Flat points

The polyhedron corresponding to the Neovius surface has the same arrangement of points as that for the infinite semi-regular polyhedral surface 6.43 discussed above, but the spaces between the points are differently filled with polygons so that each of the 48 points per cubic cell has the configuration of 8.4.8.6 and this leads to a surface of genus of 9. This surface has two kinds of flat points and is thus not regular (Mackay Terrones 1991). 12 tubes in the [110] directions connect cavities. 

A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a mapping from the minimal surface to a Riemann surface. The mapping is a composite one first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except flat points), characterised by cartesian coordinates (x,y,z) is described by the complex number (o, which... 

The Weierstrass equations allow calculation of the cartesian coordinates ((x,tf,z) with respect to an origin (xo,yOi o)) of the minimal surface at all points on the surface - except flat points - in terms of a complex analytic function R(o>). The Weierstrass equations are ... 

The "Bonnet angle", 6, is equal to 0 for the D-surface and n/2 for the P. The function R(o)) used by Schwarz for the D- and P- surfaces is simply the inverse of the square root of the product over the images of the flat points under the map from the minimal surface to the complex plane. Using fflf to denote the flat point images, the representation can be written ... 

This product form for the Weierstrass polynomial is readily generalised, and offers a useful route to the discovery and parametrisation of three- periodic minimal surfaces (IPMS). It turns out that for all "regular surfaces" (which are the topologically simplest IPMS), the distribution and character of the flat point images (the location and type of the branch points (of R(,0))) in the complex plane) alone suffice to construct the Weierstrass polynomial, and thus the complete IPMS, using the Weierstrass equations. 

Figure 1.20 A unit cell of the P-surface, embedded in a cube. The normal vectors to the P-surface at its eight flat points (one obscured) are indicated by the arrowed vectors. These vectors point towards the eight vertices of the cube.

The least number of flat points a minimal surface may possess is one. An example of such a surface is Enneper s surface, which is asymptotically flat (Fig 1.23). In this case the flat point is not an isolated point on the surface. However, only a single surface orientation is displayed by the asymptotically flat boundary. 

The next possibility is a Gauss map containing two singularities due to flat points. Examples of this case are the helicoid and the catenoid (Figs. 1.13 and 1.14). The normals of the flat points on these surfaces (at the asymptotic ends of the surfaces) are antiparallel, and hence the Weierstrass parametrisation is given by... 

Three flat points alone cannot satisfy eq. 1.21. Proceeding to four flat points we may create the Scherk surfaces (Fig. 1.15) by distributing these evenly along a great circle on the Gauss sphere. The standard Weierstrass parametrisation of these surfaces is... 

All these surfaces have one important characteristic in common. The flat "points" are not located within any finite portion of the surface. Rather, the surfaces become asymptotically flat (e.g. the trumpet-shaped "ends" in the catenoid). As the number of flat points increases beyond four, the flat points are located at fixed identifiable sites and the surface closes up to b ome periodic in three dimensions. This distinction between one- or two-periodic and three-periodic minimal surfaces is a crucial one, since it implies that the average Gaussian curvature () of one-, and two-periodic minimal surfaces is usually zero, due to the overwhelming contribution from the... 

All these IPMS have eight distinct normal vectors due to flat points. The relative distribution of these normal vectors determines the IPMS. They are described by Weierstrass parametrisations of the form (eqs. 18) ... 

Figure 2.11(b) Monkey-saddles of the gyroid (left) and the D surface (right). These saddles are related by the Bonnet transformation. Atoms, fixed to the flat points of the saddles, are transformed from the face-centred cubic to a body-centred cubic array. 

This chisel has a broad flat point and is used to cut thin sheet metal, remove rivet heads or split corroded nuts. The cutting edge is ground to an angle of approximately 60 Fig. 2.6. 

For molecular crystals in which the molecules are nearly spherical there will frequently be a high-temperature crystal phase of which the site symmetry is higher than the molecular symmetry. These crystals are referred to as plastic crystals and they have unusual properties such as very high temperatures of melting and low anisotropic properties similar to those of liquids. The theory of lattice vibrations of orientationally disordered solids has been addressed, and the results indicate that such disorder should lead to a broadening of the vibrational bands and that all modes may be active in both the infrared and Raman spectra. Spectra of this type are referred to as density-of-states spectra, since the bands correspond to the flat points in the dispersion curve. 

In Wyoming, Mike Brady opened North Fork Bullets a dozen years ago because, he says, he was obsessed with perfection. His first step was a line of premium bonded core soft point bullets. Next was a line of truncated cone flat point solids that would impact exactly the same down-range point, thus delivering a whopping amount of penetration, and he recently worked on a cup solid point as an expanding solid. These cup point solids bridge the gap between softs and solids. Bullets are now manufactured in Oregon. 

When l7(A t ) = o, x is an extremum ( flat point ), and as we move downhill in cost function at each step, x is likely a local minimum. In practice (5.4) is insufficient to... 

This yields our generalization of the condition VF = 0 to accommodate an equality constraint. Rather than finding a flat point in the cost fimction, we search for a flat point of the... 

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