EXPLORING MANIFOLDS
On the Revolutions of the Heavenly Spheres
From a distance, the horizon appears as a perfectly straight line.
Yet the Earth is in fact a sphere with a radius of about $R \approx 6{,}371 ,\text{km}$, and the horizon is part of its curved surface. Human eyes can hardly perceive this curvature, but mathematical analysis reveals the subtle difference between straightness and curvature. This can be called the signature of curvature.
With the observer’s eye height $h$ and the Earth’s radius $R$, the distance to the horizon is:
$d = \sqrt{2Rh + h^2} \approx \sqrt{2Rh}$
For example, if $h = 1.66,\text{m}$, then $d \approx 4.6,\text{km}$.
When this length of the horizon is compared to a straight line,
the Earth’s surface appears slightly lower in the center.
This vertical difference is the sagitta:
$s = R - \sqrt{R^2 - \left(\tfrac{L}{2}\right)^2} \approx \tfrac{L^2}{8R}$
Substituting $L = 4600,\text{m}$ gives $s \approx 0.415,\text{m}$, about 41.5 cm. On a photograph, this would correspond to a single pixel; on a print, to the thickness of a human hair. To the eye, the line seems straight, yet physical reality leaves a curve.
The dip angle $\gamma$, which shows how much lower the horizon appears compared to the true horizontal, is:
$\cos \gamma = \frac{R}{R+h}$
For $h = 1.66,\text{m}$, this yields $\gamma \approx 0.041^\circ$, only 1/25 of a degree—beyond the threshold of human perception.
In the language of Riemannian geometry, the Earth’s surface is a two-dimensional manifold of radius $R$, with Gaussian curvature:
$K = \frac{1}{R^2} \approx 2.5 \times 10^{-14},\text{m}^{-2}$
This extremely small value explains why the surface feels locally flat yet is globally curved. On the sphere, triangles have angle sums exceeding 180°, and the shortest path (a geodesic) is a great circle. The horizon thus embodies a duality: intrinsically a straight line, extrinsically a curve.
In Riemannian geometry, the effects of curvature manifest in various ways. One of these is the phenomenon where the sum of the interior angles of a triangle exceeds 180 degrees. If you mark three points on the Earth's surface and connect them along the surface to form a spherical triangle, the sum of its three interior angles will always be greater than 180°. The larger the area the triangle occupies, the greater this excess amount (known as spherical excess). However, for triangles of everyday size (e.g., those with sides within a few kilometers), this excess is so extremely small that it is virtually indistinguishable from planar geometry.
This is the characteristic of spherical geometry:
being locally flat but globally curved.
Our intuition perceives the horizon as flat because we are accustomed to this locally Euclidean geometry.
Another significant effect of curvature is the difference between the concepts of straight and curved lines on Earth. On a flat plane (Euclidean geometry), a straight line represents the shortest path, but the situation is slightly different in spherical geometry. On a sphere, the equivalent of a "straight line" is a geodesic, which is a path along a great circle of the sphere. For example, great-circle routes on the Earth's surface correspond to geometrically straight paths in Riemannian terms. From the perspective of an ant moving along the surface, traveling along a great circle is a path of moving straight ahead without turning at all.
However, to an external observer, that great-circle path appears as a curve following the sphere's surface. In other words, a straight line in the world of Riemannian geometry can appear curved when viewed from our three-dimensional space. The horizon is a circular boundary centered on the observer, defined as a path (part of a great circle) on the sphere of the Earth. Therefore, from a Riemannian perspective, the horizon is a "straight" boundary, but to our eyes in three-dimensional space, it is a very gently curved line. This concept is central to Riemannian geometry, showing that a line that is intrinsically straight can appear extrinsically curved.
Experiencing the Gap Between Intuition and Reality
Through the figures examined earlier, we can confirm the subtle gap that exists between our intuition and reality. Due to the limits of perception, humans hardly sense the curvature of the horizon, thus recognizing it as a flat line. However, in strict physical reality, the horizon drops tens of centimeters below eye level (an angular difference of about 0.04°), and the Earth's surface bulges upward by over 40 centimeters within that distance. This difference, though small, is a clear trace of existing curvature.
Artists and scientists can focus on this gap to present new visual and auditory experiences. For instance, by describing the horizon's curvature using the language of Riemannian geometry and then translating that data into sound or visualizing it with interactive tools, an audience could be made to experience the subtle curvature of the horizon they normally do not perceive. This can be seen as an attempt to expand the hidden curvature of a seemingly flat space into the realm of sensation. Indeed, many astronomical phenomena and navigational experiences reveal the effects of this curvature (e.g., the mast of a distant ship becoming visible before its hull). By artistically translating these facts revealed through observation and mathematical language, the audience can gain a new awareness of the world as a curved surface that is intuitively flat.
The metaphor of the horizon
In summary, the horizon possesses the paradoxical property of being a "curved surface that appears as a straight line," which is clearly understood when interpreted through Riemannian geometry. Although the difference is so minute that it is overlooked in daily life, the 41.5 cm signature of curvature is evidence that the space we inhabit is not a Euclidean plane but a spherical one. Revealing and allowing people to experience this hidden reality is an interesting meeting point for science and art. Recognizing and exploring the thin gap between our intuition and reality can transform even the trivial experience of looking at the horizon into something new.
The metaphor of the horizon connects directly to Gödel’s Incompleteness Theorem. For any consistent formal system $F$ capable of arithmetic, there exists a proposition $G$ such that:
$F \text{ is consistent } \Rightarrow F \nvdash G \quad \text{and} \quad F \nvdash \lnot G$
No system can be both complete and consistent; truth always exceeds proof.
Thus the horizon itself is incompleteness: a line that is never truly straight, always conditional, always subject to revision. The transition from the absoluteness of Euclidean geometry to the conditional openness of modern thought is inscribed within this curve.
The horizon is not only a visual motif but also a philosophical statement. It exposes the thin gap between intuition and reality, reminding us that knowledge is not a closed totality but an endless journey.