**Status of project**: PhD awarded 1993

(a) Hyberbolic LOGO (b) Surface LOGO

The purpose of this project was to implement the well-known language LOGO, often used for learning and exploring geometry, for hyperbolic and elliptic geometry first, then for user-defined surfaces in differential geometry. Both are written in C and run on a Sun Workstation.

**Hyperbolic and Elliptic LOGO**

Euclidean geometry is very well known, but it is not the only possible type of geometry. Hyperbolic and Elliptic geometries were both discovered in the 19th century. They are rather difficult to visualise, and this is where the LOGO simulation is useful. We may visualise hyperbolic geometry inside a unit disc in the Euclidian plane. Hyperbolic straight lines generalise to arcs of circles that cut the unit disc at right angles. We may visulaise elliptic geometry inside a Euclidean unit disc in a similar way. Here elliptic lines generalise to arcs of circles that cut the unit disc at the ends of a diameter.

**Surface LOGO**

Surface LOGO is a further extension that allows users to experiment with the geometry on a surface they have defined in three-dimensional space, for example a cone or a sphere. This is more difficult to implement than non-Euclidean LOGO because th turtle must be able to cope with any surface that the user defines. A *geodesic* is the generalisation of a straight line on any surface; it is represented by a nonlinear differential equation. We find that we must solve this equation numerically every time the turtle moves; so surface LOGO needs a great deal of processing power.

In Fig 1(a) we see hyperbolic LOGO, demonstrating that a seven sided of the hyperbolic plane is possible. In Fig 1(b) we see part of the screen of surface LOGO with the turtle ready to explore the sphere.