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Types
Published in Michael Hann, The Grammar of Pattern, 2019
Regular medallion patterns are those patterns where the repeating component forms a clear foreground or island, consisting possibly of a known geometrical shape such as a circle, oval, square or regular hexagon, against a contrasting background. Unlike regular spot patterns, where the repeating component is of a single flat colour, regular medallion patterns can have embellishment, texture or a figured addition of some kind within the foreground (or medallion) component. It should be noted, however, as stated above, that such descriptions are arbitrary, and there will be overlap between the two regular-pattern types. ‘Medallion’ is a term given often to circular or oval shapes, and in the case of regular medallion patterns, the term is taken to mean a regular pattern consisting of equidistant circular, or diamond, or oval or any ‘regular’ polygon (such as a regular hexagon or regular heptagon) repeated systematically, without scale change, across the plane. Irregularly shaped convex polygons may be included also within the category; the important criterion is that a series of islands (or medallions) are presented against a contrasting sea (or background).
In‐plane structure of single‐layer tissues
Published in A. Šiber, P. Ziherl, Cellular Patterns, 2018
A more detailed analysis of this process carried out by examining small clusters of cells around a heptagon as the most common dividing cell class (Figures 3.12 and 3.18) confirms the existence of correlation. The minimal‐energy states of many such clusters were computed within a mechanical model including perimeter elasticity where each edge is represented by a Hookean spring and an area elasticity due to the ideal‐gas‐type pressure within the cell [85]; although not identical, this model is really not very different from the area‐and perimeter–elasticity theory. The equilibrium cluster configurations all unequivocally show that the heptagon is polarized and that to a very good approximation, its short axis determined by approximating the polygon by an ellipse always cuts across it starting from the quadrilateral or a pentagon. On the other hand, in a cluster with no four‐and five‐sided neighbors the short axis of the dividing heptagon is “repelled” by any seven‐or eight‐sided neighbors as illustrated in Figure 3.20. These results are consistent with experimental observations [85] so that we are led to conclude that cell division is affected by the structure of the epithelium and vice versa.
Areas of common shapes
Published in John Bird, Engineering Mathematics, 2017
A polygon is a closed plane figure bounded by straight lines. A polygon, which has:3 sides is called a triangle4 sides is called a quadrilateral5 sides is called a pentagon6 sides is called a hexagon7 sides is called a heptagon8 sides is called an octagonThere are five types of quadrilateral, these being:rectanglesquareparallelogramrhombustrapezium(The properties of these are given below.)
Effects of fullerene coalescence on the thermal conductivity of carbon nanopeapods
Published in Molecular Physics, 2018
In LAMMPS, the compute of heat current calculates six quantities and stores them in a 6-component vector. The first three components are the x, y, z components of the full heat flux vector, i.e. Jx, Jy, Jz. The next three components are the x, y, z components of just the convective portion of the flux, i.e. the first term in Equation 2. Therefore, we could split the contribution of mass transfer from the full heat current. The ratio denotes the relative contribution from mass transfer to the total heat current. Figure 4 illustrates the variation of mass transfer contribution in different CNPs and a corresponding (5,5)@(10,10) DWCNT at 300 K. It can be seen that the CNPs with unpolymerised C60 molecules possess the highest mass transfer contribution ratio, which is up to 20%. As the fullerene molecule begins to coalesce, this contribution to CNPs’ heat current decreases sharply, so that the mass transfer contribution ratio is only about 5% in (5,5)@(10,10) DWCNT. Mainly, it is because the unpolymerised C60 molecules have more freedom of motion. Once the coalescence of the fullerene starts, the C60 molecules would lose the freedom of translational and rotational motions and the original non-bonded interactions between C–C atoms in adjacent C60 molecules would transform into covalent interactions. With the coalescence degree, the heptagon–heptagon and pentagon–heptagon defects disappear and the inter-atomic forces would be further enhanced, which would make the motion of carbon atoms more and more difficult.
Outer billiards with contraction: regular polygons
Published in Dynamical Systems, 2018
The behaviour of the outer billiards map outside a regular heptagon is full of mysteries. See Figure 8 which shows a series of pentagonal tiles, which have period 57,848 and diameter approximately 0.0003 (the regular heptagon has radius 1). These tiles were found by R. Schwartz.