ManyFolds in another Variety

With the bird base diagrams on a square in variation, appeared in the preceding articles: Foldable deformations of the bird base and Another foldable deformations of the bird base, area [2-dimensional volume] of the diagram on a square as flatly folded into an Euclidean plane in each case is given by a [differentiable] function of one-variable θ, the degree which is defined by:Hence its domain is a closed interval [0, 90]: 0°≦θ≦90°.
(If the angular measure is defined in radian, then the domain is [0, π/2].)




With the diagrams in the left side of above pictures:

������The maximum value of its area is S(√2 - 1)/2 at θ=45°,
������when the diagram is a standard bird base;

������The minimum volume is S(2 - √2)/4 at θ= 0°, 90°.

With the diagrams in the right side of above pictures:

������The maximum volume is S(2 - √2)/2 at θ= 0°, 90°;

������The minimum volume is S(√2 - 1)/2 at θ=45°,
������when the diagram is a standard bird base.

Here S denotes area of the square on which diagrams of the bird base in deformation are defined.

[A little bit afraid of miscalculating... By the way, these calculations might be some appropriate exercises in mathematics - geometric analysis, trigonometric functions and differential calculus -, for high school/college students, it seems.]

References [added at April 26, 2006]: