#### Document Type

Article

#### Publication Date

7-3-2013

#### Publication Title

PloS One

#### Department

Department of Mathematics

#### Additional Department

Department of Computer Science

#### Abstract

In this paper we show how the coupling of the notion of a *network with directions* with the adaptation of the *four-point probe* from materials testing gives rise to a natural geometry on such networks. This *four-point probe geometry* shares many of the properties of hyperbolic geometry wherein the network directions take the place of the sphere at infinity, enabling a navigation of the network in terms of pairs of directions: the geodesic through a pair of points is oriented from one direction to another direction, the pair of which are uniquely determined. We illustrate this in the interesting example of the pages of Wikipedia devoted to Mathematics, or “The MathWiki.” The applicability of these ideas extends beyond Wikipedia to provide a natural framework for visual search and to prescribe a natural mode of navigation for any kind of “knowledge space” in which higher order concepts aggregate various instances of information. Other examples would include genre or author organization of cultural objects such as books, movies, documents or even merchandise in an online store.

#### DOI

10.1371/journal.pone.0067508

#### Dartmouth Digital Commons Citation

Leibon, Gregory and Rockmore, Daniel N., "Orienteering in Knowledge Spaces: The Hyperbolic Geometry of Wikipedia Mathematics" (2013). *Dartmouth Scholarship*. 3026.

https://digitalcommons.dartmouth.edu/facoa/3026