# Primitive Relation for Elliptic Geometry

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration shows that the binary relation (the distance of from equals ) is a suitable primitive notion for elliptic geometry. M. Pieri showed that the ternary relation of a point being equally distant from two other points (in symbols, ) can be used as the only primitive notion of Euclidean geometry of two or more dimensions [1]. Pieri's relation can also be used as a primitive defining relation for non-Euclidean geometries.

[more]
Contributed by: Izidor Hafner (April 2018)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

(collinearity)

(perpendicularity)

(distance )

(midpoint)

(external midpoint)

(symmetric)

(Robinson's definition of Pieri's relation [2, pp. 72–73])

References

[1] M. Pieri, "La Geometria Elementare istituita sulle nozioni di punto e sfera," *Memorie di matematica e di fisica della Società italiana delle Scienze*, ser. 3(15), 1908 pp. 345–450.

[2] R. M. Robinson, "Binary Relations as Primitive Notions in Elementary Geometry: The Axiomatic Method with Special Reference to Geometry and Physics," in *Proceedings of an International Symposium Held at the University of California, Berkeley, December 26, 1957–January 4, 1958*, Amsterdam: North-Holland Publishing Company, 1959. doi:10.1017/S0022481200092690.

## Permanent Citation