The orchard problem

StefanA. Burr; Branko Gr�nbaum; N.J.A. Sloane

doi:10.1007/bf00147569

The orchard problem

Geometriae Dedicata ◽

10.1007/bf00147569 ◽

1974 ◽

Vol 2 (4) ◽

Cited By ~ 24

Author(s):

StefanA. Burr ◽

Branko Gr�nbaum ◽

N.J.A. Sloane

Keyword(s):

Orchard Problem

Ordinary hyperspheres and spherical curves

Advances in Geometry ◽

10.1515/advgeom-2020-0031 ◽

2021 ◽

Vol 21 (1) ◽

pp. 15-22

Author(s):

Aaron Lin ◽

Konrad Swanepoel

Keyword(s):

Degenerate Case ◽

Minimum Number ◽

Set Of Points ◽

Orchard Problem

Abstract An ordinary hypersphere of a set of points in real d-space, where no d + 1 points lie on a (d - 2)-sphere or a (d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a (d + 2)-point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of (d + 2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ⩾ 4.

Pólya's Orchard Problem

American Mathematical Monthly ◽

10.1080/00029890.1986.11971759 ◽

1986 ◽

Vol 93 (2) ◽

pp. 98-104 ◽

Cited By ~ 1

Author(s):

Thomas Tracy Allen

Keyword(s):

Orchard Problem

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

Periodica Mathematica Hungarica ◽

10.1007/s10998-017-0224-x ◽

2017 ◽

Vol 77 (2) ◽

pp. 164-174 ◽

Cited By ~ 2

Author(s):

Jürgen Bokowski ◽

Piotr Pokora

Keyword(s):

Pseudoline Arrangements ◽

Orchard Problem