Is Our Universe Euclidean or Non-Euclidean?
Going Beyond Euclidean Geometry With Hyperbolic and Spherical Surfaces
For the record, I am on the fence whether the universe is flat or curved (or the shape of a torus, sphere, etc.), but just like how we have discovered that the earth is not flat, it got me wondering and others questioning the familiar geometry that we know and love — Euclidean or “flat” geometry, in terms of applying it to the shape of the universe.
Derived from Euclid’s The Elements, Euclidean geometry starts off with five basic axioms and postulates. It is also known as “plane” geometry, since this type of geometry only deals with flat surfaces or planes (zero curvature). The angles in a triangle always add up to 180°, or for a quadrilateral 360°, and so on (there is a formula for this). The shortest distance between two points is a straight line, not a curved line or arc. More importantly, parallel lines remain parallel. However, the reason non-Euclidean geometry came into fruition was because of the ambiguity of Euclid’s fifth postulate.
Figure 1 summarizes this postulate, also known as the parallel postulate: two given lines, ℓ1 and ℓ2, will eventually intersect or meet if the interior angles α and β created by the third line ℓ add up to less than 180°. This implies that if α and β add up to exactly 180°, then ℓ1 and ℓ2 will not intersect; therefore, they are parallel. (By the way, if the sum is more than 180°, ℓ1 and ℓ2 will intersect on the other side of line ℓ if you imagine extending the lines on the left of the illustration.) For many centuries, several well-known mathematicians from European and Islamic communities tried and failed to prove or disprove the parallel postulate. In the 19th century, the likes of Lobachevsky and Riemann realized that the proof may lie in curved surfaces.
While Euclidean geometry deals with zero curvature, non-Euclidean deals with either positive or negative curvature — positive implies spherical or elliptic surfaces, while negative implies hyperbolic surfaces. These concepts came about after centuries of pondering on what conditions the parallel postulate, as well as other Euclidean axioms, may not be true. A triangle may end up looking differently on various curved surfaces, as the triangle’s angles in positively curved (spherical) surfaces add up to more than 180°, while in negatively curved (hyperbolic) surfaces add up to less than 180°.
Figure 2 now shows the comparisons on the behavior of straight lines by examining what happens when two bugs start crawling on a straight path on each surface, so that we can finally see how Euclid’s parallel postulate starts to break down. In flat planes, we expect parallel lines to remain parallel, so in the zero curvature surface on the left, no matter how far the two bugs travel in their respective straight paths, they will never bump into each other. However, in the spherical ball illustration in the middle, the bugs start to get closer to each other until they eventually meet, while in the rightmost illustration for the saddle surface, the bugs move farther away the more they walk their respective paths.
Figure 3 is an even more simplified illustration explaining what parallel straight lines do in the three different types of curvature. The lines would eventually diverge or move away from each other in hyperbolic surfaces, while converge or intersect in spherical or elliptic surfaces.
Basics of Spherical Geometry
- The shortest distance or “line segment” between two points is called a geodesic (Norton, 2019)
- A straight line is called a great circle, an intersection of the sphere’s surface and a plane slicing through the sphere at its center (Henderson & Taimina, undated)
- There are no such thing as “parallel” lines since each great circle intersects with at least another great circle at two “antipodal” points (spherical geometry is also known as double elliptic geometry, while elliptic geometry is known as single elliptic geometry as it treats the two intersecting points as “one”; elliptic geometry is just a generalization of spherical geometry)(Wikipedia, undated)
When we say spherical geometry, we are talking about the two-dimensional surface of a hollow sphere or all points that have the same distance from a fixed point, or the center, of the sphere. Therefore, we do not take into account the inside of that sphere when discussing spherical geometry. More importantly, straight lines here are known as great circles, similar to tracing the equatorial and (all) longitudinal lines on a map or globe. A geodesic is considered the shortest path between two given points on the sphere’s surface as long as it traces the path of any of the sphere’s great circles, or in other words, a minor arc of a great circle. Figure 4 points out a common mistake in spherical geometry, wherein the parallel of a latitude is a great circle (it is not, since it does not satisfy the proper definition), but the correct way to think of a great circle is illustrated on the sphere on the right.
Moreover, I would just think of a great circle as an intersection of a plane cutting through the hollow sphere through the sphere’s fixed center, and all “straight” paths (the shortest distance to get from one point to another) traveled along the spherical surface can only be on any one of the great circles. Traveling along paths that are not on a great circle, let’s say along the parallel of a latitude, would not be the shortest distance. Geodesics are very useful in air travel, as aircraft would follow the paths of the great circle and not of the parallel of latitude when flying from one part of the globe to the other.
Finally, while in Euclidean geometry there is exactly one straight line that connects two points, in spherical or elliptic geometry this uniqueness does not hold, since you can trace the remaining arc of the great circle to form another segment to connect these two lines. For example, for the sphere on the right in Figure 4, you can form a segment or minor arc (geodesic) from A to B, but you can also form another segment or minor arc from B to A, which goes around the opposite surface of the sphere.
Hyperbolic Geometry
Also known as saddle geometry, hyperbolic geometry can be thought of as a “unit sphere” with radius square root of negative one, or the imaginary number i. The most popular visual representation for this type of non-Euclidean geometry is in Figure 5, and others now prefer the pseudosphere due to its three-dimensional illustration to emphasize the geometry’s constant negative curvature. Below I have listed down other notable differences of hyperbolic and spherical geometry.
Going back to visually representing hyperbolic geometry, others were not satisfied with the saddle or pseudosphere illustrations and have come up with creative or inventive ways to show what hyperbolic surfaces really look like. It is proposed that the so-called edges of a hyperbolic surface would resemble that of the crochet in Figure 7 or that of the coral reef in Figure 8, wherein both exhibit how the surface area is maximized. This means that the closer a person goes to the “edge” of a (four-dimensional) saddle surface, that person would see more and more wrinkling, extending infinitely. The key word is “infinite”, so a hyperbolic universe would be “open”, as the gravitational force would be too weak to contain all matter, and matter would accelerate endlessly.
If You Were on a Hyperbolic Surface…
- If you have a friend walking away from you and towards the “wrinkled” edges, the distance between the two of you becomes exponentially larger, so they appear smaller faster from your perspective
- Shapes at the edge of the circle (also known as the Poincaré disk) in Figure 9 are “equivalent” to shapes at the center (also check out similar hyperbolic art by M.C. Escher)
If You Were on a Spherical Surface…
On a spherical surface, I personally believe it is even weirder. Let’s say you are standing on the north pole, so anything moving away from you and heading towards the south would look smaller once they get closer to the equator, but when they leave the equatorial area and get closer to the south pole, they become bigger again to your point of view, until they cover your entire field of vision. (This perspective on spherical geometry reminds me of a subplot in Cixin Liu’s science fiction novel Death’s End, the third book in his trilogy Remembrance of Earth’s Past or more popularly known as the Three-Body Problem Trilogy, wherein a giant is living on an island, but once people traveled towards the island and get closer to that giant, his size becomes more normal and no longer looks like a giant from their perspective.) Even weirder, because in spherical geometry, light travels the path of a great circle, so if you are alone on the surface and you look to the horizon, there is no horizon, and your field of vision would also go the path of the great circle around the surface of the sphere until you see the back of your head. However, even though in reality we live in a spherical planet, this would not happen due to the curvature of spacetime and gravity, which are topics for another discussion, specifically Einstein’s relativity. Experts suggest that if the universe is a (four-dimensional) sphere, the gravitational force is strong enough to hold in all matter from escaping, thus the universe becomes “finite”, and in other words, the spherical universe is “closed”. (Again, this reminds me of Death’s End.)
Things to Ponder
So, what is the shape of the universe? The consensus up until around 2019 was that the universe is flat with the standard model… but I don’t know, I guess I am, first of all, afraid to go against the experts. Measurements on the cosmic microwave background imply that the angles have not “bulged” (spherical) or “pinched” (hyperbolic), suggesting that the universe is flat, but at the same time there are doubts on the precision of the measurements. On the other hand, I am also afraid of being known as a “flat-universer” in case the Euclidean perspective turns out to be wrong, as current research now suggests that the universe appears to be a four-dimensional (hyper)sphere.
