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9.5: Non-Euclidean Geometry

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    In your geometry class, you probably learned that the sum of the three angles in any triangle is 180 degrees. This is a well-known theorem in geometry—more specifically, “plane” or “Euclidean” geometry, which has served as the foundation for all learning since the time of the Greek civilization (In fact, the word “mathematics” comes from the Greek word meaning “lessons” or “things to be learned.”) But it turns out that if you go away from the “plane,” then plane geometry may not work. This gives rise to non-Euclidean geometry.

    An example of Non-Euclidian geometry can be seen by drawing lines on a sphere or other round object; straight lines that are parallel at the equator can meet at the poles. This “triangle” has an angle sum of 90+90+50=230 degrees!

    Figure \(\PageIndex{1}\): On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean plane, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. Image is used under a CC BY-SA 3.0 license.

    It is called "Non-Euclidean" because it is different from Euclidean geometry, which was developed by an ancient Greek mathematician called Euclid.

    Some History…

    The birth of non-Euclidean geometry was REALLY a big deal. It was truly a ground-shaking event, not only in the history of mathematics and but also in philosophy. You see, Euclid (who lived over 2300 years ago!) wrote a textbook that was so popular that practically every educated person in the world used it to study geometry for the next 2000 plus years. This subject, “Euclidean geometry” (the type of geometry you studied in high school), was so popular and dominant that no one, for over two millennia, doubted its truthfulness, questioned its authority, or thought of coming up with an alternative. Well, almost no one. The conception and arrival of non-Euclidean geometry involved three mathematicians—one very famous and two completely unknown. The story is worthy of a movie or a play. It is that dramatic. It teaches us that

    • Common sense could be the greatest obstacle to finding truth.
    • What stands the “test of time” may not be absolutely true.
    • What many experts feel is offensive and repugnant may actually be true.

    Here, a very abbreviated version of the story is presented. The reader is encouraged to find out more by doing a search under “non-Euclidean geometry.”

    Euclid, who lived around 300 B.C., is best known for his book The Elements, a 13- volume masterpiece laying the foundations of geometry (and some number theory as well). This book may be the most widely read treatise in world history because no other books have been read longer or by more people, with the exception of the Bible. It was the standard book in geometry for over 2000 years, and there are over 1000 editions of the book in hundreds of languages.

    It’s too bad that the notions of copyrights and intellectual properties did not exist back then. He could have been quite wealthy all the royalties he could have earned (except he would not have cared—there is a well-known story of Euclid embarrassing and humiliating one of his students who wanted to know what he would gain by learning geometry).

    What made this book so good?

    Well, for one thing, it was the first book that laid the foundation of deductive logic—to prove general statements (called propositions) by definitions, general assumptions, and already known propositions. If you remember your high school geometry, you may recall memorizing postulates (general assumptions) and proving theorems based on known properties and other theorems. That whole thing—which is the fundamental structure of mathematics—was first established by Euclid.

    Another amazing accomplishment of Euclid was that he proved tons of propositions—465 to be exact—based on a very small number of assumptions. In fact, he started with only five axioms. What is an axiom? It is a (self-evident) statement assumed true without proof. Just as you cannot define every word you use (because each definition uses other words, each of which also needs to be defined using other words), you cannot prove everything; some statements must be assumed true at the beginning. (And they’d better be obvious to everyone so that no one would question them.) Here are the first four:

    Definition: Euclid's Axioms

    1. A line can be drawn through any two points.
    2. A line can be extended indefinitely in both directions
    3. A circle can be drawn with any center point and any radius.
    4. All right angles are congruent (equal measure of 90 degrees).

    They sound obvious, right? Join the club. Over 2200 years passed without anyone seriously challenging these.

    But the last of his five axioms was much longer and complex than the other four, and it seems that even Euclid himself hesitated using it. It is his fifth axiom (thus often called “Euclid’s Fifth,” like “Beethoven’s Fifth”):

    Euclid's Fifth Axiom (Parallel Postulate)

    Given a line L and a point P that is not on the line, there is one and only one line through P parallel to L.

    (Originally this statement was more like this: “If two lines both crossing another line form two interior angles on the same side whose sum is less than two right angles (180 degrees), then the two lines, when extended indefinitely on that side, will eventually meet.”)

    The original statement is weird, right? This is exactly what caused the controversy and, eventually, a revolution.

    Now, using the example at the beginning of this section (the “triangle” on the sphere), you may be able to see that this postulate is not true on the sphere. You saw the two vertical lines (both perpendicular to the equator) are parallel at the equator but end up meeting each other at the North Pole. In fact, on the sphere, there are no parallel lines. “Yeah, but these are not lines,” you may say. That’s understandable. You must define what a line is carefully here. But anyway, people did not figure out for a long, long time that spherical geometry is one of the models of non-Euclidean geometry. People had that much faith in Euclid.

    Because this one particular axiom was so odd, many mathematicians wanted to clean up the system—more specifically, they wanted to prove that Axiom 5 follows directly from the first four axioms so they could eliminate it altogether.

    But they all failed. Some of the most intelligent people who ever lived could not do it. It wasn’t just a few folks either. Ptolemy tried it in the 2nd century, and Girolamo Saccheri in the 18th century, and many others in between. In fact, so many had attempted this that in 1763, G. S. Glugel wrote a paper on how NOT to prove it.

    Let’s review the problem again. What were they trying to do? Remember that axioms (postulates) are assumptions we begin with (and accept as truths), perhaps what Thomas Jefferson (who, as a political philosopher and an architect, knew Euclid’s textbook extremely well) referred to as “self-evident” truths. Everyone was fine with Axioms 1 through 4. Axiom 5, on the other hand, was quite different, lengthy, and awkward. There were over 400 statements proved by Euclid based on these five axioms. Many of them depended on Axiom 5. If someone could prove Axiom 5 from the first four axioms, then we could simply take Axiom 5 as another proposition, and all of those 400+ propositions would still be true, based on four axioms. That would be really cool.

    By the way, among the more than 400 propositions were quite obvious statements like the following:

    • The sum of the three angles of any triangle equals 180 degrees.
    • There are bigger and smaller triangles with the same set of three angle measures (called similar triangles).
    • There are 4-sided polygons with four right angles (we simply call them rectangles).

    This may be hard to believe, but there were mathematicians who had dedicated their lives to solve this problem—and failed. Among them was Wolfgang Bolyai, a long-time friend and classmate of Carl F. Gauss, one of the most celebrated mathematicians of all times. After years of frustration, Wolfgang Bolyai wrote to his son John (Janos), telling him not to waste a lifetime as he himself had. These were the words of Wolfgang to his son:

    I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone. . . . I saw that no man can reach the bottom of the night. . . . I have traveled past all reefs on this infernal Dead Sea and have always come back with broken mast and torn sail. . . . I thoughtlessly risked my life and happiness.

    Pretty depressing, right? Well, what would a boy do when his father tells him NOT to do something? Yes, that’s exactly what the son Janos did. The young, unknown student Janos Bolyai worked really hard on this problem, and within a few years he wrote back to his father:

    “I have discovered such wonderful things that I was amazed…. Out of nothing I have created a strange new universe.”

    A young crazy dreamer, correct? Maybe he was. In fact, he did not solve the problem. He did not prove Axiom 5 using the first four. In fact, he did exactly the opposite; he assumed that Axiom 5 was false and pursued what difference it would make. The difference turned out to be huge. The three “obvious” facts listed above would be all false in this “strange new universe” Janos Bolyai had created. Now, that was a bold statement. Who would question the existence of a rectangle? (Remember, though, this was all theoretical, so it would be possible to imagine such a world which is consistent with the four axioms plus the negation of the fifth.)

    Recognizing his son’s marvelous work (maybe as a proud and biased dad), Wolfgang Bolyai decided to publish Janos’ results in 1832—but not as a book or a paper, but as an appendix to his own book he was publishing. Remember he had a famous friend? Wolfgang sent a copy of this book to Prof. Gauss, wondering what kind of reaction Gauss would have. Perhaps Bolyai was hoping that Gauss would be so impressed that he would praise the work of his son Janos.

    The response? Well, in a sense, Gauss was impressed. But this is what Gauss said about the “strange new universe” created by young Janos Bolyai: “To praise it would amount to praising myself.”

    What?? Gauss was saying that he had secretly been working on the same thing and came up with pretty much the same results as Janos Bolyai, except Gauss had stuffed all the notes in his drawer, hoping that perhaps someone would publish them after his death. He was afraid of ruining his reputation because it was such a crazy idea. Who would challenge the truthfulness of Axiom 5, which the entire world had accepted as absolute truth for over 2000 years? Some people claim that Gauss was especially afraid of being ridiculed by the philosophical group that had been following Immanuel Kant (1724—1804), a leading philosopher who had scoffed at anyone challenging Euclid’s axioms.

    So here we have a brilliant, well-known German mathematician trying to avoid criticism, his old Hungarian friend with a brilliant son, and that inquisitive, incredibly smart son who had developed a crazy new idea. Oh, by the way, what do you think happened to this son Janos? After receiving the reply from Gauss, this gifted young man stopped studying mathematics altogether and never published anything else, except those few pages known as “the most important appendix in the history of the world.” The fact that the whole academic world pretty much ignored the appendix did not help the situation either.

    Meanwhile, over in Russia (yes, in a totally different country), there was another person who was independently working on the same idea, coming up with pretty much the same results. His name is Nikolai Lobachevsky (1792—1856). Without electronic communications or social media, this Russian mathematician had no idea what Gauss and Bolyai were up to. In 1829, two years before Janos Bolyai’s appendix was published, Lobachevsky had published his work on non-Euclidean geometry in a little-known journal, written in Russian, published at Kazan University. Bolyai and Guass did not read Russian. Even if they had, they would not have known about this journal. But because of this publication, Lobachevsky became the first person to ever publish non-Euclidean geometry.

    What did Lobachevsky say (in Russian) that was so controversial? Well, in a sense, he basically showed the world that no one would be able to prove Axiom 5 from the first four axioms because it is possible to create a new and consistent world in which Axiom 5 is false. In that world (“imaginary” in Lobachevsky’s word), similar triangles would not exist, and neither do rectangles.

    Well, Lobachevsky did not have a famous friend to write to. All he had was a bunch of traditional, old-fashioned colleagues and scholars who thought he had gone insane. Lobachevsky was relentlessly criticized, mocked, and rejected by the academic world. His new “imaginary” geometry represented the “shamelessness of false new inventions” to them. Oh, the news eventually reached Gauss in Germany. While recognizing Lobachevsky’s work as great (because it was just like his own), Gauss still did not support or endorse this new idea in fear of criticism.

    Lobachevsky was fired from the university (perhaps due to his failing health, perhaps due to his crazy new idea), lost a child, became blind, and died in poverty. Along with Janos Bolyai, the two obscure mathematicians who had (correctly) challenged the 2000-year-old teaching and reached the surprising apex—the existence of non-Euclidean geometry—ended up leading a tragic life. Perhaps Janos’ father Wolfgang was right when he wrote of the “broken mast” and “torn sail.”

    So why does all this matter? Isn’t that just a theoretical world, totally separate from the real world where we live? Is Euclidean geometry true or false? The answer to this last question is “It depends.” If you assume all five axioms of Euclid, you get plane geometry, just as you learned in high school. If you decide not to assume Euclid’s Fifth, then… well, you get non-Euclidean geometry. One example of this is spherical geometry—the entire navigation theory is based on this. Another example is hyperbolic geometry. When Einstein was studying the structure of our universe, he needed non-Euclidean geometry. In fact, our modern astronomy would not exist without non-Euclidean geometry. Only those crazy dreamers like young Bolyai and Lobachevsky could help pioneer our space explorations. What appears ridiculous and repugnant may not be so crazy after all.

    Imagine a small village where, based on observation, everyone believes that the earth is flat. You and I could sit back and laugh at those people because, well, the earth is not a Euclidean plane; it models spherical (non-Euclidean) geometry. In addition, some research today shows that it is possible that our universe could have curvature, meaning that we may be living in a non-Euclidean universe. Imagine that! A small group of ignorant human beings who think Euclidean geometry is absolute while living on a non-Euclidean planet, which revoles around in a non-Euclidean universe. That would be a true irony.

    Math in the Real World – The 5th Postulate
1. Watch this video to get an overview of Euclidean Geometry and the 5th postulate. 
2. Watch this video to understand the basis of Non-Euclidean Geometry.

    Contributors and Attributions

    • Saburo Matsumoto

    9.5: Non-Euclidean Geometry is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

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