For centuries, mathematicians and physicists believed that the universe was accurately described by the axioms of Euclidean geometry, which now form a standard part of high school mathematics. Some of the distinctive properties of Euclidean geometry follow:

• Straight lines go on forever.
• Parallel lines are unique. That is, given a line and a point not on that line, there is one and only one parallel line that passes through the given point.
• Parallel lines are equidistant. That is, two points moving along parallel lines at the same speed will maintain a constant distance from each other.
• The angles of a triangle add to 180°.
• The circumference of a circle is proportional to its radius, r, (C = 2πr) and the area, A, of a circle is proportional to the square of the radius (A = πr²), where pi (π) is defined to be approximately 3.14 .

The last four properties...