Pieri's Ternary Relation and Euclidean 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.

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]. This Demonstration shows Robinson's definition of the quartic relation

[more]

in terms of Pieri's relation. This relation can be used to define relations that a point is between and ; that , , are collinear; and that is the midpoint of .

You can drag the points shown as locators.

[less]

Contributed by: Izidor Hafner (February 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Here are Robinson's definitions [2, pp. 71–72]:

Here, means is between and ; means , and are collinear; and means and are symmetric about (i.e. that is the midpoint of ).

References

[1] M. Pieri, "La Geometria Elementare istituita sulle nozioni di punto e sfera," Memorie di matematica e di fisica della Societ'a 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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send