Generalized Ulam Sets in the Plane

step size N

25

preset vectors

(1,φ),(φ,1)

plot transformations

None

random two-vector

random three-vector

This Demonstration plots a list of points of the generalized Ulam set

U

N

in the plane after

N

steps of iteration. The recursive rule to generate

U

N

is to append vectors to the previous set of minimal Euclidean norms that can be uniquely written as the sum of two distinct vectors from

U

N-1

. The specified list of linearly independent initial vectors completely determines the points.

Set the "step size

N

" for the

U

N

plot, keeping in mind that any setting

N≥100

may be slow.

The list of two or more initial vectors may be specified by selecting one of the preset initial vector combinations or randomly.