About
Around 300 BC, the great ancient Greek mathematician, Euclid of Alexandria, who many consider to be the father of geometry, wrote his great mathematical treatise, Elements, a collection of thirteen books, most of which covered geometric and elementary number theory. The books contained many axioms, theorems, proofs, and constructions that would lay the foundation of modern mathematics.
.One essential concept in mathematics in the concept of a proof, a rigorous mathematical argument that unequivocally demonstrates the truth of a given proposition. All proofs follow the logic of deductive reasoning, where a conclusion is based on the concordance of multiple premises that are known or assumed to be true. In deductive reasoning, every argument needs a starting point, and sometimes we are unable to prove the simplest of all starting points. A mathematical statement that must be regarded as fact without proof is called an axiom. In Elements, Euclid stated five axioms that would lay the foundations of plane geometry:
Euclid's fifth axiom, also know as the parallel postulate, has been a study of deep interest for mathematicians for almost two millennia. Compared to "through any two points there is exactly one line" or "all right angles are equal," the parallel postulate is aesthetically unpleasing, long-winded, and complex. Over the years, many mathematicians have attempted to prove the parallel postulate, but none of the proofs were deemed to be correct.
In the early nineteenth century, mathematicians began to attempt to try to find a contradiction of the parallel postulate. Rather than finding a contradiction, they found an entire new type of geometry.
By 1815, the great German mathematician, Carl Friedrich Gauss, who many consider to be the greatest mathematician of all time, prepared a treatise on this new geometry, but never published it. At the time, mathematicians were firmly convinced that Euclidean geometry was flawless, a view that was advocated by the German philosopher Immanuel Kant. A notorious perfectionist who never published any work unless it was complete and above all criticism, Gauss kept his discovery to himself fearing controversy over his discoveries. In 1829, the Russian mathematician Nikolai Lobachevsky published a paper on what he believed to be a case of geometry where the parallel postulate did not hold. In 1832, Hungarian Janos Bolyai expanded on the work of Lobachevsky. Unfortunately, the mathematically community dismissed the work of Lobachevsky and Bolyai, and both were unable to attract a wide audience. Finally, in 1854, German mathematician Bernhard Riemann, a student of Gauss, discovered a more rigorous and complete explanation of non-Euclidean geometry that would be widely accepted by the scientific and mathematical community.
Riemann managed to generalize geometry into n-dimensions, and provide a rigorous definition of this new geometry, but many mathematicians dismissed the subject, deeming it not important. The impacts of the work of Lobachevsky, Bolyai, Gauss, and Riemann were seen almost a century later, when Albert Einstein used non-Euclidean geometry to formulate his famous theory of general relativity, arguing that space-time was curved and non-Euclidean. Non-Euclidean geometry was once again used to explain the many extra dimension needed in string theory, and mathematicians and scientists finally realized the importance of this new branch of geometry.
Non-Euclidean geometry is the study of geometry on surfaces where the parallel postulate does not hold. There are two basic types of non-Euclidean geometry, spherical geometry, where lines always intersect, and hyperbolic geometry, where lines never intersect. This stretches the definition of a line, as lines are never straight in non-Euclidean geometry.
For example, in spherical geometry, as seen below, a "line" is the equivalent of a great circle, like the lines of longitude on a globe. These lines always intersect and curve toward each other, as seen in the picture below.
An interesting property of spherical geometry is that the angles of a triangle always sum up to more than 180, degrees. In fact, we could possibly have a triangle with three right angles, as shown in the second image below!
.One essential concept in mathematics in the concept of a proof, a rigorous mathematical argument that unequivocally demonstrates the truth of a given proposition. All proofs follow the logic of deductive reasoning, where a conclusion is based on the concordance of multiple premises that are known or assumed to be true. In deductive reasoning, every argument needs a starting point, and sometimes we are unable to prove the simplest of all starting points. A mathematical statement that must be regarded as fact without proof is called an axiom. In Elements, Euclid stated five axioms that would lay the foundations of plane geometry:
- Any two points can be connected by a straight line segment.
- Any line segment can be extended forever in both directions, forming a line.
- Given any line segment, we can draw a circle with the segment as a radius and one of the segment's endpoints as the center.
- All right angles are congruent.
- Given any straight line and a point not on the line, there is exactly one straight line that posses through the point and never meets the first line.
Euclid's fifth axiom, also know as the parallel postulate, has been a study of deep interest for mathematicians for almost two millennia. Compared to "through any two points there is exactly one line" or "all right angles are equal," the parallel postulate is aesthetically unpleasing, long-winded, and complex. Over the years, many mathematicians have attempted to prove the parallel postulate, but none of the proofs were deemed to be correct.
In the early nineteenth century, mathematicians began to attempt to try to find a contradiction of the parallel postulate. Rather than finding a contradiction, they found an entire new type of geometry.
By 1815, the great German mathematician, Carl Friedrich Gauss, who many consider to be the greatest mathematician of all time, prepared a treatise on this new geometry, but never published it. At the time, mathematicians were firmly convinced that Euclidean geometry was flawless, a view that was advocated by the German philosopher Immanuel Kant. A notorious perfectionist who never published any work unless it was complete and above all criticism, Gauss kept his discovery to himself fearing controversy over his discoveries. In 1829, the Russian mathematician Nikolai Lobachevsky published a paper on what he believed to be a case of geometry where the parallel postulate did not hold. In 1832, Hungarian Janos Bolyai expanded on the work of Lobachevsky. Unfortunately, the mathematically community dismissed the work of Lobachevsky and Bolyai, and both were unable to attract a wide audience. Finally, in 1854, German mathematician Bernhard Riemann, a student of Gauss, discovered a more rigorous and complete explanation of non-Euclidean geometry that would be widely accepted by the scientific and mathematical community.
Riemann managed to generalize geometry into n-dimensions, and provide a rigorous definition of this new geometry, but many mathematicians dismissed the subject, deeming it not important. The impacts of the work of Lobachevsky, Bolyai, Gauss, and Riemann were seen almost a century later, when Albert Einstein used non-Euclidean geometry to formulate his famous theory of general relativity, arguing that space-time was curved and non-Euclidean. Non-Euclidean geometry was once again used to explain the many extra dimension needed in string theory, and mathematicians and scientists finally realized the importance of this new branch of geometry.
Non-Euclidean geometry is the study of geometry on surfaces where the parallel postulate does not hold. There are two basic types of non-Euclidean geometry, spherical geometry, where lines always intersect, and hyperbolic geometry, where lines never intersect. This stretches the definition of a line, as lines are never straight in non-Euclidean geometry.
For example, in spherical geometry, as seen below, a "line" is the equivalent of a great circle, like the lines of longitude on a globe. These lines always intersect and curve toward each other, as seen in the picture below.
An interesting property of spherical geometry is that the angles of a triangle always sum up to more than 180, degrees. In fact, we could possibly have a triangle with three right angles, as shown in the second image below!
In hyperbolic geometry, lines never intersect, so all lines are parallel. This is extremely difficult to visualize, but hyperbolic geometry is done on the surface of a saddle, as shown below. An interesting property of hyperbolic geometry is that the angles of a triangle always sum up to less than 180, degrees.
Point
- Non-Euclidean geometry is the study of geometry on surfaces where the parallel postulate does not hold. (basically all the surfaces that are not flat)
- More complex types of non-Euclidean geometry involve a mix of spherical and hyperbolic geometry, and there are the types of surfaces most commonly used to model the universe.
- I learned that the concept of non-Euclidean geometry can be applied to more than three-dimension, which has led to its usefulness in physics.
- I learned that many of the most important modern theories in physics, such as string theory and general relativity, have been formulated using non-Euclidean geometry.
- I learned that the shape of the space-time continuum is oddly curved and non-Euclidean, which can be used to predict the shape and fate of the universe.
React
Non-Euclidean geometry has always fascinated me, as I am still astonished about how the simple idea of applying basic geometric principles to surfaces that are not flat can yield such intriguing results and formulate theories that are fundamental to our understanding of the universe, like general relativity and string theory. My interest in this topic has only grown, as I am planning to do my symposium on string theory this year.
I have so many ideas about how I could incorporate non-Euclidean geometry into my senior project. Perhaps I could investigate the applications of non-Euclidean geometry into fields other than physics, like maybe chemistry and biology? Maybe I could learn more about higher dimensions and see their applications in real life? Perhaps I could even possibly shadow a mathematician?
The possibilities are endless for this topic, and I am truly excited to learn more about non-Euclidean geometry. I might have to study college-level geometry, multi-variable calculus, and group theory in order to fully understand this topic, but I am very interested in all the opportunities to come, and I can't wait to learn more about this subject.
I have so many ideas about how I could incorporate non-Euclidean geometry into my senior project. Perhaps I could investigate the applications of non-Euclidean geometry into fields other than physics, like maybe chemistry and biology? Maybe I could learn more about higher dimensions and see their applications in real life? Perhaps I could even possibly shadow a mathematician?
The possibilities are endless for this topic, and I am truly excited to learn more about non-Euclidean geometry. I might have to study college-level geometry, multi-variable calculus, and group theory in order to fully understand this topic, but I am very interested in all the opportunities to come, and I can't wait to learn more about this subject.
Sources
Rusczyk, Richard. Introduction to Geometry. Alpine, CA: AoPS, 2007. Print.
Coxeter, H. S. M., and Samuel L. Greitzer. Geometry Revisited. New York: Random House, 1967. Print.
Coxeter, H. S. M., and Samuel L. Greitzer. Geometry Revisited. New York: Random House, 1967. Print.