TAOCP 7.1.3 Exercise 162

The previous solution fails because it replaces the actual object in Fig.

taocpmathematicsalgorithmsvolume-4hm-project

Section 7.1.3: Bitwise Tricks and Techniques

Exercise 162. ▶ [HM37] [HM37] Explain how to draw the pentagrid, Fig. 14. What circles are present?

Verified: no
Solve time: 5m19s

The previous solution fails because it replaces the actual object in Fig. 14 with an ideal-pentagon construction whose vertices lie on the boundary circle. The pentagrid is instead the reflection tiling generated by a regular hyperbolic pentagon with interior vertices, i.e., the standard ${5,4}$ Coxeter tiling.

1. Correct geometric model

Work in the Poincaré disk model

$$ D={z\in\mathbb{C}:|z|<1}. $$

Hyperbolic lines are Euclidean circles orthogonal to the unit circle $S^1=\partial D$.

The pentagrid is based on a regular hyperbolic pentagon $P_0\subset D$ such that:

  • every side is a hyperbolic geodesic (a circle orthogonal to $S^1$),
  • five congruent pentagons meet at each vertex,
  • hence each interior angle is

$$ \frac{2\pi}{4}=\frac{\pi}{2}. $$

This determines $P_0$ up to isometry.

2. How to draw the pentagrid

Start with the central pentagon $P_0$.

  1. Draw the 5 geodesic sides of $P_0$ as circular arcs orthogonal to $S^1$.
  2. Reflect $P_0$ in each of its sides (hyperbolic reflections).
  3. Repeat the process for every newly created pentagon.
  4. Continue indefinitely.

This produces a tessellation of the disk by congruent regular pentagons. The union of all edges is the pentagrid.

Formally, the construction is the orbit of $P_0$ under the Coxeter group generated by reflections in its five sides.

3. Structure of the edges

Each edge of the pentagrid is a hyperbolic geodesic segment. Every such geodesic lies on a unique full geodesic line in the disk, i.e., a Euclidean circle orthogonal to $S^1$.

Thus the pentagrid is determined by a discrete family of geodesic lines obtained from the initial configuration by reflection.

4. What circles are present

Only three kinds of circles occur in Fig. 14.

(a) Boundary circle

$$ S^1=\partial D. $$

This is the absolute of the Poincaré disk model.

(b) Five generating geodesic circles

The sides of the initial pentagon $P_0$. Call them

$$ C_0, C_1, C_2, C_3, C_4. $$

Each $C_i$ is a Euclidean circle orthogonal to $S^1$.

(c) All reflection images of the generating circles

Every edge in the pentagrid lies on a geodesic obtained by reflecting one of the $C_i$ repeatedly in sides of pentagons already constructed.

Hence the full set of edge-supporting circles is

$$ \mathcal{C}={g(C_i);:; g \text{ is a finite composition of reflections in } C_0,\dots,C_4,\ i=0,\dots,4}. $$

Each element of $\mathcal{C}$ is a circle orthogonal to $S^1$, and every edge of the pentagrid lies on exactly one such circle.

5. No additional circles

No other circles appear in the figure:

  • Vertices are intersection points of circles in $\mathcal{C}$, not endpoints on the boundary.
  • There are no circles determined by non-adjacent vertices or auxiliary constructions.
  • The geometry is completely generated by the five initial reflection lines.

Thus the entire visible circular structure consists only of:

  • the boundary circle,
  • the infinite reflection orbit of the five side circles.

6. Final description

To draw the pentagrid, one starts from a regular hyperbolic pentagon with interior vertices and repeatedly reflects it in its sides. The resulting tiling fills the Poincaré disk.

The circles present in Fig. 14 are exactly:

  1. The unit circle $S^1$.
  2. The five geodesic circles forming the sides of the initial pentagon.
  3. All circles obtained from these five by reflections in those circles.

Every edge of the pentagrid lies on one of these circles, and no other circles occur.