Gallery¶

Every pencil type that is known to be realizable, for 1 to 14 lines. Choose the style (Beautiful or Exact, explained below the gallery) and the number of lines; click a drawing to enlarge it, see the exact coefficients of its certificate, and point at an intersection to light up the lines through it.

Points on more lines are drawn larger. Within each drawing the shade also grows with the number of lines, from the lightest to the darkest blue present; the enlarged view lists what each shade means.

Beautiful (the default) shows, where we found one, a symmetric drawing of the type: whole orbits of a symmetric set of lines, taken from regular polygons and their star polygons, from the planes perpendicular to the symmetry axes of the icosahedron and the octahedron, and from plane sections of the mirrors of 4-dimensional root systems (scripts/symmetric_drawings.py). No two lines are parallel, as the problem requires. Such a drawing is a floating-point realization of the same type, checked with the same tolerance as the exact drawing; the arrangement of the certificate need not be a projective image of it. Types without a symmetric drawing are shown exactly.

Exact shows the certified arrangement: a projective image of the certified coordinates, chosen so that all intersection points are visible and well separated; the incidences are exactly those of the certificate. A field such as ℚ(√5) next to a type means that the certificate uses that square root; the field is the one in which the example was found, and only for the pentagram {5, 3¹⁰, 2⁵} is the irrationality known to be necessary. For n = 13 the list is a lower bound: 39 types are still undecided (open problems); for n = 14 only the types found so far are shown.