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Chasing the Parallel Postulate

The views expressed are those of the author and are not necessarily those of Scientific American.


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Euclidean geometry, codified around 300 BCE by Euclid of Alexandria in one of the most influential textbooks in history, is based on 23 definitions, 5 postulates, and 5 axioms, or “common notions.” But as I mentioned in my recent post on hyperbolic geometry, one of the postulates, the parallel postulate, is not like the others.

In Thomas Heath’s translation of Euclid’s Elements (also known as the translation I have), the five postulates are stated as:

“Let the following be postulated:
1) To draw a straight line from any point to any point.
2) To produce a finite straight line continuously in a straight line.
3) To describe a circle with any centre and distance.
4) That all right angles are equal to one another.
5) That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

The first four are short and sweet, but the fifth is a mouthful. It’s easier to see in a picture.

The parallel postulate says that if the angle measures of α and β add up to less than 180 degrees, then the dotted lines eventually intersect. Image by 6054, via Wikimedia Commons.

The parallel postulate seems natural enough, but it’s the kind of statement that seems like it should be a theorem—something we prove using other axioms and postulates—rather than a postulate. For 2000 years, mathematicians tried to prove this, to show that the parallel postulate could be derived from the other axioms and postulates. (For the fascinating story of Giovanni Girolamo Saccheri, one of the mathematicians who tried to do this, check out Thony Christie’s blog post the problem with parallels.)

Every attempt at proving the parallel postulate as a theorem was doomed to failure because the parallel postulate is independent from the other axioms and postulates. We can formulate geometry without the parallel postulate, or with a different version of the postulate, in a way that adheres to all the other axioms. Hyperbolic geometry, the subject of my last post, uses a different version of the parallel postulate and therefore ends up with a completely different looking geometry.

I am not a math historian, so my ideas of how it felt to try to prove the parallel postulate are speculative. But I imagine the parallel postulate as a wrinkle in a sheet that moves around when you try to smooth it out but never goes away. People tried to “smooth out” the parallel postulate by proving it, but they only pushed the wrinkle over into some other statement. These statements are fascinating to me. Any one of them can take the place of the parallel postulate, but like the parallel postulate, they can’t be proven using only the other postulates and axioms.

Playfair’s axiom is probably the simplest statement that is equivalent to the parallel postulate. In fact, I learned it as the parallel postulate: In a plane, given a line and a point not on it, exactly one line parallel to the given line can be drawn through the point.

Playfair's Axiom states that there is only one line through the point P that does not intersect the line L. Image: Evelyn Lamb.

Playfair’s axiom is simple and direct, but some of the most interesting statements that are equivalent to the parallel postulate involve triangles. You probably remember from high school geometry class that the angles of a triangle add up to 180 degrees. That is only true if you assume the parallel postulate. In fact, it is equivalent to the parallel postulate. But wait, there’s more: the statement that all triangles have the same sum of angles is also equivalent to the parallel postulate. I think this is fascinating because it means that in other geometries, there are triangles that have different sums of interior angles! In hyperbolic geometry, for example, the sum of angles can be any number that is less than 180 degrees and greater than or equal to 0 degrees.

These hyperbolic triangles have angles adding up to 0 degrees. Image: Public domain, via Wikimedia Commons.

It’s not too hard to see that the properties of triangles in different geometries lead to properties of other shapes. For example, Euclidean geometry is the only geometry with rectangles. Why? If you divide a rectangle into two triangles by drawing in one of the diagonals, you end up with two congruent triangles. The rectangle’s angles sum up to 360 degrees, so each triangle must have angles that sum up to 180 degrees. If you like rectangles, stick with Euclid.

If you'd like to see these again, you'd better accept the parallel postulate. Image: Webber, via Wikimedia Commons.

Getting back to triangles, you may remember the ideas of “similar” triangles and “congruent” triangles.Two triangles are similar if they have the same angles and congruent if they are similar and have the same side lengths.

Euclidean triangles that are similar but not congruent. The existence of similar, non-congruent triangles is equivalent to the parallel postulate. Image: Nguyenthephuc, via Wikimedia Commons.

In Euclidean geometry, there are similar triangles that are not congruent. In fact, there are tons of them. No matter what triangle you start with, you can scale it up or down to whatever size you want. However, the fact that there are similar triangles that are not congruent is very special. It’s yet another statement that is equivalent to the parallel postulate! In hyperbolic geometry, for example, the angles of a triangle uniquely determine the lengths of its sides. This has an even more surprising consequence: Euclidean geometry is the only geometry in which triangles can be arbitrarily large. In hyperbolic geometry, there is a largest triangle.

Another statement about triangles that is equivalent to the parallel postulate is the Pythagorean theorem: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. It’s one of the first theorems we learn in geometry, but it’s only true if we assume the parallel postulate.

An illustration of the Pythagorean Theorem from Oliver Byrne's 1847 translation of Euclid's Elements. The Pythagorean Theorem states that the sum of the areas of the black and red squares is equal to the area of the yellow and blue square below. Image: Public domain, via Wikimedia Commons.

As Alexander Bogomolny writes on his website Cut the Knot, some of the statements that are equivalent to the parallel postulate seem obvious and some do not at all: “By all accounts, the Pythagorean Theorem is far from obvious. It is amazing that the Parallel Postulate, being equivalent to such intuitive statements as [there exists a pair of similar non-congruent triangles] and [there is no upper limit to the area of a triangle], is also equivalent to the Pythagorean Theorem.”

As centuries of mathematicians failed to prove the parallel postulate using the other postulates and axioms, I imagine them chugging along, trying to figure out what would happen if they assumed the parallel postulate to be false. They pushed and pushed, but they could never definitively find a contradiction. While they were chugging along, they discovered these statements and many others that are equivalent to the parallel postulate. If you assume any one of them, you end up with Euclidean geometry, but without them, you can find your way to the fantastical land of non-Euclidean geometry where angles determine lengths and there are no squares!

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.





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  1. 1. Will_in_BC 10:58 am 02/28/2014

    Nice little article. Thanks.

    Link to this
  2. 2. waterbergs 3:32 pm 02/28/2014

    Great article. Concise, fascinating and full of juicy maths, thanks.

    Link to this
  3. 3. stargene 2:32 am 03/1/2014

    What I’m about to say may seem blitheringly obvious
    to others, but here goes:

    “…the parallel postulate is independent from the
    other axioms and postulates. We can formulate geometry without the parallel postulate, or with a different version of the postulate, in a way that adheres to
    all the other axioms.”

    I am struck by the particular wording you have
    given to this famous development, inaugurating non-Euclidian geometries. The existence of two
    opposing solutions to the same logical question,
    both of which are compatible with all other axioms/
    postulates reminds me of the current flurry of
    papers on the Continuum Hypothesis, where two
    effectively competing ideas give opposing answers,
    and yet oddly each one is, as far as anyone knows, compatible with the rest of mathematics.
    A discussion of this debate is at

    http://www.scientificamerican.com/article/infinity-logic-law/

    … where, quoting the article: “…scholars largely
    agreed upon two main contenders for additions
    to ZFC: forcing axioms and the inner-model axiom “V=ultimate L.” …”

    It occurs to me that it may yet be found that
    the two different approaches, with different
    answers, are analogous to the crucial differences
    between Euclidean and Non-Euclidean geometries.
    Ie: neither truly contradicts the other when
    seen from a larger framework in the (dare I say it…Infinite?) math universe.

    Link to this
  4. 4. David Cummings 6:39 am 03/1/2014

    Hi Evelyn,

    You say:

    ” the angles of a triangle add up to 180 degrees. That is only true if you assume the parallel postulate. In fact, it is equivalent to the parallel postulate.”

    How so?

    You repeat that assertion several other times, in various congruent forms in the rest of your article, but you don’t explain why.

    I always liked working with math but I was never very good at following these “proof” arguments. Your piece reminds me of the headaches I used to get back in my school days trying to follow the “it follows” arguments from professors who don’t really explain how “it follows”.

    Thanks,

    Dave

    Link to this
  5. 5. m 6:32 am 03/2/2014

    David

    Well first, i must say i find this all to easy.

    Second the reason and the solution to this initial issue is in number 4.

    Two right angles from 2 parallel and 1 perpendicular produce an open box. The sides do not close, so 180 deg cannot produce a triangle.

    Any angle of either less than 90 will, therefore a triangle must have one angle less than 90 deg. Notice what I did here…. one angle… of those initial intersections. The reason being we don’t need to do the triangle on a plane.

    Link to this
  6. 6. patrick 4:01 am 03/3/2014

    The parallel postulate,is the strong room door, to penetrate the fifth dimension,and calls for eight pairs of parallels to
    be perfectly ,geometrized in synchronized dynamical orbital triangulation, while bisecting celestial orbital points,
    sweeping through equal angles, at one fixed moment of time.
    around a star.

    Link to this
  7. 7. hjmglass 11:27 pm 04/23/2014

    I always thought this is a fun postulate. He doesn’t really say parallel lines will never cross. He says that non-parallel lines will cross. He alludes to the concept of the parallel but never

    When people talk about non-Euclidian geometry, is this the postulate that is changed?

    Link to this
  8. 8. Evelyn Lamb in reply to Evelyn Lamb 11:04 am 04/24/2014

    We often think of parallel lines as lines that have the same direction, or that stay the same distance apart from each other, but that characterization only works in Euclidean geometry. The real definition of parallel lines is lines that never intersect. Therefore the definition of non-parallel is two lines that do intersect. The 5th postulate as Euclid stated it is therefore more along the lines of “lines that do not have the same direction are not parallel.”
    The easiest 5th postulate equivalent to work with is Playfair’s axiom, “given a line L and a point P not on L, there is exactly one line parallel (=non-intersecting) to L that contains P.” We change that postulate when we move to non-euclidean geometry. Either we say “given a line L and a point P not on L, there are no lines parallel to L that contain P” (elliptic/spherical geometry) or “…there are infinitely many lines parallel to L that contain P” (hyperbolic geometry). Interestingly, these are the only options for how many parallel lines we get: 0, 1, infinity. We can’t have just 2, or 17, or any finite number (other than 1) of them. Once we get 2, we can find infinitely many lines between those 2 lines.

    Link to this

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