From: veritasium
The understanding of our universe was significantly advanced by challenging a single sentence in Euclid’s “Elements,” one of the oldest and most widely published math books, second only to the Bible [00:00:00]. For over 2,000 years, “The Elements” served as the primary mathematics textbook [00:00:12], yet mathematicians were skeptical of one specific line that seemed out of place [00:00:18]. Slight alterations to this line eventually led to the discovery of new geometries [00:00:32], which, 80 years later, proved fundamental to comprehending our universe [00:00:41].
Euclid’s Elements and the Problem of Postulate 5
Around 300 BC, the Greek mathematician Euclid undertook the monumental task of summarizing all known mathematics in “The Elements” [00:00:52]. Prior to Euclid, mathematical proofs often suffered from circular reasoning, lacking a fundamental starting point [00:01:09]. Euclid resolved this by establishing a set of simple, basic truths, called postulates, from which all other theorems could be logically derived [00:01:36]. This method perfected the rigorous mathematical proof system still used today [00:02:02].
“The Elements” consisted of 13 books containing 465 theorems covering geometry and number theory, all based on definitions, common notions, and five postulates [00:02:11].
The first four postulates are straightforward [00:03:00]:
- A straight line can be drawn between any two points [00:03:02].
- A straight line can be extended indefinitely [00:03:06].
- A circle can be drawn with any center and radius [00:03:11].
- All right angles are equal to each other [00:03:15].
However, the fifth postulate stood out due to its complexity and length [00:03:19]: “If a straight line falling on two straight lines makes 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” [00:03:22]. Its convoluted nature made mathematicians suspicious, leading them to believe it might be a theorem that could be proven from the first four [00:03:51].
Attempts to Prove the Fifth Postulate
Many mathematicians, including Ptolemy and Proclus, attempted to prove the fifth postulate from the first four, but they only managed to restate it in different words [00:04:10]. One common formulation, known as the Parallel Postulate, states: if you have a line and a point not on that line, then there is a single unique line parallel to the first line [00:04:23].
When direct proof failed, others like al-Haytham and Omar Khayyam tried proof by contradiction [00:04:41]. This involved assuming the fifth postulate was false while keeping the first four true [00:04:51]. Two alternatives to Euclid’s fifth postulate were considered:
- No parallel lines: This alternative was ruled out because it implied that lines must be finite in length, contradicting Euclid’s second postulate that lines can be extended indefinitely [00:05:23].
- More than one parallel line: Mathematicians couldn’t find a contradiction when assuming that more than one parallel line could be drawn through a point not on the first line [00:05:42].
After over 2,000 years of failed attempts to prove it, the fifth postulate remained a mystery [00:05:58].
Discovery of Non-Euclidean Geometries
János Bolyai and Hyperbolic Geometry
Around 1820, a 17-year-old student named János Bolyai became engrossed in the problem of the fifth postulate, despite his father’s warning about the “bottomless night” it represented [00:06:09]. After years of work, Bolyai realized that the fifth postulate might be completely independent of the other four [00:06:46]. He envisioned a world where more than one parallel line could exist through a point not on a given line, which necessitated a curved surface [00:06:57].
In this curved space, “straight lines” are the shortest paths between two points, known as geodesics [00:07:20]. On such a surface, these paths might appear bent to an external observer because the surface itself is curved [00:07:25]. This concept led to what is now known as hyperbolic geometry [00:07:55].
Hyperbolic geometry can be visualized as an “infinite crumpling mess” of saddles [00:08:00]. As one moves outward from the center, more and more “fabric” is created, pushing parallel lines apart exponentially, causing a crumpling effect [00:08:15].
The Poincare Disk Model
To represent the infinite hyperbolic plane, a map was developed that fits the entire plane into a disk, known as the Poincare Disk Model [00:08:45]. In this model:
- The entire plane is filled with triangles that appear smaller further from the center but are actually the same size [00:08:53].
- Straight lines are arcs of circles that intersect the disk at 90 degrees [00:09:32].
- Lines infinitely close to the edge of the disk appear infinitely small, never quite reaching the boundary [00:09:14].
Bolyai discovered that the behavior in hyperbolic geometry was mathematically consistent, despite being very different from Euclidean geometry [00:09:56]. In 1823, he wrote to his father, “Out of nothing, I have created a strange new universe” [00:10:07]. Bolyai published his findings in 1832 as a 24-page appendix to his father’s textbook [00:11:17].
Carl Friedrich Gauss and the Independent Discovery
Farkas Bolyai sent his son’s work to Carl Friedrich Gauss, one of the greatest mathematicians [00:11:32]. Gauss responded that the work coincided almost exactly with his own meditations over the past 30-35 years [00:11:42]. Years prior, in 1824, Gauss had privately described a “curious geometry” with paradoxical theorems [00:11:56]. For example, in this geometry, the angles of a triangle can become as small as desired if the sides are large enough, yet the area of the triangle can never exceed a definite limit [00:12:12]. This can be seen in the Poincare Disk Model where infinitely long lines form triangles with finite areas [00:13:33].
Gauss found this geometry thoroughly consistent, naming it non-Euclidean geometry [00:13:01]. However, he feared ridicule and chose not to publish his findings [00:13:24]. Bolyai, upon receiving Gauss’s response, was devastated, believing Gauss was trying to undermine and steal his ideas, leading him to never publish again [00:15:06]. Bolyai later learned that Nikolai Lobachevsky had also independently discovered non-Euclidean geometry [00:15:20].
Spherical Geometry (Elliptic Geometry)
Another geometry that deviates from Euclidean principles is spherical geometry, which we are all familiar with because we live on a sphere (Earth) [00:13:31]. In spherical geometry:
- Straight lines are parts of great circles (circles with the largest possible circumference, like the equator or lines of longitude) [00:13:42].
- Any two great circles on a sphere will always intersect, meaning there are no parallel lines on a sphere [00:14:08].
Initially, spherical geometry wasn’t considered a non-Euclidean geometry because lines on a sphere cannot be extended indefinitely, violating Euclid’s second postulate [00:16:06]. However, in 1854, Bernhard Riemann generalized the second postulate from “infinite extension” to “unbounded,” allowing spherical geometry to be classified as a valid non-Euclidean geometry [00:16:25]. By accepting the generalized first four postulates and assuming no parallel lines (the opposite of Euclid’s fifth), spherical or elliptic geometry can be derived [00:16:43].
Consistency of Non-Euclidean Geometries
In 1868, Eugenio Beltrami definitively proved that hyperbolic and spherical geometries were just as consistent as Euclid’s flat geometry [00:19:57]. This means that if any inconsistencies were found in these non-Euclidean geometries, they would also have to be present in Euclidean geometry [00:20:08].
Geometries as Games: Riemann’s Vision
Geometry can be thought of as a game where the first four postulates are the minimum rules [00:19:01]. The fifth postulate then determines the world you play in [00:19:05]:
- Spherical Geometry: No parallel lines [00:19:09].
- Flat (Euclidean) Geometry: One parallel line [00:19:13].
- Hyperbolic Geometry: More than one parallel line [00:19:17].
Riemann, in 1854, took this further by proposing a geometry where curvature could vary from place to place, extending to three or more dimensions [00:19:23].
Modern Relevance: Understanding Our Universe
Einstein’s General Relativity and Curved spacetime and geodesics
The seemingly “strange new universes” of non-Euclidean geometries became crucial to understanding our own universe with Einstein’s General Theory of Relativity, published in 1915 [00:20:05]. Einstein realized that gravity is not a force, but rather a manifestation of spacetime itself being curved by massive objects [00:22:35]. Objects moving through this curved spacetime follow the shortest path, or geodesic [00:22:43].
For example, astronauts in the space station follow a straight line (a geodesic) through Earth’s spacetime, which appears as an orbit to a distant observer because of the curvature [00:22:50].
Evidence for curved spacetime includes:
- Gravitational Lensing: In 2014, astronomers observed the same supernova in four different places due to a massive galaxy between the supernova and Earth curving spacetime, causing light to take multiple paths [00:23:15]. This gravitational lens even allowed astronomers to predict and observe a replay of the supernova a year later [00:24:00].
- Gravitational Waves: We can now measure ripples in spacetime itself, formed by cosmic events like black hole mergers [00:24:19].
Measuring the Shape of the Universe
The overall shape of the universe can be determined by measuring the angles of a triangle [00:25:00]:
- Flat Geometry: Angles add up to exactly 180 degrees [00:25:04].
- Spherical Geometry: Angles add up to more than 180 degrees [00:25:11].
- Hyperbolic Geometry: Angles add up to less than 180 degrees [00:25:16].
Gauss attempted such a measurement with a triangle formed by three mountains, finding its angles summed to 180 degrees within observational error [00:25:27]. However, this triangle was far too small relative to the universe’s scale to detect its curvature [00:26:10].
To overcome this scale issue, scientists use the largest possible triangles by observing the Cosmic Microwave Background (CMB), the earliest light we can see from when the universe was 380,000 years old [00:26:26]. By predicting the expected size of temperature variations in the CMB (if the universe is flat) and comparing it to measured angles, scientists can infer the universe’s curvature [00:27:00].
Data from the Plank mission shows that the universe’s curvature is approximately 0.0007 ± 0.0019, which is essentially zero within the margin of error [00:28:16]. This strongly suggests that the universe we live in is flat [00:28:35].
This flatness is remarkably serendipitous, as a slight increase or decrease in the universe’s average mass-energy density (equivalent to just one hydrogen atom per cubic meter) would result in a spherically or hyperbolically curved universe, respectively [00:28:46]. The reason for this precise mass-energy density remains an open question [00:29:03].
The non-Euclidean geometries, born from over 2,000 years of inquiry into a single sentence in Euclid’s “Elements,” are at the very heart of General Relativity, one of our best physical theories of reality [00:29:09].