About

The Tribonacci word is a fixed point of the morphism σ defined on the alphabet {a, b, c} by σ(a) = ab, σ(b) = ac, σ(c) = a. It generalizes the Fibonacci word to a three-letter alphabet and exhibits factor complexity C(n) = 2n + 1, making it a Rauzy fractal generator and a canonical example of an episturmian word. Miscounting iteration depth or confusing the morphism rules produces words with incorrect combinatorial properties, invalidating any downstream analysis in tiling theory or numeration systems.

The length of the word at iteration n follows the Tribonacci number sequence T(n), where T(n) = T(n−1) + T(n−2) + T(n−3). This tool constructs the actual word string at each iteration, computes symbol frequencies, and reports growth metrics. Note: the word length grows exponentially (ratio converging to the tribonacci constant β ≈ 1.839286), so iterations beyond 28 produce words exceeding 10 million characters and are capped to prevent browser memory exhaustion.

Formulas

The Tribonacci word is generated by iterated application of a morphism σ over the alphabet {a, b, c}. The substitution rules are:

σ(a) = ab , σ(b) = ac , σ(c) = a

Starting from w₀ = a, each subsequent word is w_n+1 = σ(w_n). The length satisfies the Tribonacci recurrence:

T(n) = T(n − 1) + T(n − 2) + T(n − 3)

with initial conditions T(0) = 1, T(1) = 2, T(2) = 4. The growth ratio converges to the tribonacci constant β, the real root of:

x³ = x² + x + 1

which yields β ≈ 1.839286755214161. The substitution matrix M of σ is:

M = 111100010

Where σ = the morphism (substitution function), w_n = the Tribonacci word at iteration n, T(n) = length of w_n (Tribonacci number), β = tribonacci constant (1.839286...), M = the incidence matrix whose columns encode how many of each letter each rule produces.

Reference Data

Iteration n	Word Length T(n)	Count of a	Count of b	Count of c	Ratio T(n)/T(n−1)
0	1	1	0	0	-
1	2	1	1	0	2.000
2	4	2	1	1	2.000
3	7	4	2	1	1.750
4	13	7	4	2	1.857
5	24	13	7	4	1.846
6	44	24	13	7	1.833
7	81	44	24	13	1.841
8	149	81	44	24	1.840
9	274	149	81	44	1.839
10	504	274	149	81	1.839
11	927	504	274	149	1.839
12	1,705	927	504	274	1.840
13	3,136	1,705	927	504	1.839
14	5,768	3,136	1,705	927	1.839
15	10,609	5,768	3,136	1,705	1.839
16	19,513	10,609	5,768	3,136	1.839
17	35,890	19,513	10,609	5,768	1.839
18	66,012	35,890	19,513	10,609	1.839
19	121,415	66,012	35,890	19,513	1.839
20	223,317	121,415	66,012	35,890	1.839

Frequently Asked Questions

The Fibonacci word uses a 2-letter alphabet {a, b} with morphism σ(a) = ab, σ(b) = a, producing lengths following the Fibonacci sequence. The Tribonacci word extends this to a 3-letter alphabet {a, b, c} with σ(a) = ab, σ(b) = ac, σ(c) = a. Its lengths follow the Tribonacci recurrence T(n) = T(n−1) + T(n−2) + T(n−3), and it generates the Rauzy fractal in ℝ³ rather than the simpler 1D quasiperiodic tiling of the Fibonacci case.

The factor complexity C(n), counting the number of distinct subwords of length n, equals 2n + 1 for all n ≥ 1. This is the minimum possible for a 3-letter aperiodic word, classifying it as an Arnoux-Rauzy (episturmian) word. By comparison, a Sturmian word on 2 letters has complexity n + 1.

The substitution matrix M has characteristic polynomial x³ − x² − x − 1 = 0. Its dominant real root β ≈ 1.839286755214161 is the tribonacci constant. By the Perron-Frobenius theorem, the ratio T(n)/T(n−1) converges to this dominant eigenvalue. The other two roots are complex conjugates with modulus less than 1, so their contribution decays exponentially.

Each character in JavaScript occupies 2 bytes (UTF-16). At iteration 25, the word length is approximately 1.3 million characters (~2.6 MB). At iteration 28, it reaches ~8.1 million characters (~16 MB). Beyond iteration 30 (~27.5 million characters, ~55 MB), most browsers will struggle with string allocation. This tool caps at iteration 28 to prevent tab crashes.

Yes. This tool allows custom morphism rules. For example, setting σ(a) = abc, σ(b) = a, σ(c) = b yields a different substitutive sequence with its own growth rate determined by the dominant eigenvalue of the corresponding 3×3 incidence matrix. The tool recomputes all statistics for any valid custom morphism.

The Rauzy fractal is the projection of the stepped surface (the "broken line" of the Tribonacci word embedded in ℤ³) onto the contracting plane of the substitution matrix M. It is a connected, simply connected compact set that tiles the plane by translation. The fractal has Hausdorff dimension equal to the fractal boundary dimension, approximately 1.0933. The Tribonacci word encodes the symbolic dynamics of the toral rotation associated with this fractal.