Non-Euclidean Geometry: Yes or B-S? Quiz

Answer: False

Euclid did not actually have a triangle postulate. The postulate that was not accepted by other mathematicians was the parallel postulate, which states: "If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles [i.e. less than 180°], then the two lines intersect at some point."

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|B
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If that picture is clear, the parallel postulate states that those lines (lines a and b) will intersect at some point if angle A and angle B add up to less than 180 degrees. It seems obvious, but there really is no way to prove that it's true. Mathematicians such as Saccheri and Legendre tried to prove the postulate, but failed to do so because it required them to assume things that they couldn't.

Answer: False

There only two types of non-Euclidean geometry and they are hyperbolic and elliptical. There is actually a proof by Hilton proving this conjecture. Note that spherical geometry is just a more specific case of elliptical geometry (just as a circle is a more specific ellipse and a sphere is a more specific ellipsoid).

Answer: True

In hyperbolic geometry, the area of a triangle is given by its defect. In hyperbolic geometry, the sum of the angles in a triangle is always less than 180°. The defect of a triangle ABC -- written as δ(ΔABC) and calculated as 180°-(sum of the angles) -- is equal to its area in hyperbolic geometry. Thus, the maximum area is when all the angles are as close to zero as possible, thus making the the defect approximately 180°. Thus the maximum area of a triangle approaches 180° = π. (The area can also be some multiple of this, depending on what units of measurement you're using.)

As a corollary of this, because one would expect the area to get larger as you increase the side lengths, this theorem also implies that the angles in a triangle get smaller as the side lengths get longer.

Building off that last statement, you can now say that there is no such thing as similar triangles in hyperbolic geometry. Even if the side lengths of two triangles are in a ratio, the angles of the larger triangle will be smaller, thus making them not similar. In fact the only way two triangles could have their angles the same and their sides in a ratio is if they are the same triangle altogether (the ratio would be 1:1). Thus, in hyperbolic geometry, congruence is the same as similarity, eliminating the need for similarity entirely.

Answer: False

There is, however, a maximum line length in elliptical geometry. Think of yourself standing on a sphere or ellipsoid. If you keep walking in any one direction, you will eventually end up right back where you started. The maximum line length is equal to the maximum circumference of the surface you're on.

Answer: True

If you are on a sphere, you can start at a point (call it the top-most point) and draw two lines that intersect at a right angle at that point. Draw them down to the circumference (the one at which they intersect with at a right angle). Then connect them along the circumference and you have your triangle with three right angles.

For a good picture, look here: http://aether.lbl.gov/www/classes/p10/gr/img247.gif

Answer: True

Think about this as an example of how this can be true. If you are at the North Pole and you want to draw a line to the South Pole, you can follow any of the lines of longitude down to it. Hence there are an infinite amount of lines connecting those two points.

This is not always the case, though. In fact, it's only the case when a Euclidean line (through the sphere) that connects the two points is the diameter of a great circle of the sphere.

Answer: False

The defining factor of non-Euclidean geometry is that it violates the parallel postulate. The parallel postulate is necessary in order to prove the Pythagorean theorem. In fact, it was Einstein who found that, for large distances in the universe, the Pythagorean theorem does not actually hold due to the curvature of space.

Answer: True

At small scales (small as in millions of miles), non-Euclidean geometry can be approximated by Euclidean geometry. It's only when you get into huge distances that you have to take into account the curvature of space.

Answer: True

On a sphere, the only lines which are considered 'lines' are lines of longitude. The reason for this is that any curve connecting two points that does not lie on a line of longitude is not the shortest distance between those two points, and the definition of a line is the shortest distance between two points. Given that fact, all the lines of longitude intersect at both polar ends of the sphere, thus making none of them parallel.

Answer: False

The maximum sum of angles of a triangle in elliptical geometry actually approaches 540°, as each angle can approach a maximum of 180°. (If they were exactly 180°, you'd have a line.)

The maximum sum of angles of a triangle in hyperbolic geometry approaches 0°.

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