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Geometry Playground gives a canvas to experiment with the variations of postulates on Euclid's Postulates in seven different geometries.
Instructions for the installation and use of Geometry Playground.
- Any two points (.) can be connected with a straight line segment (s).
- Any straight line segment can be extended indefinitely in a straight line (l).
- Given any straight line segment, a circle can be drawn having the segment as the radius and one point as the center (c).
- All Right Angles are congruent.
- Given any straight line and a point not on it, there exists one and only one straight line that passes through that point and is parallel (q) to the first line.
This is the geometry of Euclid, and the postulates are those of Euclid.
Fundamentally, changing Postulate 5 to eliminate the existence of parallel lines. Other implications are explained.
Except that there are no antipodal points in projective geometry, this is similar to spherical geometry. Projective geometry also breaks the "Plane Separation Postulate," an unstated postulate of Euclid's. (He was probably unaware that he was assuming this.)
This geometry assumes all of the postulates of Euclid except for the fifth. In hyperbolic geometry, given any line and point not on it, there are infinitely many straight lines that pass through the point and are parallel to the first line.
This geometry, like projective, breaks an unstated postulate of Euclid concerning how distances are measured, or equivalently, about the invariance of distance under certain movements (isometries, such as reflections).
This is the geometry of the surface of a doughnut, or equivalently, the real plane modulo the integer lattice (as Lie groups). Certain tools are disabled.
This is the orbifold of a standard cone. Certain tools are disabled.