Subdigons in Geometry and Combinatorics

• Subdigons are polygonal substructures that appear as regular subpolygons, subdivided roofed polygons, or minimal cells in tiling and circle arrangements.
• They are constructed via symmetric chord systems in regular polygons and recursive paneling operations, resulting in exact area relations and generating function identities.
• In circle and tiling arrangements, subdigons serve as local configurations that govern global combinatorial bounds, influencing area ratios and structural parity.

Subdigons denotes several distinct but related objects in recent geometric and combinatorial literature. In one usage, a subdigon is a regular sub-$n$-gon $T$ embedded inside a larger regular $n$-gon $P$, produced by a rotationally symmetric system of chords and often exhibiting integer area divisibility. In another, it is a convex planar roofed polygon subdivided by non-intersecting diagonals, with a type vector recording the numbers of faces of each size; these objects underlie the hyper-Catalan numbers and Wildberger’s polynomial formula. Other papers do not use the term formally, but treat digons or 2-gons as local substructures of circle arrangements or surface tilings, where they play an analogous structural role as minimal embedded polygonal components (Parks, 31 Dec 2025, Rubine, 16 Aug 2025, Ackerman et al., 2024).

1. Terminological scope

The literature records more than one explicit meaning of subdigon. In the chord-construction setting of regular polygons, the term refers to a regular subpolygon with the same number of sides as the ambient polygon. In the algebraic-combinatorial setting of Wildberger and Rubine, it refers instead to a roofed subdivided polygon whose internal faces may have varying sizes but whose outer boundary is a single convex polygon. In arrangements of circles, pseudocircles, and tilings, the term is not standard, yet digons and 2-gons are repeatedly treated as local substructures, boundary components, or minimal cells that govern global combinatorics (Parks, 31 Dec 2025, Rubine et al., 8 Aug 2025, Yan, 2019).

These meanings are not interchangeable. The regular-polygon usage is fundamentally about similarity, rotational symmetry, and area ratios. The roofed-polygon usage is fundamentally about recursive decomposition, multiset specifications, and generating functions. The arrangement-theoretic and tiling-theoretic usages are local and cell-complex based: a digon is a 2-sided face, often a lens, lune, or boundary component of a degenerate tile. A common pattern is that subdigons are treated as embedded polygonal substructures whose constrained local geometry supports exact counting or global structural theorems (Felsner et al., 2022, Ackerman et al., 2024).

2. Regular sub-$n$-gons inside regular $n$-gons

In "Polygons in Polygons with a Twist" (Parks, 31 Dec 2025), a regular $n$-gon $P$ has $n$ equal sides of length $s$, equal interior angles, and center $T$0. A regular sub-$T$1-gon $T$2 is another regular $T$3-gon inside $T$4, obtained purely as the polygon formed by intersections of a system of chords drawn inside $T$5. The chordal system is encoded by $T$6, meaning that from each vertex one draws a chord to a point on a side that is $T$7 sides away along the perimeter, and then rotates this construction by the central angle $T$8. Because the construction is generated by equal rotations, the intersection points form a smaller concentric regular $T$9-gon. A chordal triple is written

$n$0

with $n$1 the area ratio (Parks, 31 Dec 2025).

The central phenomenon is integer area divisibility: $n$2 For a regular $n$3-gon with side length $n$4, the paper quotes

$n$5

If $n$6 and $n$7 are similar regular $n$8-gons with side lengths $n$9 and $P$0, then the paper proves that

$P$1

Thus an integer area ratio fixes the linear scale factor $P$2, and the inner polygon is a scaled copy of the outer one (Parks, 31 Dec 2025).

The paper develops this through explicit examples. For the square triple $P$3, the outer square has side $P$4, the inner square has side $P$5, the inner area is $P$6, and the outer square area is $P$7, so the area ratio is $P$8. For $P$9, the surrounding region can be rearranged into six more hexagons congruent to the central one, yielding seven congruent hexagons in total. For $n$0, the outer octagon decomposes into three congruent octagons. The paper also lists additional triples for squares, hexagons, octagons, and decagons, including $n$1, $n$2, $n$3, $n$4, and $n$5 (Parks, 31 Dec 2025).

Dynamic geometry software is central to this program. Sketchpad or GeoGebra is used to construct $n$6, place a movable point $n$7 on an opposite side, rotate the chord $n$8, measure $n$9, and search numerically for positions of $n$0 that produce integer $n$1. The paper begins with special midpoint-based constructions but argues that a much more general situation exists, especially for legal rotationally symmetric chord systems. Odd $n$2, particularly the pentagon, are reported to be more difficult, with approximate positions and conjectured integer ratios rather than complete exact formulas (Parks, 31 Dec 2025).

3. Roofed subdivided polygons and hyper-Catalan enumeration

In the Wildberger–Rubine framework, a subdigon $n$3 of type $n$4 is a convex planar polygon with a distinguished side called its roof, subdivided by non-intersecting diagonals into $n$5 triangles, $n$6 quadrilaterals, $n$7 pentagons, and so on, with only finitely many $n$8 nonzero. There is also a null subdigon $n$9, consisting of two vertices, one edge, no faces, and type $n$0. The type vector records face counts, and the accounting monomial is

$n$1

For type $n$2, one paper writes

$n$3

satisfying Euler’s relation $n$4 (Rubine et al., 8 Aug 2025).

The recursive structure is encoded by $n$5-ary paneling operators, written either $n$6 or $n$7. Given subdigons $n$8, the object $n$9 is formed by a central roofed $P$0-gon with $P$1 adjoined along their roofs in counterclockwise order. Every non-null subdigon has a unique decomposition of this form. At the level of monomials,

$P$2

and the multiset $P$3 of all subdigons satisfies

$P$4

This yields the functional equation

$P$5

for the subdigon generating series $P$6 (Rubine, 16 Aug 2025).

These objects are in bijection with plane trees with no unary nodes: internal nodes of degree $P$7 correspond to central $P$8-gons, and the null subdigon corresponds to a leaf. Their enumeration is given by the hyper-Catalan numbers. One paper denotes them $P$9; another writes $n$0 and gives the closed form

$n$1

The corresponding generating series is the formal series zero of the geometric polynomial

$n$2

so that $n$3 (Rubine et al., 8 Aug 2025).

A further refinement introduces layering variables $n$4 for vertices, edges, and faces, and studies truncations by level. Vertex layers, edge layers, and face layers are obtained by truncating $n$5 modulo powers of $n$6, $n$7, or $n$8, and the resulting truncated series still satisfy the relevant polynomial equation modulo the chosen level. This finite interpretation converts the formal series identity into a family of finite identities indexed by bounded numbers of vertices, edges, or faces (Rubine et al., 8 Aug 2025).

4. Tubdigons and polynomial identities

"Subdigons" (Rubine, 16 Aug 2025) generalizes the Wildberger–Rubine objects to tubdigons by allowing 2-gons. A tubdigon is identical to a subdigon except that 2-gons are allowed; because 2-gons cannot be drawn with straight edges, the paper uses at least one curved edge for every 2-gon. The corresponding bijection with trees now includes unary nodes. Types are written

$n$9

where $s$0 is the number of 2-gons and $s$1 is the underlying subdigon type (Rubine, 16 Aug 2025).

The tubdigon multiset $s$2 satisfies

$s$3

and the corresponding generating series

$s$4

obeys

$s$5

Rearranged, this becomes

$s$6

which is of the same form as Wildberger’s soft polynomial formula for formal series solutions of general polynomial equations (Rubine, 16 Aug 2025).

The paper counts tubdigons of fixed type in two ways. First, by a stars-and-bars argument, if the underlying subdigon has $s$7 edges, then distributing $s$8 additional 2-gons among those edges gives

$s$9

Second, applying Wildberger’s polynomial formula and expanding via the multinomial theorem expresses the same coefficient as a sum of multinomial coefficients over constrained integer vectors. Equating the two expressions yields Fine’s identity

$T$00

In this derivation, subdigons supply the recursive algebra, tubdigons provide the unary extension needed for 2-gons, and the generating function identity becomes a combinatorial proof of an arithmetic formula (Rubine, 16 Aug 2025).

5. Digons as local substructures in arrangements and tilings

Several papers treat digons as substructures without adopting the term subdigon formally. In arrangements of pairwise intersecting circles, a digon is a face whose boundary consists of exactly two edges, each lying on different circles; the two principal geometric types are lenses and lunes. The paper "On the number of digons in arrangements of pairwise intersecting circles" proves Grünbaum’s conjectured bound for circles: every non-trivial simple arrangement of $T$01 pairwise intersecting circles has at most $T$02 digons. Its proof encodes digons as edges of a geometric graph on circle centers, classifies circles as internal or external, establishes three forbidden edge-configuration lemmas, and uses a spherical doubling trick to construct a planar bipartite graph $T$03 with $T$04 vertices and $T$05 edges, yielding $T$06 (Ackerman et al., 2024).

In arrangements of pairwise intersecting pseudocircles, digons are likewise 2-cells. "Arrangements of Pseudocircles: On Digons and Triangles" proves that if the touching graph contains a triangle, then the number of touchings, equivalently digons after contraction, satisfies $T$07. The same paper constructs arrangements with $T$08 and no triangle in the touching graph, and also shows that for every $T$09 there exists a simple digon-free arrangement with

$T$10

triangles. In that setting, digons are the smallest cells whose presence or absence controls the triangular face count and the global extremal behavior of the arrangement (Felsner et al., 2022).

In tiling theory, 2-gons appear as boundaries of degenerate tiles and as minimal cycles. "Pentagonal Subdivision" does not define subdigons as a formal class, but 2-gons occur as boundary components of degenerate quadrilateral tiles and as embedded regions produced by identifications of edges or vertices. The tile $T$11, for example, becomes a Möbius band and has boundary a 2-gon. These 2-gons affect orientability, bipartiteness, and parity constraints in subdivisible quadrilateral tilings: Proposition 3 states that on an orientable surface each fundamental cycle is represented by an even cycle, while on a non-orientable surface each fundamental cycle is represented by an odd cycle, and Proposition 4 derives the bipartite criterion in the orientable case (Yan, 2019).

6. Structural themes and open directions

Across these literatures, subdigons are controlled by one of three mechanisms: symmetry, recursion, or local forbidden configurations. In regular polygons, rotational symmetry forces the chord intersections onto a concentric regular $T$12-gon and produces exact area relations. In the Wildberger–Rubine program, recursive decomposition by central polygons yields multiset equations and exact generating functions. In circle and pseudocircle arrangements, digons are tracked through center graphs, touching graphs, or cell decompositions, and global bounds follow from restrictions on local patterns. This suggests that the term subdigon is best understood not as a single universal object but as a family of polygonal substructures whose local constraints are unusually rigid (Parks, 31 Dec 2025, Rubine et al., 8 Aug 2025, Ackerman et al., 2024).

The open problems are correspondingly diverse. For regular $T$13-gons, the main unresolved directions include classifying all chordal systems $T$14 that yield integer area ratios, understanding odd $T$15, and determining when iterated constructions produce infinite families of nested subdigons with areas $T$16 of the original polygon (Parks, 31 Dec 2025). For arrangements, the full Grünbaum conjecture for simple arrangements of pairwise intersecting pseudocircles remains open, as do non-simple arrangements of circles and unified spherical forms of the key geometric lemmas (Ackerman et al., 2024). For hyper-Catalan subdigons, finite-level truncations, powers $T$17, and their relations to central polygons, Raney’s lists of words, and typed noncrossing structures remain an active algebraic-combinatorial interface (Rubine et al., 8 Aug 2025).

The resulting picture is technically heterogeneous but conceptually coherent. In every major usage, subdigons are not arbitrary polygons: they are embedded, typed, or recursively generated polygonal units whose constrained incidence structure makes exact formulas possible. Their role ranges from similarity-based Euclidean constructions, to formal series zeros of geometric polynomials, to extremal bounds for 2-cells in arrangements, and to parity-sensitive local structures in surface tilings.