stereochemical formulae (or models) into coincidence (or to be exactly superposable in space, and for the corresponding molecular entities or objects to become exact replicas of each other) by nothing more than translation and rigid rotation. 

An object is chiral and a chiroid if and only if it cannot be superposed on its mirror image by a proper congruence, otherwise it is achiral two chiroids are heterochiral and enantiomorphs if and only if they are improperly congruent and two chiroids are homochiral and homomorphs if and only if they are properly congruent. 

In these definitions, object refers to any rigid array, such as a set of points, or to a geometrical figure, or to a model of a molecule ideally realized (of which more below) properly congruent can be replaced by superposable and homomorphs, which may be either chiral or achiral, are congruent counterparts in Kant s terminology. 

An object is chiral if and only if it is not superposable on its mirror image otherwise... 

The breadth in scope of this definition comes at the cost of requiring flexibility and judgment in the interpretation of object, superposable, and mirror image. The model must suit the occasion of its use. Chirality, with respect to an isolated molecule, is a quantum-mechanically undefined concept, but, because we... 

Identification of structural units. This involves the search for secondary structural units (more precisely, for units that are superposable on standard units to within a specified error), or for combinations of them. The operator will want to specify the range of the search — within a certain molecule, or over a class of proteins, or over the entire available set of coordinates. He or she will also need to be able to qualify the object of the search, specifying perhaps a range of lengths of... 

There are many objects, both animate and inanimate, which have no symmetry planes but which occur in pairs related by a symmetry plane and whose mirror images cannot be superposed. W. H. Thompson, Lord Kelvin, wrote I call any geometrical figure or group of points... 

Before we attempt to answer the above questions, we need to examine briefly the terminology relevant to a discussion of chiral drugs. Specifically, the definition and usage of two important terms need to be clarified. Chiral was defined in one recent leading monograph on stereochemistry as follows Not superposable. .. with its mirror image, as applied to molecules, conformations, as well as macroscopic objects, such as crystals [3]. Mislow gave a shorter but essentially equivalent definition An object is chiral if and only if it is not superposable on its mir-... 

Structure comparison methods are a way to compare three-dimensional structures. They are important for at least two reasons. First, they allow for inferring a similarity or distance measure to be used for the construction of structural classifications of proteins. Second, they can be used to assess the success of prediction procedures by measuring the deviation from a given standard-of-truth, usually given via the experimentally determined native protein structure. Formally, the problem of structure superposition is given as two sets of points in 3D space each connected as a linear chain. The objective is to provide a maximum number of point pairs, one from each of the two sets such that an optimal translation and rotation of one of the point sets (structural superposition) minimizes the rms (root mean square deviation) between the matched points. Obviously, there are two contrary criteria to be optimized the rms to be minimized and the number of matched residues to be maximized. Clearly, a smaller number of residue pairs can be superposed with a smaller rms and, clearly, a larger number of equivalent residues with a certain rms is more indicative of significant overall structural similarity. 

Thomson (Lord Kelvin) coined a word for this property. He defined an object as chiral if it is not superposable on its mirror image. Applying Thomson s term to chemistry, we say that a molecule is chiral if its two mirror-image forms are not superposable in three dimensions. The work chiral is derived from the Greek word cheir, meaning hand, and it is entirely appropriate to speak of the handedness of molecules. The opposite of chiral is achiral. A molecule that is superposable on its mirror image is achiral. 

The surest test for chirality is a careful examination of mirror-image forms for superposability. Working with models provides the best practice in dealing with molecules as three-dimensional objects and is strongly recommended. 

Superimposable, superposable Two objects are superimposable if they can be brought into coincidence by translation and rotation. For chemical structures, free rotation around single bonds is permissible. Thus, two molecules of R-2-butanol are considered superimposable independent of their conformations. 

Achiral An object that is superposable on its mirror image... 

We have performed Monte Carlo simulations of the telescope design using a radius of 0.5 cm for the central zone and total of 400 zones. The finest zone width is 125 x 10 cm and separation between the plates is 1 meter. Fig. 1 shows the simulation done with identical zone plates. The pattern at the image plane reveals parallel fringes for a single source at infinity, while for two sources there are superposed fringe patterns. The deconvolved objects are shown in Fig. 2. 

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