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Geometry Playground gives a canvas to experiment with the implications of postulates similar to Euclid's in seven different geometries.

Euclidean Geometry: This is the geometry of Euclid, and the postulates are those of Euclid:

1. Any two points (.) can be connected with a straight line segment (s).
2. Any straight line segment can be extended indefinitely in a straight line (l).
3. Given any straight line segment, a circle can be drawn having the segment as the radius and one point as the center (c).
4. All Right Angles are congruent.
5. 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.

Spherical Geometry: In this geometry, any pair of non-antipodal points can be connected with a unique line segment; this is not the case for antipodal points like the N and S poles (Euclid's Postulate 1). Straight lines can still be extended indefinitely (Postulate 2). However, this does not mean that they are infinitely long, as Euclid probably intended: they eventually wrap back atop themselves. In addition, there are no such things as parallel lines in spherical geometry (Postulate 5).

Projective Geometry: 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.)

Hyperbolic Geometry: 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.

Manhattan Geometry: 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).

Toroidal Geometry: 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.

Conical Geometry: This is the orbifold of a standard cone. Certain tools are disabled.