9459

15-Point Projective Space

This Demonstration shows a representation of the smallest projective 3-space, that is, the smallest geometry that satisfies the postulates of incidence and existence of synthetic projective geometry and that can be coordinatized by four homogeneous coordinates. It contains 15 points, 15 planes, and 35 lines. Each line is incident with exactly three points. Each plane is a Fano plane. No particular analytic geometric meaning should be attached to the diagram, although it is useful for visualizing symmetries. This space is known as , the 3D projective space over the field of integers mod 2, 2.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Synthetic projective geometry is a system based on a set of postulates, where points and lines are undefined elements. Postulates of incidence define how points and lines are related, and postulates of existence eliminate trivial cases. It can be shown that the smallest 3-space that satisfies the set of postulates contains 15 points. The smallest projective plane (i.e. 2-space) contains seven points and seven lines and is known as the Fano plane.
Some of the lines are shown as "circles"; one such as CIJ merely means that it is the unique line incident with points C, I, and J.
References
[1] B. Meserve, Fundamental Concepts of Geometry, New York: Dover, 1983.
[2] A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge: Cambridge University Press, 1998.
[3] B. Polster, A Geometrical Picture Book, New York: Springer, 1998.
[4] "Barycentric Coordinates from Cross- and Dot-Products" in "Circumscribed Circle." (Feb 9, 2011) http://en.wikipedia.org/wiki/Circumcenter.
[5] "Parametric Equation" in "Circumscribed Circle." Wikia. (Aug 14, 2009) http://math.wikia.com/wiki/Circumscribed_circle.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+