Why should spacetime be Euclidean? Because you learned 2-D plane geometry in high school? Euclid won't even work on the surface of the Earth (surveying, navigation). You have a choice of eight fundamental 3-D geometries,
WP Thurston, "Three-dimensional geometry and topology," Vol. 1. Princeton Mathematical Press, Princeton, NJ, 1997
Given a nice laser and two mirrors. You make an arbitrary optical triangle with sides each a few million miles long in free fall orbiting the sun. The laser shoots a light beam to the first plane mirror, it reflects to the second plane mirror, and that reflects back to the laser. You add up the sum of the interior angles of the triangle. Three possiblities:
1) The sum is exactly 180 degrees. Plane (Euclid) geometry. The circumference divided by the diameter of a circle is exactly pi. Given a point not on a given line, there can be drawn only ONE line through that point parallel to the given line. Flat sheet of paper.
2) The sum is more than 180 degrees. Elliptic (Bolyai-Lobechevsky) geometry. The circumference divided by the diameter of a circle is less than pi. Given a point not on a given line, there can be drawn ZERO lines through that point parallel to the given line. Surface of the Earth.
3) The sum is less than 180 degrees. Hyperbolic (Riemann) geometry. The circumference divided by the diameter of a circle is more than pi. Given a point not on a given line, there can be drawn an INFINITE number of lines through that point parallel to the given line. Surface of a nuclear power plant cooling tower.
If you do the laser experiment, (2) not (1) will obtain. This can be modeled as gravitation arising from either spacetime curvature (Einstein, metric theories of gravitation) or spacetime torsion (Weitzenböck, affine/teleparallel theories of gravitation). Spacetime curvature and spacetime torsion give identical predictions for all gravitational phenomena at all scales. However, they are measurably different based on their assumptions.
http://arxiv.org/abs/gr-qc/0505092
http://arxiv.org/abs/gr-qc/0412034
Metric gravitation postulates the Equivalence Principle: all local bodies fall identically in vacuum. Affine gravitation makes no additional postulate. If you can provide two lumps that fall differently in vacuum, Einstein was wrong. Folks have looked,
http://www.mazepath.com/uncleal/eotvos.htm#b22
given the menu of ways of being fundamentally different,
http://www.mazepath.com/uncleal/eotvos.htm#b21
The last possiblity for measurable metric/affine theory divergence will have its empirical experiment completed in mid-September. What makes you think there are only three material dimensions?
http://arXiv.org/abs/astro-ph/0508572
