Circle packings

Figure 4.25 Circle packing, (left) In two dimensions, no more than four circles can be placed so that each circle touches all the others, with every pair touching at a different point. What happens in higher dimensions (right) An attractive computer-graphic study of circle packing.

Given a map M, its circle-packing representation (see [Moh97]) is a set of disks on a Riemann surface E of constant curvature, one disk D(v, rv) for each vertex v of M, such that the following conditions are fulfilled ... 

Simultaneous circle-packing representations of a map M and its dual M are called primal-dual circle representation of M if it holds ... 

A map M is called reduced (see [Moh97, Section 3]) if its universal cover is 3-connected and is a cell-complex. It is shown in [Moh97, Corollary 5.4] that reduced maps admit unique primal-dual circle packing representations on a Riemann surface of the same genus moreover, a polynomial time algorithm allows one to find the coordinates of those points relatively easily. This means that the combinatorics of the map determines the structure of the Riemann surface. 

Moh97] B. Mohar, Circle packing of map in polynomial time, European Journal of Combinatorics 18 (1997) 785-805. 

Table 16.2. Shape parameters (equations (16.1) and (16.4)) and approximate swelling exponents s and s, cf. equation (16.8)) for known lyotropic mesophases. The (variable) constants /i and depend on the specific symmetry of the phase ( /i is the homogeneity index, ideal equal to 3/4 (cf. Table 16.1) / is the interstitial packing fraction for dense sphere and circle packings, equal to unity for ideal homogeneous packings)...

Figure 3.7 Number of times out of = 10 trials that each of the 20 distinct MS packings at dpj is obtained using the LS (squares) and MD (circles) packing-generation methods for 2D bidisperse systems with N = 6.

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

The furnace and thermostatic mortar. For heating the tube packing, a small electric furnace N has been found to be more satisfactory than a row of gas burners. The type used consists of a silica tube (I s cm. in diameter and 25 cm. long) wound with nichrome wire and contained in an asbestos cylinder, the annular space being lagged the ends of the asbestos cylinder being closed by asbestos semi-circles built round the porcelain furnace tube. The furnace is controlled by a Simmerstat that has been calibrated at 680 against a bimetal pyrometer, and the furnace temperature is checked by this method from time to time. The furnace is equipped with a small steel bar attached to the asbestos and is thus mounted on an ordinary laboratory stand the Simmerstat may then be placed immediately underneath it on the baseplate of this stand, or alternatively the furnace may be built on to the top of the Simmerstat box. 

Fig. 1.21. Ratio of free-path distribution Xp to the scaled zero-density free-path distribution plotted as a function of reduced free-path length r/X for two-dimensional (a) and three-dimensional (b) liquids. Circles, inverted triangles and upright triangles refer to reduced volumes V/V0 of 1.6, 2, and 3, respectively (V0 is the volume of the system at close packing) [74].

Fig. 2.—A diagram showing the ranges of values of the neutron number N in which successive uubsubshells of the mantle, outer core, and inner core are occupied by neutrons, as calculated with use of the packing equation. Observed values of spin and parity of odd N and odd Z nuclei are indicated by circles and squares.

Fig. 8.—Packing arrangement of four symmetry-related 2-fold helices of mannan II (6). (a) Stereo view of two unit cells approximately normal to flic frc-plane. The two chains in the back (open bonds) and the two in the front (filled bonds) are linked successively by 6-0H-- 0-6 bonds. The front and back chains, both at left and right, are further connected by 0-2 -1V -0-2 bridges, (h) Projection of the unit cell along the c-axis the a-axis is down the page. This highlights the two sets of interchain hydrogen bonds between antiparallel chains, distinguished by filled and open bonds. The crossed circles are water molecules at special positions.

Fig. 9. — Antiparallel packing arrangement of the 3-fold helices of (1— 4)-(3-D-xylan (7). (a) Stereo view of two unit cells roughly normal to the helix axis and along the short diagonal of the ab-plane. The two helices, distinguished by filled and open bonds, are connected via water (crossed circles) bridges. Cellulose type 3-0H-0-5 hydrogen bonds stabilize each helix, (b) A view of the unit cell projected along the r-axis highlights that the closeness of the water molecules to the helix axis enables them to link adjacent helices.

Fig. 13.—Packing arrangement of extended, 6-fold, KOH-amylose (11) helices, (a) Stereo view of two unit cells approximately normal to the ftc-plane. The helix (filled bonds) at the center is antiparallel to the two helices (open bonds) at the comers in the back. Potassium ions (crossed circles) have water molecules (open circles) and hydroxyl groups from amylose helices as ligands.

Fig. 19.—Antiparallel packing arrangement of 2-fold poly(GulA) (16) helices, a) Stereo view of two unit cells roughly normal to the fee-plane. The helix at the center (filled bonds) is antiparallel to the two in the back (open bonds). Intrachain hydrogen bonds stabilize each helix. Association of antiparallel helices involves the carboxylate groups and water molecules (crossed circles).

Fig. 28.—Antiparallel packing arrangement of 4-fold helices of sodium hyaluronate (26). (a) Stereo view of a unit cell approximately normal to the hc-plane. The two comer chains in the front (filled bonds) are linked directly by hydrogen bonds. The chain at the center (open bonds) interacts with die comer chains via sodium ions (crosses circles) and hydrogen bonds.

Fig. 30. — Packing arrangement of 4-fold antiparallel double helices of potassium hyaluronate (32). (a) Stereo view of a unit cell approximately normal to the line of separation of the two helices. The two chains in each duplex, drawn in open and filled bonds for distinction, are linked by not only direct hydrogen bonds, but also water bridges. Inter double-helix hydrogen bonds are mediated between hydroxymethyl and iV-acetyl groups. Potassium ions (crossed circles) at special positions have only a passive role in the association of hyaluronate chains.

FlC. 32.—Antiparallel packing arrangement of the 2-fold helices of calcium chondroitin 4-sulfate (35). (a) Stereo view of two unit cells approximately normal to the he-plane. The two comer chains, drawn in filled bonds are hydrogen bonded to the antiparallel center chain (open bonds). Calcium ions (crossed circles), associating with sulfate and carboxylate groups and water molecules link adjacent antiparallel chains, which ate also directly hydrogen bonded. 

Fig. 37. (continued)—(b) An axial view projected along the r-axis shows the packing arrangement of three welan double helices in the trigonal unit cell. The helix drawn in solid bonds is antiparallel to the remaining helices (open bonds). Note that calcium ions are positioned between the helices and each water molecule (large open circle) shown here is connected to all three surrounding helices. The interstitial space is occupied by several other ordered water molecules (not shown). 

Fig. 14.3 Polyhedral packing plots for the two-dimensional layers of [RE(P2S6),/2(PS4)P in the series of solids A2RE(P2S6)i/2(PS4), where A=K, Cs RE = Y, La. Rare-earth polyhedra are striped PS4 polyhedra are black phosphorous atoms in P2S6 are shown as black circles. Alkali atoms are not shown for clarity. Although these phases have distinctly different structures based on space group symmetry and atomic positions, the compounds are clearly related upon close inspection of the building blocks.

FIG. 8 Salt exclusion as a function of surface charge in a cylindrical pore in equilibrium with a 0.1 molar electrolyte. The open circles are GCMC results for 1 1 RPM electrolyte in a pore ofR = 5d The circles with a centered cross are results for a 2 1 electrolyte in a pore of = 5d. The up-trian-gles are results for a 2 1 electrolyte in a pore ofR = lOd. The solid circles are results for a 1 1 SPM model with 0.3 solvent packing fraction in a pore of = 5d. The solid squares are the same results for a pore of R = Id. 

FIG. 5 Schematic representation of packing arrangements of natural amphipathic double-chain lipids with different headgroup size in crystalline bilayers. The small filled circles indicate the accommodation of spacer molecules, such as water or ions. (Reprinted by permission from Ref 14, copyright 1992, Elsevier Science.)... 

Fig. 43. Packing in the crystal structure of the 1 imidazole 2 H,() associate viewed from the c direction1111. Observe the layer-like arrangement of water molecules near x = 0 (H-bonds are indicated as broken lines only relevant H atoms are shown O atoms of the host are dotted water oxygen as a bold circle N atoms are hatched the hatched segments signify the imidazole rings)...

Fig. 13. Projection view of Vh-amylose on the a, b plane. The amylose chains are packed in an antiparallel manner in space group />212121. Guest water molecules are represented as filled circles in the helical canals and in interstitial sites between the helices...

The heat transfer data discussed above refer only to the average behavior of a vial of a given type which is surrounded by other vials in a hexagonal packing array of vials. We now consider intervial variations in heat transfer in a set of nominally equivalent vials and variations in heat transfer arising from variations in the position of the vial in the array. An experiment demonstrating such variations is described by Figure 35. Each circle represents a vial placed on a temperature-controlled shelf in a small laboratory freeze dryer. The vials contained pure... 

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