We are quite familiar with plane geometry (aka. the two-dimensional Euclidean Geometry) starting from our elementary school days. However, our universe is abundant in curved objects and shapes which are not flat, and hence to study those kind of geometries, we need to take resort to a different kind of geometry which would allow us to study curved surfaces.
Differential Geometry is the application of Calculus to study the geometry of curved spaces. But first let us try to understand a bit about what flat means. This would allow us to have a better understanding of spaces which are not flat for example, the surface of earth, horse-saddles and many more.
Let us start from a layman’s point of view. We inhabit a natural world pervaded by curved objects, and if a child asks us the meaning of the word flat, how would we describe it? Most likely, we would answer in terms of the absence of curvature, a smooth surface without bumps or hollows.
Nevertheless, the very earliest mathematicians seem to have been drawn to the singular simplicity and uniformity of the flat plane, and they were rewarded with the discovery of startlingly beautiful facts about geometric figures constructed within it (for example, a triangle). With the benefit of enormous hindsight, some of these facts can be seen to characterize the plane’s flatness.
One of the earliest and most profound such facts to be discovered was the Pythagoras Theorem. The ancients must have been awed to learn the fact that a seemingly unalloyed fact about numbers,
in fact has a geometric meaning, as seen in the following diagram:
Contrary to the name of Pythagoras Theorem, it was actually discovered and used in ancient civilizations much earlier than Pythagoras. While Pythagoras lived in Greece around 500 BCE, the earliest known example of the knowledge of this theorem is recorded in the Babylonian clay tablet (catalogued as ”Plimpton 322”) shown below:
It was found buried in what is now Iraq, and it dates from about 1800 BCE (Almost 1300 years earlier than Pythagoras).
The tablet lists Pythagorean Triples, which are non-negative integers such that
is the hypotenuse of a right triangle with sides
and
and therefore,
Some of these ancient examples are impressively large, and it seems clear that they did not stumble upon them randomly. Instead, they possessed a mathematical process for generating Pythagorean triples (e.g., the fourth row of the tablet records the fact that (13500, 12709, 18541) is a Pythagorean triple!).
Another Babylonian clay tablet namely, Si. 427 (dating from 1600 BCE) serves as an evidence that the ancients of Babylon knew a number of applications of Pythagorean triples.
We find the Pythagoras Theorem stated in the Indian text Śulbasūtra by Baudhāyana, written in Sanskrit, dating from 800 BCE. In fact, it is the exact same version that we find in our modern day textbooks, along with several other fascinating geometric constructions (Śulba means ‘cord’ or ‘rope’ while Sūtra means ‘rule’ in Sanskrit. )
Jumping 200 years forward in time, it was the Greek philosopher Thales of Miletus (around 600 BCE) who, according to scholars, first pioneered the deductive approach, i.e. getting new results from previously established ones. This logical chain begins at a few clearly articulated assumptions, which we now call axioms.
We find one of the most perfect examples of this new axiomatic approach in Euclid’s Elements, dating from 300 BCE (almost 300 years beyond Thales).
Euclid postulated five axioms from which the entire Euclidean geometry could be derived as theorems, through this deductive approach. Although, each one of the axioms are simple, the most intriguing is the fifth and last axiom of Euclid, which is the Parallel postulate or Parallel axiom. As we will see, this fifth axiom is the seed from which the concept of alternative Non-Euclidean geometries is about to germinate. Thus the parallel postulate is one of the milestones in the discovery of Non-Euclidean Geometry as well will see very soon. First let us go through the fifth axiom.
Define two lines to be parallel if they do not meet even if produced indefinitely in either directions. Euclid’s fifth axiom states that:
See the illustration below for the exact version of the fifth axiom given by Euclid, which is of course equivalent to the version stated above.
The character of this axiom was more complex and less immediate than the first four. Mathematicians struggled for a long time trying to derive the fifth axiom from the first four axioms as they believed for a long time that the fifth axiom is actually a theorem, and it is possible to get rid of this axiom.
However, this tension went unresolved for the next 2000 years. For centuries, several people attempted to prove the parallel axiom, and the number of attempts and intensity of efforts reached a crescendo in the 1700s, all meeting with failure.
However, these efforts were not completely in vain. Several equivalent and useful versions of the fifth axiom emerged along the way. Some of them are mentioned below:
- There exist similar triangles of different sizes. (This version was Due to Wallis in 1663)
- Angles in a triangle add up to two right angles. (Given by Euclid himself in his Elements)
The equivalent version given by Euclid is so straightforward that it is still taught to every school kid. This deceptively simple look of this axiom might be one of the reasons why it kept mathematicians wondering for centuries, that it can be logically deduced from the first four axioms.
It wasn’t until 1830 that the explanation for the failure of the earlier attempts surfaced. Completing a quest that had begun 4000 years earlier, Russian mathematician Nikolai Lobachevsky and Hungarian mathematician János Bolyai independently announced the discovery of a new alternative form of geometry, which we now call Hyperbolic Geometry. This new geometry takes place in a new kind of plane (now called the hyperbolic plane). In this Geometry the first four Euclidean axioms hold, but the fifth axiom i.e. the parallel axiom does not! Instead, the following is true:
These pioneers delved into the logical consequences of this revolutionary axiom, guided by purely abstract reasoning. Their efforts unearthed a wealth of fascinating results, revealing a geometric framework that defied the familiar principles of Euclidean Geometry and ventured into uncharted mathematical territory.
In the upcoming articles, we will dive deeper into these radical departures of Non-Euclidean Geometry from its Euclidean counterpart. Together, we will strive to formulate, quantify, and truly grasp the nature of these anomalies. Ultimately, this series aspires to uncover the Geometry of spaces in its most general and profound sense.
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