Math is an indispensable part of human life. It is the cornerstone for understanding the physical world and is used in mathematical models and theories.
From basic arithmetic to complex calculus, there are many types of mathematics that people study, and one that stands out as the most difficult of all is non-Euclidean mathematics. This type of mathematics is considered to be the hardest math of all time due to its difficulty in being understood and applied in everyday life. In this article, we will delve into what non-Euclidean mathematics is, its evolution, its difficulty level and its potential uses.
What is Non-Euclidean Math?
Non-Euclidean math refers to any type of math that does not follow Euclid’s fifth postulate, which states that for any given line and point not on the line, at least one line parallel to the given line can be drawn through the point. This type of math breaks from much of the work of Euclid, as it states that such a line does not exist.
This type of math is further divided into two types: hyperbolic geometry, and elliptic geometry. Hyperbolic geometry, which is derived from the German mathematician Georg Friedrich Bernhard Riemann is commonly referred to as ‘Lobachevsky’s geometry. ’ It is an alternative form of non-Euclidean mathematics, where the fifth postulate does not apply.
In this type of geometry, Euclid’s postulate of parallel lines is not a necessary truth, and straight lines and curves can be treated as distinct shapes, with one curving towards or away from itself. On the other hand, elliptic geometry originated from the work of John Wallis in 1658 and distinguishes itself from hyperbolic geometry due to its commitment to the postulate that, for any given line L, through a point not on it, there lies an infinite number of lines parallel to L.
Evolution of Non-Euclidean Math
The concept of non-Euclidean mathematics first came to be when the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Lobachevsky first independently formulated the properties of hyperbolic geometry in 1829-1830. Both mathematicians were attempting to consider Euclid’s postulate and contradict it if it were false.
This was the beginning of an era where Euclid’s postulates were challenged and that would eventually overturn the entire existing notion of geometry. To this day, mathematicians still actively contribute to the development of non-Euclidean math. For example, another German mathematician named Felix Klein contributed to advancing the idea of elliptic geometry in 1873, proposing the idea for a proper mathematical theory for it.
Such a theory would introduce concepts such as a triangle sum, angles of a triangle and the concept of distance.
Difficulty Level of Non-Euclidean Math
The difficulty of understanding and mastering non-Euclidean mathematics is debatable. Each type of non-Euclidean geometry; hyperbolic and elliptic, are considered to be quite difficult and challenging to understand. For example, hyperbolic geometry is considered to be more difficult than Euclidean geometry and Elliptic geometry is considered to be more difficult than Hyperbolic geometry.
It can be argued that the non-intuitive aspects of non-Euclidean mathematics make it difficult to comprehend, as there are often too many conflicting interpretations and rules. Furthermore, the complexity of its derivation of formulae, proof techniques and definitions also make it a particularly pesky field of mathematics.
Potential Uses of Non-Euclidean Math
The complexity and often counterintuitive nature of non-Euclidean mathematics make it difficult to understand and apply to everyday life. However, it is these very features that make it so powerful and useful.
From a theoretical standpoint, this type of mathematics has been applied in theoretic physics, such as Albert Einstein’s General Theory of Relativity. As a result, this type of mathematical theory is used to describe the physical universe, its structure and its motion. On an applied level, non-Euclidean mathematics has also been used to solve various real-world problems.
For example, in robotics and computer vision, non-Euclidean mathematics allows machines to make intelligent decisions based on spatial concepts and other visual cues. In addition, this type of math is also used in geographical information systems (GIS) to identify land use, create maps and determine other key environmental data.
Conclusion
Non-Euclidean maths is a fascinating and complex field of mathematics, that while difficult to understand and master, encourages creative, expansive thinking and exploration. Its applications are far-reaching, as it can be found in domains as diverse as physics and robotics, to GIS and computer vision. Its difficultly level and counterintuitive nature makes it one of the hardest math of all time, and it is this very feature which continues to make it the source of ongoing inquiry today.