The Fish Do Not Shrink

In 1959 M. C. Escher filled a circle with fish.

Circle Limit III is the third of four woodcuts he made between 1958 and 1960. The fish at the rim are tiny, and the gaps between them look tiny too. The natural reading is that the edge holds smaller copies of the fish packed into a smaller space.

It does not. The disk is a picture of a hyperbolic plane, where the fish stay congruent while the Euclidean drawing compresses them toward a boundary that is infinitely far away in the geometry being pictured.

Escher had seen the construction in H. S. M. Coxeter’s 1957 article on crystal symmetry, and in a letter the following year he named the figure that started the work: a tiling by triangles of 30, 45 and 90 degrees, a set of angles no Euclidean triangle can have.

The picture keeps a relation and throws away a measurement. That is precisely what makes it useful.

The disk preserves the arrangement and not the ruler #

The Poincaré disk is not defective because its rim lies. The lie is the instrument.

A bounded Euclidean disk holds an unbounded amount of hyperbolic distance, because distance stretches without limit as the boundary is approached. What looks dense in the drawing need not be dense in the space, and what looks adjacent may be arbitrarily far apart.

Non-Euclidean geometry states the missing ruler rather than hiding it. In the hyperbolic plane the circumference and area of a circle grow exponentially with radius rather than linearly and quadratically, so a circle of moderate radius already exceeds its Euclidean counterpart by orders of magnitude. A triangle’s area is fixed entirely by how far its angles fall short of a straight line.

Those are not decorative corrections to a familiar picture. They say that a length, an angle and an area are claims a reader may only make after the metric and the curvature have been named. A disk that leaves them implicit is a display rather than a measuring instrument.

Topology asks a narrower question on purpose #

Topology gets brought in after the fact because it keeps something solid while a shape bends. The solid thing is not distance.

The Euler characteristic of a triangulated surface does not depend on which triangulation was chosen. A sphere gives two, a torus gives zero, and those counts survive deformations that would render every coordinate and every drawn angle useless. For compact surfaces, orientability, Euler characteristic and boundary components classify the surface completely.

The classification is exact because it declines to say anything about how the surface sits on the page.

Land records apply the same restraint in software. A GIS topology model stores shared nodes, edges and faces, so that two adjoining parcels cannot hold separate coordinate lists for the boundary they have in common. It establishes that two parcels meet along the same edge. It has no opinion about which pair is economically near.

That is the first discipline any map owes a reader: say whether the claim is that two things connect, that they share a boundary, that a loop survives, or that they lie a certain distance apart. The first three are topological. The fourth has already left topology.

The bargain can be struck deliberately #

In 2010 Dmitri Krioukov, Marián Boguñá and Fragkiskos Papadopoulos reported that router-level Internet structure — heavily skewed degree distribution, strong clustering — corresponds naturally to hyperbolic rather than Euclidean geometry.

The operational payoff was greedy routing. A router uses its own coordinates and those of its immediate neighbours to forward a packet, holds no map of the whole network, and the local rule performs close to optimal path-finding.

That is the right geometry chosen for a stated task. It is not permission to read every visual separation as a physical length.

Nickel and Kiela made the same choice for symbolic data in 2017, placing the WordNet noun hierarchy in the Poincaré ball and reporting lower distortion than Euclidean embeddings of the same dimensionality. Hyperbolic space has room for an exponentially branching tree because its circles grow exponentially. A fixed-dimensional Euclidean space does not, and the tree has to be crushed to fit.

The result explains why the map can be good. It does not enlarge the set of questions the map is allowed to answer. Hierarchy, nearest relation and routing survive the embedding while the Euclidean distances visible on the screen remain the wrong quantities to feed a diversification calculation.

What survives the representation #

Persistent homology separates the two states without needing to be told which is which. It builds nested complexes as a scale parameter grows and records when components, loops and voids appear and disappear. A feature that persists across a wide range of scales counts as structure. One that appears and vanishes almost at once counts as noise.

That is a better question to ask a map than whether its clusters have clean outlines, because an outline is an artefact of a projection. A component surviving many scales has passed a structural test. The size of the gap around it still belongs to a metric, and the metric may well have been replaced when the data was flattened.

Here is where the transfer stops.

Escher’s disk has a curvature fixed before the first fish is drawn, so the distance the picture hides is recoverable exactly by anyone who knows what the curvature is. A learned financial embedding takes its metric from the sample and the objective. Change either and the neighbourhood itself can move, not merely the apparent distances within it.

Physical space supplies a ruler that sits outside the representation. The financial map usually has no such thing.

The clusters on the screen are the part everyone looks at, and they were produced by a projection somebody chose.

The native exposure matrix is the part nobody draws, and it is still what decides which positions move together on the day it matters.