Phi Space

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The set of 3D points given by triplets of numbers of the form , where and are integers and is the golden ratio (~ 1.61803…), forms a lattice that contains most of the regular and Archimedean polyhedra. The snub dodecahedron is one of the exceptions. The lattice is the basis of the Zome construction system, in which the so-called "blue struts" would have length two here. In this Demonstration, and are limited to -1, 0, 1, and the lengths of the segments connecting pairs of points in the lattice are sorted by the number of segments of that length. The Demonstration also computes a polynomial with integer coefficients of minimal degree of which this length is a root. An icosahedron is well hidden within the 1023 lines of length two in the last figure.

Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA



The ordering for the roots of the polynomials takes real roots to come before complex ones, and takes complex conjugate pairs of roots to be adjacent.

C. J. Henrich, "A Look at Zome," 2009.

T. Davis, "The Mathematics of Zome," 2007.

Zometool, Inc.

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