Two star polygons appear in Danny Carey’s visual vocabulary, and they are not the same star. One is open, almost friendly — seven points connected by relatively shallow angles. The other is sharper, the points more acute. They look like variations on a theme, which is accurate: both are drawn on seven equally spaced vertices, but one connects every second vertex and the other connects every third.
In Schläfli notation — the system for naming regular star polygons — these are $\{7/2\}$ and $\{7/3\}$ [1]. Both appear in Tool’s artwork, in Thelemic symbolism, in medieval Islamic geometric patterns, and on the floor plans of cathedrals. They are the most visually intricate star polygons that can be drawn in a single closed stroke before the figure becomes illegible.
Both of them have a property that five-pointed and six-pointed stars do not share: they visit every vertex before closing. This is a consequence of 7 being prime. And it turns out to matter for how rhythmic accent cycles are built.
The Schläfli Symbol
A regular star polygon $\{n/k\}$ is constructed by placing $n$ points evenly on a circle and connecting every $k$-th point in sequence until the path closes. The structural key is a single number:
$$d = \gcd(n, k).$$If $d = 1$, the traversal visits all $n$ vertices before returning to the start — a single connected figure. If $d > 1$, the path visits only $n/d$ vertices before closing, and the full figure consists of $d$ separate copies of the smaller star $\{(n/d)\,/\,(k/d)\}$.
The most familiar example of the disconnected case: $\{6/2\}$, the Star of David. Here $\gcd(6,2) = 2$, so the figure breaks into two copies of $\{3/1\} = \{3\}$ — two overlapping equilateral triangles. The traversal starting at vertex 1 visits $1 \to 3 \to 5 \to 1$, leaving vertices 2, 4, 6 entirely unvisited.
The pentagram $\{5/2\}$ is connected: $\gcd(5,2)=1$, traversal $1 \to 3 \to 5 \to 2 \to 4 \to 1$, all five vertices.
For $n=7$:
- $\{7/2\}$: $\gcd(7,2)=1$, traversal $1 \to 3 \to 5 \to 7 \to 2 \to 4 \to 6 \to 1$, all seven vertices.
- $\{7/3\}$: $\gcd(7,3)=1$, traversal $1 \to 4 \to 7 \to 3 \to 6 \to 2 \to 5 \to 1$, all seven vertices.
Both connected. Neither leaves any vertex unvisited.
The Group Theory
The traversal of $\{n/k\}$ is an instance of a standard construction in modular arithmetic: the orbit of an element under repeated addition in $\mathbb{Z}/n\mathbb{Z}$.
Label the $n$ vertices $0, 1, \ldots, n-1$. Starting at vertex 0, the traversal visits:
$$0, \quad k \bmod n, \quad 2k \bmod n, \quad 3k \bmod n, \quad \ldots$$The orbit of 0 under the action of $+k$ is the subgroup of $\mathbb{Z}/n\mathbb{Z}$ generated by $k$. By a standard result, this subgroup has size $n / \gcd(n,k)$.
- When $\gcd(n,k) = 1$: orbit size $= n$. The traversal visits every vertex.
- When $\gcd(n,k) = d > 1$: orbit size $= n/d$. The traversal visits only a fraction of the vertices.
For prime $n$: $\gcd(n,k) = 1$ for every $1 \leq k \leq n-1$, without exception. Every traversal is complete. There is no step size that traps the path in a proper sub-orbit before visiting all vertices. This follows directly from the fact that a prime has no divisors other than 1 and itself, so $\mathbb{Z}/p\mathbb{Z}$ has no non-trivial subgroups (Lagrange’s theorem: any subgroup of a group of prime order must have order 1 or $p$).
This is the specific property that makes 7 — and any prime — rhythmically fertile.
The Contrast with Six
The comparison with $n = 6$ is the clearest illustration.
In $\mathbb{Z}/6\mathbb{Z}$, the possible step sizes are 1, 2, 3, 4, 5. Their orbits:
| Step $k$ | $\gcd(6,k)$ | Orbit size | Vertices visited |
|---|---|---|---|
| 1 | 1 | 6 | 0,1,2,3,4,5 (the hexagon) |
| 2 | 2 | 3 | 0,2,4 only |
| 3 | 3 | 2 | 0,3 only |
| 4 | 2 | 3 | 0,2,4 only |
| 5 | 1 | 6 | 0,5,4,3,2,1 (the hexagon reversed) |
The only step sizes that visit all six vertices are 1 and 5 — both of which just traverse the hexagon itself, not a star. Every non-trivial star polygon on six points gets trapped. $\{6/2\}$ visits only half the vertices. $\{6/3\}$ visits only two. There is no connected six-pointed star that isn’t either the hexagon or a compound figure.
In $\mathbb{Z}/7\mathbb{Z}$, every step from 2 to 5 generates the full group:
| Step $k$ | $\gcd(7,k)$ | Orbit size | Traversal |
|---|---|---|---|
| 2 | 1 | 7 | 1,3,5,7,2,4,6 |
| 3 | 1 | 7 | 1,4,7,3,6,2,5 |
| 4 | 1 | 7 | 1,5,2,6,3,7,4 |
| 5 | 1 | 7 | 1,6,4,2,7,5,3 |
All four non-trivial step sizes give connected traversals. Both are stars. Both visit every vertex. This is not a coincidence: it is the algebraic signature of primality.
From Geometry to Rhythm
The connection to drumming is direct. Here is the mechanism.
Consider a repeating rhythmic figure of 7 beats — a bar of 7/8, say, with positions 1 through 7. An earlier post discussed Euclidean rhythms: the algorithm that distributes $k$ onset positions as evenly as possible among $n$ slots. That is a problem of selection — which of the $n$ positions to activate.
The star polygon traversal asks a different question. Given that all $n$ positions are present, in what order of emphasis should they be related, such that each accent is a fixed distance from the last? The traversal of $\{n/k\}$ answers this: accent position $1$, then $1+k$, then $1+2k$, and so on modulo $n$.
For $\{7/2\}$: the accent cycle within a single bar runs $1 \to 3 \to 5 \to 7 \to 2 \to 4 \to 6$. Each featured beat is two positions ahead of the last.
Now project this across multiple bars. In bar 1, the primary accent sits on beat 1. In bar 2, if the accent shifts by 2, it lands on beat 3. Bar 3: beat 5. Bar 4: beat 7. Bar 5: beat 2. Bar 6: beat 4. Bar 7: beat 6. Bar 8: beat 1 again.
The accent takes seven bars to return to its starting position. Because $\gcd(2,7) = 1$, the step of 2 generates all of $\mathbb{Z}/7\mathbb{Z}$: every beat position receives the accent exactly once before the cycle resets. The resulting large-scale figure is $7 \times 7 = 49$ beats long — a super-phrase built from a single local rule.
The $\{7/3\}$ traversal generates the same exhaustiveness with a different path. Step 3 gives $1 \to 4 \to 7 \to 3 \to 6 \to 2 \to 5$: a seven-bar accent cycle that visits every position before repeating, but with wider spacing between accented beats, creating a different feel over the same underlying meter.
A six-beat figure with step 2 cannot do this. The accent visits only beats 1, 3, 5 — half the cycle — and loops back without touching beats 2, 4, 6. A drummer building phrase-level architecture from a six-beat grid is working with a more fragmented material.
Two Problems, One Prime
It is worth stating the relationship between the star polygon approach and Euclidean rhythms precisely, because the two are sometimes conflated [2].
The Euclidean algorithm distributes $k$ onsets among $n$ positions with maximal evenness. The result is a subset of the $n$ positions — a selection. The primality of $n$ matters here too: because $\gcd(k,p) = 1$ for prime $p$ and any $1 \leq k \leq p-1$, the Euclidean rhythm $E(k,p)$ always achieves its theoretical maximum of evenness. There are no divisibility shortcuts that cause clumping.
The star polygon traversal selects no subset — it relates all $n$ positions via a cyclic permutation. The primality of $n$ matters here because it guarantees that every non-trivial cyclic permutation (every step size $k$ with $1 < k < n$) generates the full group, visiting all positions before repeating.
Same arithmetic property — $\gcd(k,p) = 1$ for all non-zero $k$ — but the two problems ask different things of it. Euclidean rhythms use it to guarantee dense coverage. Star polygon traversals use it to guarantee no sub-orbit trapping.
The Compound Structure
Written out explicitly, the $\{7/2\}$ accent pattern over seven bars looks like this — with bold marking the featured beat in each bar:
$$\begin{array}{rccccccc} \text{bar 1:} & \mathbf{1} & 2 & 3 & 4 & 5 & 6 & 7 \\ \text{bar 2:} & 1 & 2 & \mathbf{3} & 4 & 5 & 6 & 7 \\ \text{bar 3:} & 1 & 2 & 3 & 4 & \mathbf{5} & 6 & 7 \\ \text{bar 4:} & 1 & 2 & 3 & 4 & 5 & 6 & \mathbf{7} \\ \text{bar 5:} & 1 & \mathbf{2} & 3 & 4 & 5 & 6 & 7 \\ \text{bar 6:} & 1 & 2 & 3 & \mathbf{4} & 5 & 6 & 7 \\ \text{bar 7:} & 1 & 2 & 3 & 4 & 5 & \mathbf{6} & 7 \\ \end{array}$$Each bar is metrically identical. The large-scale accent — which beat carries the phrase-level emphasis — traces the traversal path of the $\{7/2\}$ star polygon across the seven-bar cycle.
This is the kind of large-scale rhythmic architecture visible in a great deal of Tool’s output. Whether Danny Carey explicitly constructs accent cycles from star polygon traversal paths, or whether the same structure emerges from his intuitive sense of how prime time signatures behave, produces the same result. The mathematics and the musical instinct point toward the same pattern.
Why the Heptagram
The full mathematical picture of why seven-fold symmetry is special — why the regular heptagon cannot be constructed by compass and straightedge, what the minimal polynomial of $\cos(2\pi/7)$ implies about the heptagon’s position outside the constructible world, and how the Galois group of the cyclotomic field over $\mathbb{Q}$ carries the obstruction — is developed in the companion post The Impossible Heptagon.
The short version, for the purposes of this post: seven is the smallest odd prime that is not a Fermat prime ($2^{2^j}+1$). This algebraic accident places it outside the reach of ruler-and-compass construction — the heptagon exists as an ideal but cannot be manifested by the classical tools. Its star polygons are the accessible shadows of an inaccessible form. And its primality, in both the constructibility sense and the traversal sense, is precisely what makes it inexhaustible as a rhythmic resource.
The Fibonacci structure in “Lateralus” [3], the group theory underlying twelve-tone equal temperament [4], and the Euclidean rhythm algorithm [5] are all different facets of the same observation: mathematical structure, introduced as compositional constraint, generates musical complexity that cannot easily be produced by intuition alone. The star polygon is another instance. The drummer who keeps a heptagram on his kit has found, by a non-mathematical route, an object with a precise and interesting mathematical identity.
References
[1] Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover. Ch. 2.
[2] Toussaint, G. (2013). The Geometry of Musical Rhythm: What Makes a “Good” Rhythm Good? CRC Press.
[3] See Fibonacci and Lateralus on this blog.
[4] See Twelve-TET and Group Theory on this blog.
[5] See Euclidean Rhythms on this blog.