

A088896


Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.


1



125, 1000, 2197, 3375, 4913, 8000, 15625, 17576, 24389, 27000, 39304, 42875, 50653, 59319, 64000, 68921, 91125, 125000, 132651, 140608, 148877, 166375, 195112, 216000, 226981, 274625, 314432, 343000, 389017, 405224, 421875, 474552, 512000
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OFFSET

1,1


COMMENTS

The set of values for the integralwidths corridors and longest ladder are merely the cubes of Pythagorean triples, viz. (A046083, A046084, A009000).
The corridors' widths may be parametrically expressed as d*(sin x)^3 and d*(cos x)^3, for a longest ladder length d making an angle x with one of the corridors.
A given ladder, however, is maximumcornerbending for a family of infinite pairs of perpendicular corridor widths and that the envelope of the maximum bending positions is that of a sliding rod against the outer wall, which is a branch of an astroid or fourcusped hypocycloid.


REFERENCES

E. Mendelson, 3000 Solved Problems in Calculus, Chapter 16 Problem 16.56 pp. 131, Mc GrawHill 1988.
M. Spiegel, Theory and Problems of Advanced Calculus, Chapter 4 Problem 40 pp. 75, Mc GrawHill 1974.


LINKS

J. J. O'Connor and E. R. Robertson, Astroid
Eric Weisstein's World of Mathematics, Astroid


FORMULA



CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



