Tribonacci Constant

The tribonacci constant is the limiting ratio of successive terms of the tribonacci sequence. It is the unique real root of and is given by

			(1)
			(2)
			(3)

(OEIS A058265).

The tribonacci constant satisfies the identities

			(4)
			(5)
			(6)
			(7)

(P. Moses, pers. comm., Feb. 21, 2005).

The tribonacci constant is extremely prominent in the properties of the snub cube.

The constant also describes a division of a line segment of length 1 into 3 parts in continuous proportion (Mustonen 2005, p. 2). The lengths are , , and , whose sum is 1. This is the case of the geometric interpretation of the limiting ratios of Fibonacci n-step numbers.

This division also gives a three-piece refinement of the unit interval. At each stage, replace every longest interval by three consecutive subintervals with relative lengths , , and , leaving all shorter intervals unchanged. Starting from the unit interval at stage 0, the distinct lengths at stage are exactly , , and . Indeed, replacing length creates lengths , , and , while the other lengths persist. Thus exactly three sizes coexist after the first refinement. This is the , instance of the general size bound for largest-first substitution tiling rules, and generalizes the Kakutani partition construction.