Circles maximizing area

Figure 4.5 The octahedral arrangement of six equal circles on a sphere maximizes the area covered by the circles.

Fig. 15. Exclusive action of porcine lipase on emulsified esters (146). Ordinates activity in perj cent of maximal activity on triolein emulsion. Abscissas lower axis, substrate amounts expressed in fractions of saturation for the solutions (on the left of the vertical dotted line) or in multiples of saturation for the emulsions (on the right of the line) upper axis, interfacial area expressed in 10 X cm in 100 ml. White circles, impure lipase containing some esterases. Black triangles, purified lipase. The substrate is triacetin on the left and methylbutyrate on the right. 

Figure 3. Maximal urine osmolality after fluid deprivation and vasopressin administration during (closed circles) and before (open circles) amiloride administration. The shaded area shows the range of urine osmolality in normal subjects tested after fluid deprivation and vasopressin administration in our laboratory. Li denotes lithium and Li + A lithium plus amiloride (reproduced from Batlle et al [26]).

The calculus of variations is applied, in Appendix V, to the problem of the maximum area occupied by a closed perimeter of fixed length. The perimeter forms the circumference of a circle in order to maximize the area. This was demonstrated experimentally, in section 1.3, using a loose loop of thread placed inside a ring containing a disc of soap film. The soap film inside the loop was broken with the result that the loop was pulled by the surrounding soap film into a circle (Plates 1.1(a) and (b)). The hole formed inside the loop takes up its maximum area because the area of the surrounding soap film is minimized. It can also be shown that if the area remains fixed and the perimeter is allowed to vary the minimum length of perimeter occurs when the area is contained by a circle. 

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