Abstract
Nothing beyond the horizon can be seen, it would seem. This paper shows that tall things are seen there routinely——a tower $8.6697$ times one’s own horizon away. No new theorem or law is claimed.
First, the distance to the horizon is $\sqrt{2Rh+h^{2}}$, or $\mathbf{4.6542}$ km at a height of $1.7$ m (Section 2).
Second, being a square root, doubling one’s height buys only $\mathbf{1.4142}$ (Section 2).
Third, and this is the core of the paper. Tall things are visible past the horizon: a $100$ m tower from $\mathbf{40.3503}$ km, $\mathbf{8.6697}$ times one’s own horizon (Section 3).
Fourth, Mount Fuji ($3776$ m) is visible from $\mathbf{224.0352}$ km (Section 3).
Fifth, refraction multiplies the distance by $\mathbf{1.063015}$, the square root of the $1.13$ effective-radius factor (Section 4).
Sixth, the separator is whether one is looking at the ground or at an object (Section 5).
Scope (what is not claimed)
No new theorem or law is claimed——the horizon distance, the sum formula for two objects and the effective-radius factor for atmospheric refraction are all known. No surveying is done——real visibility is decided by terrain and buildings. Only a sphere and straight lines of sight. No atmosphere is built——the vertical profile of refractive index, mirages and looming are not the subject. One representative value $k=1.13$ is used. The Earth is taken as a sphere——oblateness, swell and tide are not the subject. Whether a thing is actually seen is not claimed——eyesight, atmospheric clarity and contrast are not the subject. Only whether the sight line is geometrically blocked. Horizon over land and sea are not distinguished——an unobstructed spherical horizon is treated. Relation to earlier papers: Paper 395 showed that sound travels far because it bends——here light bends too and gains $1.063$. Paper 389 showed that length, not thickness, is what binds——here the other object’s height, not one’s own. Paper 391 showed that a representative value does not represent——here “distance to the horizon” does not represent visibility. Paper 300 showed that same-root and different-root can be decided——one’s own height and the object’s height are different-root by that criterion. What is added is putting numbers on the horizon distance at four heights, confirming the square root at factors of two and four, obtaining the visibility distances of a tower and of a mountain, giving the refraction factor $1.063015$, and placing the separator at “ground or object”.
1. Notation and the Question
Setting: take the Earth as a sphere of radius $R=6371$ km and find the distance from an eye at height $h$ to the horizon.
$$d(h)=\sqrt{2Rh+h^{2}}\ \approx\ \sqrt{2Rh}$$An object of height $h_{2}$ is visible from an eye at height $h_{1}$ out to $d(h_{1})+d(h_{2})$.
Observation 1.1 $d$ is the square root of $h$. The $h^{2}$ term is negligible for $h\ll R$——raising the height buys little.
Observation 1.2 the two-object formula is a sum. $d(h_{1})+d(h_{2})$——one’s own height and the object’s act separately.
Observation 1.3 if the object is tall, a low eye still reaches. One term of the sum is large——“distance to the horizon” is only one half of that sum.
Question: is nothing beyond the horizon visible?
2. It Grows Only as the Square Root of Height
Computation 2.1: horizon distance against height (computed in this paper).
| Eye height [m] | To the horizon [km] | With refraction [km] |
|---|---|---|
| $\mathbf{1.7}$ (a person) | $\mathbf{4.6542}$ | $4.9475$ |
| $10$ | $11.2881$ | $11.9994$ |
| $100$ | $35.6961$ | $37.9455$ |
| $10000$ (an airliner) | $\mathbf{357.0994}$ | $379.6019$ |
Observation 2.2 the horizon of a person $1.7$ m tall is only $4.6542$ km. Under five kilometres——an hour’s walk.
Observation 2.3 four times the height gives exactly twice the distance. $1.7\to6.8$ m gives $4.6542\to9.3084$ km——the square root confirmed numerically.
Observation 2.4 twice the height buys only $1.414214$. $\sqrt{2}$——standing on tiptoe is inefficient.
Observation 2.5 at an airliner’s $10$ km it is $357.0994$ km. Here the $h^{2}$ term acts——$\sqrt{2Rh}$ alone gives $356.9$ km, off by $0.06$ per cent.
Observation 2.6 at a person’s height the $h^{2}$ term acts at under $10^{-6}$. The range where the approximation holds——where it starts to matter is settled by putting in the numbers.
3. A $100$ m Tower Is Visible $8.6697$ Horizons Away
Computation 3.1: visibility from an eye at $1.7$ m (computed in this paper).
| Object | Visible from [km] | In units of one’s own horizon |
|---|---|---|
| Sea surface (the horizon) | $4.6542$ | $1.0000$ |
| A $10$ m boat | $15.9423$ | $3.4253$ |
| A $100$ m tower | $\mathbf{40.3503}$ | $\mathbf{8.6697}$ |
| Mount Fuji, $3776$ m | $\mathbf{224.0352}$ | $\mathbf{48.1361}$ |
Observation 3.2 this is the core of the paper. A $100$ m tower is visible from $40.3503$ km——$8.6697$ times one’s own horizon of $4.6542$ km. Seeing past the horizon is not an exception.
Observation 3.3 Mount Fuji from $224.0352$ km, $48.1361$ times. Nearly fifty horizons——consistent with its being seen from far away.
Observation 3.4 what is seen is only the summit. The base is hidden by the Earth——“visible” and “visible entire” are different.
Observation 3.5 the object’s height matters more than one’s own. The larger term dominates the sum——the tower’s $35.6961$ km is $7.67$ times one’s own $4.6542$ km.
Observation 3.6 the shape is that of Paper 389. There length rather than thickness was what bound——here the object’s height rather than one’s own. In both, the quantity attended to is not the one that acts.
4. Light Bends a Little Too
Observation 4.1 air thins with height, so rays bend downward. The same sense as the Earth’s curvature——putting the horizon a little further off.
Observation 4.2 by convention the Earth’s radius is multiplied by $1.13$ and the rays treated as straight. The effective radius——the distance grows by $\sqrt{1.13}=1.063015$.
Observation 4.3 at a person’s height $4.6542$ km becomes $4.9475$ km. $293$ m further——a $6.3$ per cent effect.
Observation 4.4 the shape is that of Paper 395. There sound travelled far because it bent——here light is seen a little further because it bends. In both, not travelling straight extends the reach.
Observation 4.5 but the magnitudes differ by orders. Sound is trapped and carries thousands of kilometres; light gains $6.3$ per cent——the same word “bends”, with gradients that differ by orders of magnitude.
Observation 4.6 $1.13$ is representative, not constant. A temperature inversion raises it——and a ship beyond the horizon appears to float.
5. Separator: The Ground, or an Object
Observation 5.1 separator——does the thing being looked at have height or not.
| Object | Height | Visible from | At an eye height of $1.7$ m |
|---|---|---|---|
| Sea or ground | $0$ | $d(h_{1})$ | $4.6542$ km |
| Anything with height | $h_{2}>0$ | $d(h_{1})+d(h_{2})$ | unbounded |
Observation 5.2 only the first row is “the distance to the horizon”. An object of height zero——confuse it with visibility in general and the error is orders of magnitude.
Observation 5.3 the second row has no ceiling. As far as the object is tall——the Moon is $380{,}000$ km away and is seen.
Observation 5.4 the separator can be moved. Let the object’s height go to zero and the visibility falls to one’s own horizon——the two are connected continuously.
Observation 5.5 to be honest——both $40.3503$ km and $224.0352$ km are pure geometry without refraction. In practice clarity and contrast decide——what does not move is that the distance goes as the square root of height and that two objects add.
Observation 5.6 one last thing——“the distance to the horizon” is a quantity about oneself. It says nothing about the other object——call it visibility and one attends only to one’s own height.
6. Roll Call and Closing
- $d=\sqrt{2Rh+h^{2}}$ (Section 1): the square root of height.
- $4.6542$ km at $1.7$ m (Computation 2.1): under five kilometres.
- Four times the height, twice the distance (Observation 2.3): the square root confirmed.
- Twice the height buys $1.414214$ (Observation 2.4): tiptoe is inefficient.
- $357.0994$ km from an airliner (Computation 2.1): the $h^{2}$ term starts to act.
- A $100$ m tower from $40.3503$ km (Computation 3.1): $8.6697$ horizons.
- Mount Fuji from $224.0352$ km (Computation 3.1): $48.1361$ horizons.
- Only the summit is seen (Observation 3.4): the base is hidden.
- Refraction gives $1.063015$ (Observation 4.2): the root of $1.13$.
- Only a zero-height object is the horizon (Observation 5.2): confusing them costs orders.
Closing: nothing beyond the horizon can be seen, it would seem——but tall things are seen there routinely. The distance to the horizon is $d(h)=\sqrt{2Rh+h^{2}}$, the square root of the height——only $\mathbf{4.6542}$ km at $1.7$ m, an hour’s walk. Four times the height gives exactly twice the distance, and twice the height buys only $1.414214$——standing on tiptoe is inefficient. At an airliner’s $10$ km it is $357.0994$ km, where the $h^{2}$ term acts and $\sqrt{2Rh}$ alone is off by $0.06$ per cent; at a person’s height that term acts at under $10^{-6}$. But visibility between two objects is a sum——$d(h_{1})+d(h_{2})$, so a $100$ m tower is visible from $\mathbf{40.3503}$ km, which is $\mathbf{8.6697}$ times one’s own horizon. Mount Fuji ($3776$ m) from $\mathbf{224.0352}$ km, $48.1361$ times, consistent with its being seen from far away. What is seen is only the summit, the base being hidden by the Earth——“visible” and “visible entire” are different. The larger term dominates: the tower’s $35.6961$ km is $7.67$ times one’s own $4.6542$ km, so the object’s height matters more than one’s own. As in Paper 389, the quantity attended to is not the one that acts. Light bends a little too——air thins with height so rays bend in the same sense as the Earth’s curvature, and by convention the radius is multiplied by $1.13$, giving a distance factor of $\sqrt{1.13}=\mathbf{1.063015}$. At a person’s height $4.6542$ km becomes $4.9475$ km, $293$ m further, a $6.3$ per cent effect. As in Paper 395, not travelling straight extends the reach——but the magnitudes differ by orders: sound is trapped and carries thousands of kilometres, light gains $6.3$ per cent. And $1.13$ is representative rather than constant: a temperature inversion raises it, and a ship beyond the horizon appears to float. One thing does the separating——does the thing being looked at have height or not. Sea or ground, of height zero, is visible out to $d(h_{1})=4.6542$ km; anything with height, out to $d(h_{1})+d(h_{2})$, without a ceiling. Only the first is “the distance to the horizon”, and confusing it with visibility in general costs orders of magnitude. Let the object’s height go to zero and the visibility falls to one’s own horizon: the two are connected continuously. To be honest——both $40.3503$ km and $224.0352$ km are pure geometry without refraction, and in practice clarity and contrast decide. What does not move is that the distance goes as the square root of height and that two objects add. One last thing——“the distance to the horizon” is a quantity about oneself. It says nothing about the other object. Call it visibility and one attends only to one’s own height.
References
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- Young, A. T. (2006). Understanding Astronomical Refraction. The Observatory.
- van der Werf, S. Y. (2003). Ray tracing and refraction in the modified US1976 atmosphere. Applied Optics.
- Lynch, D. K., Livingston, W. (2001). Color and Light in Nature (2nd ed.). Cambridge: Cambridge University Press.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Paper Stops Folding Not Because It Gets Thick, but Because It Runs Out of Length [Paper 389]. Zenodo. 10.5281/zenodo.22699183
- Yamagishi, Y. (2026). The Significant Wave Height Is Neither the Mean Nor the Maximum [Paper 391]. Zenodo. 10.5281/zenodo.22699212
- Yamagishi, Y. (2026). Sound Travels Far Because It Does Not Travel Straight [Paper 395]. Zenodo. 10.5281/zenodo.22699289
Reference
Yamagishi, Y. (2026). The Horizon Is Not the Limit of What Can Be Seen [Paper 404]. YOI: https://www.yamagishiyuuki.com/yoi/404
DOI: 10.5281/zenodo.22699418
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 404. This paper is a preprint and has not been peer reviewed.