Hyperbolic Geometry

Definition

"In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P that does not intersect l; i.e., that is parallel to l. In hyperbolic geometry there are at least two distinct lines through P which do not intersect l, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid."

The geometry was developed independently by Nikolai Lobachevsky and János Bolyai in the 1820s and 1830s, with Carl Friedrich Gauss having reached similar conclusions privately and declined to publish them.

The Property That Matters

For purposes other than mathematics, one consequence of the axioms does almost all the work: in hyperbolic space, area grows exponentially with radius.

In the Euclidean plane, a disc of radius r has area proportional to r², so doubling the radius quadruples the room available. In the hyperbolic plane, area grows like eʳ. There is exponentially more space a little further out. The standard visual demonstration is the Poincaré disk model, in which the whole infinite hyperbolic plane is squeezed into a finite circle: tilings of identical shapes appear to shrink toward the boundary, but every tile is in fact the same size, and there is no edge to reach.

Hyperbolic Trees and Focus-plus-Context

This exponential room is exactly what a hierarchy needs. A tree in which each node has k children has exponentially many nodes at depth d — the same growth rate as hyperbolic area — which means a tree embeds naturally in the hyperbolic plane and very badly in the Euclidean one.

John Lamping and Ramana Rao exploited this at Xerox PARC in 1995 with the hyperbolic browser, laying out large hierarchies in the hyperbolic plane and projecting them into the Poincaré disk. The result is a focus-plus-context display: whatever is dragged to the centre is shown in detail, while everything else remains visible, compressed toward the rim, and can be brought to the centre by a smooth transformation. Nothing is ever hidden or scrolled away.

Hyperbolic Social Relations

The same growth argument applies to social networks, and this is where the geometry becomes anthropologically interesting.

The number of people reachable at social distance d — friends, friends of friends, and so on — grows roughly exponentially in d, while the number of people to whom any individual is close remains small and roughly constant. That is the signature of hyperbolic rather than flat geometry. Dmitri Krioukov and colleagues formalised this in 2010, showing that the scale-free degree distributions and strong clustering observed in real networks arise naturally if the network is generated by connecting nodes that are near each other in an underlying hyperbolic space, where distance combines similarity with popularity.

The everyday consequence is a familiar social experience with a precise geometric description. Each person sits at the centre of their own disk, and everyone else is compressed toward the rim: an acquaintance two steps away is barely distinguishable from one ten steps away, even though the population at those two distances differs by orders of magnitude. This is why the world feels simultaneously small — any two people are joined by a short path — and impossibly large: the number of people at each additional step explodes, so almost everyone is crowded into the same peripheral band of near-invisibility.

Related Reading

• The Social Logic of Space
• Space Syntax
• Ambient Intimacy
• Persistent Paleontology