Polyhedral billiards

Closed billiard paths in the regular tetrahedron

We start with one exact path. A point moves in straight segments inside the tetrahedron, reflects from face to face, and returns to where it started after four bounces. From that one example, the unfolding trick gives a practical calculation, and then the same language points toward what should happen at edges and vertices.

Start with the path Jump to singular paths

A period-four closed path

The red orbit has face word DABC. It hits face D, then face A, then B, then C, and closes up. The stratum notation introduced below keeps this ordinary face word exactly as it is.

Closed path DABC

Word: DABC
View: folded tetrahedron
Path: closed broken line

The first red segment stays in the original tetrahedron. Each click reflects the remaining tail across the next hit face, and the last copy carries the segment identified with the first.

The path

Label the vertices of the regular tetrahedron by A,B,C,D, and write a point in barycentric coordinates relative to those vertices. The face with barycentric coordinate d=0 is denoted by the same letter D. The period-four path above has these four hit points:

\begin{aligned} x_0 &= \frac{1}{10}(3A+4B+3C) &&\in F_D,\\ x_1 &= \frac{1}{10}(3B+4C+3D) &&\in F_A,\\ x_2 &= \frac{1}{10}(3A+3C+4D) &&\in F_B,\\ x_3 &= \frac{1}{10}(4A+3B+3D) &&\in F_C. \end{aligned}

Thus x_0 lies on face D because the coefficient of vertex D is zero, while the other three coefficients are positive. Joining x_0,x_1,x_2,x_3 cyclically gives a broken line with four equal segments, so its total length is \(4\sqrt{5}/5\).

At each hit, the incoming and outgoing unit directions differ by a multiple of the face normal:

\frac{x_{i+1}-x_i}{\lVert x_{i+1}-x_i\rVert} - \frac{x_i-x_{i-1}}{\lVert x_i-x_{i-1}\rVert} \in \mathbb{R}\, n_{F_i}, \qquad i \in \mathbb{Z}/4\mathbb{Z}.

This is just the equal-angle reflection law written as a vector equation. The path is ordinary because every bounce is in the relative interior of a face.

The unfolding trick

Reflection at a face can be moved from the velocity to the room. When the path hits a face, reflect the next copy of the tetrahedron across that face and let the point continue straight. If the folded path has face word \(F_0F_1\cdots F_{k-1}\), the unfolded picture is a straight segment through copies

T_0=T,\qquad T_{i+1}=R_{F_i}(T_i),

where \(R_{F_i}\) is reflection in the hit face of the current copy. Folding all the reflected copies back to \(T\) recovers the original billiard path. In the unfolded picture there are no angle conditions left: a candidate orbit is found by asking for one line to pass through the prescribed faces in order.

Finding the example

For the word DABC, unfold four copies. In the \(i\)-th copy, the straight line must meet the reflected face labeled by the \(i\)-th letter. If \(q(s)=u+s(v-u)\) is the line in unfolded space and \(0=s_0<s_1<s_2<s_3<s_4=1\), the unknowns are constrained by

q(s_i)\in R_{F_{i-1}}\cdots R_{F_0}(F_i), \qquad q(s_4)=R_{F_3}R_{F_2}R_{F_1}R_{F_0}(q(s_0)).

These are linear incidence equations once the face word is fixed. Solving them for DABC, then folding the hit points back to the original tetrahedron, gives the barycentric matrix

\frac{1}{10} \begin{pmatrix} 3 & 4 & 3 & 0\\ 0 & 3 & 4 & 3\\ 3 & 0 & 3 & 4\\ 4 & 3 & 0 & 3 \end{pmatrix}.

Each row has exactly one zero, and the zeroes occur in the order D,A,B,C. That is the face word. The same computation, repeated over many canonical face words, produces the inventory.

Open the full inventory Ordinary data

Why ordinary paths avoid singular points

The calculation above only used face-interior hits. That is deliberate. At a face there is one tangent plane and one normal direction, so reflection is unambiguous. At an edge there are two incident faces. At a vertex there are three. A literal billiard ball striking such a point has no single face normal to reflect across.

One option would be to throw these paths away. Another would be to choose an incident face by hand. Neither feels intrinsic to the tetrahedron. The normal-cone rule keeps the variational reflection law, but replaces the single face normal by the cone generated by all active face normals.

A rule for singular paths

The normal cone is one ray in a face interior, a wedge along an edge, and a trihedral cone at a vertex. The rule below agrees with ordinary reflection on faces, and still has a precise meaning on the nonsmooth boundary.

Smooth face reflection

Active faces: 1
Normal cone: one ray
Stationary jump: supported by the face normal

Red segments are incoming and outgoing directions. Amber geometry is the admissible normal cone.

The normal-cone condition

At a boundary point \(x\), let \(N_P(x)\) be the cone generated by the outward normals of all faces containing \(x\). In a face interior this cone is a single ray. On an edge it is generated by the adjacent face normals. At a vertex it is generated by the normals of all incident faces.

v_{\mathrm{out}} - v_{\mathrm{in}} \in N_P(x).

Equivalently, the broken path is stationary for length under variations that keep each bounce on its assigned face, edge, or vertex stratum. The momentum jump has no allowed tangential component; it is carried by the active boundary constraints.

Stratum words

The notation records the smallest boundary stratum containing each bounce. The clean convention is to let A,B,C,D denote the four faces, then write a token as the set of active faces at the bounce. A one-letter token is an ordinary face hit. A two-letter token is an edge. A three-letter token is a vertex.

More generally, this is the natural notation for simple polytopes, including every simplex. In dimension \(n\), a vertex of a simple polytope is the transverse meeting point of exactly \(n\) facets, and no more. A token with \(r\) face labels names a codimension-\(r\) stratum.

Token	Stratum	Barycentric test
`D`	Face `F_D`	`d = 0`, while `a,b,c > 0`
`BC`	Edge `F_B \cap F_C`, joining vertices `A` and `D`	`b = c = 0`, while `a,d > 0`
`ABC`	Vertex `F_A \cap F_B \cap F_C`, namely vertex `D`	`a = b = c = 0`, while `d > 0`

Two ways to arrive at the same rule

Round the tetrahedron

Replace an edge by a smooth strip, or a vertex by a smooth cap. Ordinary reflection applies on every rounded surface. As the rounding shrinks, the normal direction converges into the normal cone of the sharp tetrahedron.

Stay inside the tetrahedron

Keep the tetrahedron fixed and follow ordinary closed paths whose face hits approach the boundary of their faces. In barycentric coordinates, this is exactly the process of additional coordinates tending to zero.

Rounding the sharp point

On a rounded edge or vertex there is again a unique tangent plane at the bounce. As the rounding radius goes to zero, the smooth normal approaches a direction inside the normal cone.

Rounding an edge

Limit stratum: edge
Radius: decreasing
Reflection: ordinary on the rounded surface

Sharp limit

The teal normal of the rounded surface converges into the amber normal cone. The red incoming and outgoing segments keep a meaningful limiting bounce.

Let ordinary hits approach a stratum

This scene keeps the tetrahedron fixed. The red face-interior hits move toward the boundary of their faces. In the limit, the same geometric path is recorded by an edge or vertex stratum rather than by an arbitrary incident face.

Approaching opposite edges

Limit stratum: BC AD
Margin: decreasing
Tetrahedron: fixed

Boundary margin

The amber object is the singular limit. In edge mode, two pairs of ordinary face hits merge onto the opposite edges with active-face tokens BC and AD. In vertex mode, ordinary hits on incident faces collapse to the vertex token ABC.

How the singular inventory is found

The singular search enumerates primitive cyclic words in the fourteen boundary strata: four faces, six edges, and four vertices. Consecutive tokens are required to have disjoint active face sets, so the open segment between two bounces lies inside the tetrahedron rather than along a face.

For each stratum word, the algorithm minimizes the polygonal length over the product of the assigned closed strata. A relative-interior minimizer is then checked against the normal-cone equation at every bounce and reduced modulo cyclic shift, reversal, and tetrahedral relabeling. The published singular file is exhaustive through period 7; it contains 15 exact rational representatives and 24 numeric normal-cone representatives.

Singular data

How it appears in the inventory

The inventory display uses this active-face convention. A word like A BC D BC means a face hit, then an edge hit, then another face hit, then the same edge stratum again.

Row	Word	What it shows
`p04_DABC`	`DABC`	The basic period-4 face orbit, unfoldable through reflected tetrahedra.
`s02_AB_CD`	`AB CD`	A symmetric edge-to-opposite-edge path with total length `2`.
`s04_A_BC_D_BC`	`A BC D BC`	A one-parameter normal-cone family whose length is constant.

What the convention does not claim

A physical ball striking a perfectly sharp edge need not have a deterministic outgoing direction. The inventory is making a geometric and spectral choice: include the closed broken geodesics that are stationary under the constraints of the polyhedron, especially when they arise as stable limits of ordinary billiards on smooth approximations.