On 10/14/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 10/12/07, Bill Thurston <wpthurston@mac.com> wrote:
... You can make a similar origami type embedded construction for the square. Make concave-upward creases on the diagonal lines, and concave-downward creases on the altitudes. Add extra vertices on the altitudes at some fixed distance from the center, with new concave- downward creases from these vertices to the two adjacent corners, and reverse the direction of the fold on the distal portions of the altitudes. This should fold into a figure in upper half-space whose base looks like a 4-pointed star in the plane. (But I haven't actually done it)
The optimum position for this "extra" vertex is 0.21628 units approx from the edge of the flat bag; it lies at that height vertically above a point 0.18452 inboard from the edge of the square hull of edge 0.92941 occupied by the inflated bag on its horizontal plane of symmetry.
In this configuration the (maximised) volume equals 0.1656982335 --- not bad at all for a straight-off-the-cuff construction, when compared with the best attempt of 0.2055 claimed by Andrew Kepert! WFL
In practice, 0.21628 is sufficiently close to tan(\pi/8)/2 that the "extra" creases can be placed nearly optimally via the traditional origami trick of (twice) halving the angle at the corner of the flat square. All of which --- to a (former) outdoor enthusiast --- prompts the unexpected observation that, using just 5 poles, 8 pegs, 13 reinforcement patches and a 5-metre or so square of mylar, it is possible to fashion a pretty serviceable tent. Entry and exit might prove problematic, of course ... Details, my dear chap, details! WFL