- The Olsen Twins Revisited: 'So Little Time'
- Twinsburg Revisited
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- The Olsen Twins Revisited: 'So Little Time'

God willing. In other words, the re-education of a new generation of children has begun! And we must talk about it. First a refresher! Lead characters Chloe and Riley are just your typical teens who live in a Malibu mansion with their divorced-but-still-besties parents.

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Chloe or Riley? On the plus side? We began in the kitchen, with Riley dreading the school day ahead. Chloe, while being completely different from her sister in literally every way, had to agree: This day was not going to be a great one. Even male maid Manuelo took a break from mopping to commiserate about the true depths of their collective despair. Then, without any warning whatsoever, everybody in the kitchen began to bang on things and dance out of the room. I mean, come on.

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## The Olsen Twins Revisited: 'So Little Time'

That title! Was it supposed to be stressful as hell? So little time because we are all hurtling toward the grave? So yes, they got to see the boy band, but now they had to somehow replenish that lock box. But how! Ever heard of something called dark magic? First they attempted to recite forbidden exultations to the Old Gods.

### Twinsburg Revisited

The involved groups are , unit cell U , with the intersection group with the same unit cell U , as shown in Fig. This leads to , meaning that the twin operation connects two individuals and different crystals — corresponding to the families of 1, 1, 1 planes, A , B or C — can be formed around one single crystal.

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Finally, the twin index that corresponds to the indices of the lattices only is , thus the term used to designate this kind of grain boundary. Two equivalent ways of describing the classical twin in f. On the left, emphasis is put on the sublattice conservation between twinned crystals, whereas on the right, emphasis is put on the twinned crystals sharing a common plane. This same twin can be as well defined through its epitaxy property and using point groups. The two adjacent crystals share the same 1, 1, 1 plane; thus, , with an intersection group. This means that the twin , that keeps a 1, 1, 1 plane invariant, is between variants and different individuals — the four orientation families of 1, 1, 1 planes — that can be formed around one given variant.

All known types of twins enter the general group—subgroup tree of Fig. For instance, merohedral twins are characterized by and having the same lattice; coincidence grain boundaries are twins by reticular merohedry with a grain boundary index being the order of the lattice of in the lattice of. As we will show here, there are cases where the scheme of Fig. The geometric nature of the twin interface h in green is shown in Fig. It is built with six adjacent regular pentagons and has the crystallographic two-dimensional space group.

Taking the radius of the elementary pentagon as the unit length see Fig. The whole structure is described by only two Wyckoff positions generated by the positions and drawn in green and blue in Fig. This periodic subset of the -module has unit cell and and two Wyckoff positions and. This structure is thus a periodic decoration of a -module 2 of rank 5 4 in fact, because the sum of the five unit vectors gives the zero vector generated by the five vectors defined by the regular pentagon. The point symmetry operations are signed permutation matrices given by.

These operations together with the translation group generated by form a faithful representation of the group in five-dimensional Euclidean space. Now, we choose the underlying five-dimensional lattice generated by the five mutually orthogonal vectors whose projections are the five basic vectors of the pentagon, as the geometric object that should be left invariant. The group of all operations that keep the five-dimensional lattice invariant together with the two-dimensional cut space is generated by.

Thus, the general group—subgroup tree Fig. For example, one among the possible cosets of equivalent twin operations is given by. It can be easily verified that this twin is perfectly coherent although it has no two-dimensional coincidence lattice. The boundary is defined by a sinuous row of adjacent pentagons that belong to both structures. This is the habit direction of the twin: we have a perfect epitaxy with no two-dimensional coincident lattice.

All these structures have the basic property of being defined by Wyckoff positions that are all on the same -module and can thus be described as two-dimensional cut-and-projections of n -dimensional structures. We discuss here some of the simplest of these kinds of polygonal tilings where the n -gons in the unit cell are all crystallographically equivalent. We shall designate these patterns as monogeneous n -gon patterns. An efficient way of characterizing these patterns consists of reporting in a vector the sequence of the number of free edges between each connected edge around an n -gon as exemplified in Fig.

We call it the vector of free edges , the length of which is equal to the coordination of the n -gon. The local configurations of n -gons around a central one are characterized by the sequence of the number of free edges of the central n -gon that are between two consecutive connections.

Here, for example, 9-gon tilings are shown with coordination of configurations from a to c : , and. For the coordination , there is only one configuration issued from a and shown in d ; but it leads to a non-monogeneous pattern since there are two kinds of n -gons, one in grey of coordination 4 and the other in white of coordination 2.

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The search for possible periodic solutions is significantly simplified by observing that, for an n -gon surrounded by p identical n -gons, the vector of free edges is such that with. Also, the maximum possible number of non-overlapping n -gons sharing an edge of a central identical n -gon is given by. For the simple case , monogeneous non-overlapping n -gon periodic patterns are generated only if the centre of the central n -gon is inside the triangle formed by the centres of the three adjacent n -gons, in which case the triangle characterizes the unit cell of the structure see Fig.

The vector of free edges has the form. All twins in these structures are merohedral twins built with the same n -gon. They are characterized by the symmetry elements of the n -dimensional lattice that leave the projected two-dimensional space invariant and that do not belong to the symmetry group of the structure as dictated by the general group tree of Fig.

In two-dimensional space, these twin operations are symmetry operations of the central n -gon that are not symmetry elements of the two-dimensional periodic structure and that leave invariant a row of the structure to form a perfect plane of epitaxy. For , these elements are signed permutations of the n basic vectors generating the n -gon that transform into each other two of the other adjacent n -gons and put the third one in a new position.

This translates in the vector of free edges in exchanging two symbols while keeping the third constant. Any questions? Renew Close. Looks like Javascript is disabled in your browser. Malaysiakini requires Javascript to run normally. Click here to enable Javascript in your browser.

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## The Olsen Twins Revisited: 'So Little Time'

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