Mathematics of Escher's Circle Limit

What is Circle Limit?

M. C. Escher’s Circle Limit series (I–IV) fills a disk with fishes, bats, or angels that shrink toward the boundary. The figures meet edge-to-edge without gaps, yet never leave the circle. That picture is not Euclidean: it is a drawing of the hyperbolic plane in the Poincaré disk model.

To recreate the art, you need four layers of mathematics:

1. Hyperbolic geometry (what space you are drawing in)
2. The Poincaré disk (how that space sits in the unit disk)
3. Regular tessellations $\{p,q\}$ (how tiles fit together)
4. Discrete isometry groups (how to generate infinitely many copies of a motif)

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Euclidean Geometry vs. Hyperbolic Geometry

Euclidean geometry rests on the parallel postulate: through a point not on a line $\ell$, there is exactly one line parallel to $\ell$.

Hyperbolic geometry replaces that axiom.

Consequences that matter for Circle Limit:

• The sum of angles in a triangle is strictly less than $\pi$.
• Area is proportional to angle defect: for a triangle with angles $\alpha,\beta,\gamma$,

$$\operatorname{Area}(\triangle) = K\bigl(\pi - (\alpha+\beta+\gamma)\bigr),$$

where $K > 0$ depends on the curvature normalization (often $K = 1$).

• There exist regular $n$-gons with arbitrarily small interior angles, so you can pack more than four squares around a vertex — which Euclidean geometry forbids.

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Models of the Hyperbolic Plane

Several models realize the same abstract geometry. For Escher-style art, the Poincaré disk is the natural choice.

Model	Underlying set	Geodesics look like
Poincaré disk	Open unit disk $\mathbb{D}$	Circular arcs orthogonal to the boundary, or diameters
Klein–Beltrami	Open unit disk	Straight Euclidean chords
Upper half-plane	$\{z : \operatorname{Im} z > 0\}$	Vertical rays and semicircles on the real axis
Hyperboloid	Sheet of $x^2+y^2-z^2=-1$	Intersections with planes through the origin

Klein makes straight lines easy but distorts angles. The Poincaré disk is conformal (angles are true), which is why Escher’s motifs keep recognizable shapes near the center and only shrink near the rim.

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The Poincaré Disk

Distance

Integrating the metric along a geodesic between $z,w \in \mathbb{D}$ gives

$$d_{\mathbb{D}}(z,w) = \operatorname{acosh}\!\left( 1 + \frac{2|z-w|^2}{(1-|z|^2)(1-|w|^2)} \right) = 2\operatorname{atanh}\left|\frac{z-w}{1-\bar{z}w}\right|.$$

As $|z| \to 1$, Euclidean distance to the boundary stays finite, but hyperbolic distance to the origin

$$d_{\mathbb{D}}(0,z) = \log\frac{1+|z|}{1-|z|}$$

tends to infinity. That is why motifs shrink toward the circle: equal hyperbolic size becomes tiny Euclidean size near $\partial\mathbb{D}$.

Geodesics

Two circles $|z-a|=r$ and $|z|=1$ are orthogonal when $|a|^2 = r^2 + 1$. Solving for an arc through two interior points $z_1,z_2$ is the basic drawing primitive for hyperbolic polygons.

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Isometries: Möbius Transformations of the Disk

Orientation-preserving isometries of $\mathbb{D}$ are the Möbius transformations that map $\mathbb{D}$ to itself:

$$f(z) = e^{i\theta}\,\frac{z-a}{1-\bar{a}z}, \qquad a \in \mathbb{D},\ \theta \in \mathbb{R}.$$

They form the group $\operatorname{PSU}(1,1) \cong \operatorname{PSL}(2,\mathbb{R})$.

Including reflections (anti-holomorphic maps $z \mapsto e^{i\theta}\overline{\frac{z-a}{1-\bar{a}z}}$) gives the full isometry group. Circle Limit compositions are typically generated by reflections in the sides of a fundamental hyperbolic polygon (a Coxeter / kaleidoscopic construction).

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Regular Tessellations $\{p,q\}$

A regular tessellation $\{p,q\}$ places regular $p$-gons so that $q$ of them meet at each vertex.

In Euclidean geometry the only possibilities are

$$\{3,6\},\quad \{4,4\},\quad \{6,3\}$$

(triangular grid, square grid, hexagonal grid).

Almost all integer pairs $(p,q)$ are hyperbolic. Classic Circle Limit choices:

Work	Tessellation	Motif idea
Circle Limit I	related to $\{6,4\}$ / fish along geodesics	fish
Circle Limit III	$\{8,3\}$ (with hyperbolic “white lines”)	fish
Circle Limit IV	$\{6,4\}$	angels and demons

For a regular hyperbolic $p$-gon with $q$ at a vertex, each interior angle is $2\pi/q$. The hyperbolic area of one tile is

$$\operatorname{Area} = (p-2)\pi - p\cdot\frac{2\pi}{q} = \pi\left(p - 2 - \frac{2p}{q}\right).$$

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From One Tile to the Whole Disk

Fundamental domain

Choose a closed hyperbolic polygon $F \subset \mathbb{D}$ (often a triangle or a regular $\{p,q\}$ tile, or a union of them that carries the art motif). The discrete group $\Gamma$ of isometries generated by reflections in the sides of $F$ (or by suitable rotations/translations) should act so that the copies

$$\{\, g(F) : g \in \Gamma \,\}$$

cover $\mathbb{D}$ without overlapping interiors. Then $F$ is a fundamental domain.

Coxeter triangle (kaleidoscope)

Many Circle Limit patterns reduce to a hyperbolic triangle with angles

$$\frac{\pi}{\ell},\ \frac{\pi}{m},\ \frac{\pi}{n} \qquad\text{where}\quad \frac{1}{\ell}+\frac{1}{m}+\frac{1}{n} < 1.$$

Reflecting across the three sides generates the triangle group $(\ell,m,n)$. Coloring alternating reflected copies — or drawing a motif inside a larger fundamental domain assembled from several triangles — produces the interlocking figures.

Algorithmic recipe

A practical generation loop for code (Manim, SVG, canvas, …):

1. Build one seed polygon in the Poincaré disk: compute the Euclidean centers and radii of $p$ orthogonal arcs that form a regular $\{p,q\}$ tile centered at $0$ (or a Coxeter triangle).
2. Place a motif inside the seed (curves, fish outline, …) in hyperbolic coordinates, or draw it in the Euclidean chart knowing that only the hyperbolic structure is preserved by later maps.
3. Generate group elements by composing reflections in the sides (BFS / queue over words in the generators), discarding duplicates up to a depth or until tiles become smaller than a pixel.
4. Apply each $g \in \Gamma$ to every edge and motif curve: for a Möbius map $f$, send sample points $z$ to $f(z)$ (or $f(\bar{z})$ for orientation-reversing maps).
5. Stop near the boundary when Euclidean diameter of a tile is below a threshold; the infinite hyperbolic tiling becomes a finite drawable set.

Inversion in a circle of center $c$ and radius $r$ (the Euclidean description of reflection in a hyperbolic geodesic) is

$$z \mapsto c + \frac{r^2}{\overline{z-c}}.$$

Composing such inversions implements the reflection generators without leaving the complex plane.

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Ideal Vertices and Circle Limit III

Some tilings use ideal polygons, whose vertices lie on $\partial\mathbb{D}$. An ideal triangle has all three vertices at infinity and angles $0$; its area is $\pi$.

In Circle Limit III, the white circular arcs are not the tile edges themselves but equidistant curves (hypercycles) relative to the underlying $\{8,3\}$ skeleton — a reminder that the decorative lines in Escher’s print need not coincide with the geodesics of the tiling group. When recreating a specific print, separate:

• the symmetry group (which copies of the fish are congruent),
• the drawn curves (geodesics, horocycles, or hypercycles).

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Complex Analysis Viewpoint

The group of biholomorphic automorphisms of $\mathbb{D}$ is exactly

$$\operatorname{Aut}(\mathbb{D}) = \left\{ z \mapsto e^{i\theta}\frac{z-a}{1-\bar{a}z} \right\}.$$

Thus Circle Limit art is also a visualization of a Fuchsian group: a discrete subgroup $\Gamma \subset \operatorname{Aut}(\mathbb{D})$ whose quotient $\mathbb{D}/\Gamma$ is a hyperbolic orbifold (often a sphere with cone points, or a higher-genus surface). Drawing one fundamental domain and painting all $\Gamma$-translates is the same operation as unfolding that orbifold back onto the disk.

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Minimal Checklist for Making the Art

Ingredient	Role
Poincaré metric $ds = 2\|dz\|/(1-\|z\|^2)$	Explains shrinking toward the rim
Orthogonal circular arcs	Edges of hyperbolic polygons
Condition $(p-2)(q-2) > 4$	Chooses a hyperbolic $\{p,q\}$
Disk automorphisms / circle inversions	Copy the motif across the tiling
Finite truncation by Euclidean size	Makes the infinite pattern drawable

With those pieces, “Escher’s circle” is no longer a mysterious illustration: it is the Poincaré disk, tiled by a discrete hyperbolic symmetry group, with a motif painted on a fundamental domain and propagated by isometries.