Abstract
How the volume of a ball grows with its radius splits into three behaviours according to the curvature alone. This paper measures the three and shows numerically that the crowding produced by curvature has the same shape as the crowding produced by dimension. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, the growth splits three ways. In the three-dimensional models the volume of a ball of radius $4$ is $4657.344588$ at curvature $-1$ and $268.0825731$ at curvature $0$, while at curvature $+1$ it has already finished at radius $\pi$ (Section 2).
Second, in hyperbolic space only the outside grows. The share of the ball lying between $r-1$ and $r$ is $0.8646647168$ at $r=20$, agreeing with $1-e^{-2}=0.864664716763$ to ten digits — however large the radius is made, eighty-six per cent goes on living in the outermost unit shell (Section 4).
Third, this is the core. The same crowding can be bought in two different ways. The dimension needed to reproduce that share in flat space is $4.93$ at $r=3$, $18.98$ at $r=10$ and $98.997$ at $r=50$ — the dimension has to keep growing like twice the radius (Section 5).
Fourth, the ratio never turns back. Dividing the volume at greater curvature by the volume at lesser, the ratio fell at every one of six radii, checked by an assertion — the content of the comparison theorem is not its value at one radius but that it never reverses (Section 6).
Fifth, a hyperbolic ball never becomes mostly interior. The ratio of surface to volume goes to zero as $3/r$ in flat space, and settles at $2.0$ by $r=20$ in hyperbolic space (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the volumes of the constant-curvature models and the Bishop–Gromov comparison theorem are standard. The comparison theorem is not proved — what is checked here is that the ratio between two models falls monotonically, and nothing is claimed about a general manifold with a lower bound on its Ricci curvature. Only three dimensions are measured — Section 5 alone varies the dimension, and what it varies is the dimension of flat space. Nothing is claimed about any real space. Relation to Paper 182: 182 measured the crowding produced by dimension — this paper measures the same crowding produced by curvature. One phenomenon with two separate causes, and Section 5 is the demonstration that either will buy it. Relation to Papers 60 and 63: the corpus already treats hyperbolic geometry in velocity space — that is about where relativistic velocities live, and this is about how volume grows. What is added is obtaining the three growth laws by quadrature and in closed form, measuring at six radii that the outer shell settles on a constant share in hyperbolic space, showing that the dimension needed to reproduce that share in flat space is twice the radius, and confirming the monotonicity of the ratio with an assertion.
1. Notation and Position
Consider the volume of a geodesic ball of radius $r$ in the $n$-dimensional model of constant curvature $K$. In three dimensions there are closed forms: $\pi(\sinh 2r-2r)$ at $K=-1$, $\tfrac43\pi r^3$ at $K=0$, and $\pi(2r-\sin 2r)$ at $K=+1$ for $r\le\pi$. The numbers below come from multiprecision quadrature, checked against those forms.
Position: Paper 182 treated the curse of dimensionality and measured that raising the dimension gathers the volume toward the outside. This paper measures the same gathering from the side of curvature — and Section 5 shows that the two can be exchanged.
2. The Growth Splits Three Ways
Check 2.1: measure the volume at several radii in the three-dimensional models.
| $r$ | $K=-1$ | $K=0$ | $K=+1$ |
|---|---|---|---|
| $0.5$ | $0.5504107829$ | $0.5235987756$ | $0.4980335895$ |
| $1$ | $5.110932706$ | $4.188790205$ | $3.426543191$ |
| $2$ | $73.16743277$ | $33.51032164$ | $14.94393577$ |
| $3$ | $614.8510174$ | $113.0973355$ | $19.7273656$ |
| $4$ | $\mathbf{4657.344588}$ | $268.0825731$ | (finished at $r=\pi$) |
Observation 2.2 One grows exponentially, one as a cube, and one stops partway. The same phrase, “the volume of a ball”, points at three separate behaviours according to the sign of the curvature alone.
3. Checking the Tool
Check 3.1: set the quadrature against the closed forms.
| $r$ | Curvature | Quadrature | Closed form |
|---|---|---|---|
| $1$ | $-1$ | $5.11093270570829$ | $5.11093270570829$ |
| $1$ | $0$ | $4.18879020478639$ | $4.18879020478639$ |
| $3$ | $-1$ | $614.851017405332$ | $614.851017405332$ |
| $3$ | $0$ | $113.097335529233$ | $113.097335529233$ |
They agree to fifteen digits. Every number below comes from that quadrature.
4. In Hyperbolic Space Only the Outside Grows
Check 4.1: measure the share of the whole ball taken by the shell between $r-1$ and $r$.
| $r$ | $K=-1$ | $K=0$ | $K=+1$ |
|---|---|---|---|
| $2$ | $0.930147437$ | $0.875$ | $0.7707067775$ |
| $3$ | $0.880999737$ | $0.7037037037$ | $0.242476868$ |
| $5$ | $0.8652687948$ | $0.488$ | (finished) |
| $8$ | $0.8646673804$ | $0.330078125$ | (finished) |
| $12$ | $0.8646647182$ | $0.2297453704$ | (finished) |
| $20$ | $\mathbf{0.8646647168}$ | $0.142625$ | (finished) |
($1-e^{-2}=0.864664716763$.)
Observation 4.2 In flat space the shell empties out, from $0.875$ down to $0.142625$. In hyperbolic space it settles on a constant share, agreeing with $1-e^{-2}$ to ten digits by $r=20$. However large the radius is made, more than eighty-six per cent goes on living in the outermost unit shell.
Observation 4.3 The spherical column loses its meaning partway — it is already down to $0.242476868$ at $r=3$, because the ball covers the whole sphere at $r=\pi$.
5. The Core: the Same Crowding, Bought Two Ways
In flat $n$-dimensional space the outer unit shell takes a share of $1-\left(\frac{r-1}{r}\right)^{n}$. What must $n$ be to match the hyperbolic three-dimensional value of $1-e^{-2}$?
Check 5.1:
| $r$ | Dimension $n$ giving the same share | $2r$ |
|---|---|---|
| $3$ | $4.932606925$ | $6$ |
| $5$ | $8.962840235$ | $10$ |
| $10$ | $18.98244316$ | $20$ |
| $20$ | $38.99145149$ | $40$ |
| $50$ | $\mathbf{98.99663291}$ | $100$ |
Observation 5.2 To produce the same crowding in flat space, the dimension must keep growing like twice the radius. The curvature side was set to $-1$ once and increased nothing thereafter. One phenomenon, bought by paying in dimension on one side and in curvature on the other.
Observation 5.3 So the observation “the volume is gathered toward the outside” does not by itself say what caused it. Paper 182 measured the gathering caused by dimension and this paper the gathering caused by curvature — the shape that shows is the same, and the roots are separate. To tell which, vary the dimension and see whether the share moves.
6. The Ratio Never Turns Back
Check 6.1: divide the volume at the greater curvature by the volume at the lesser.
| $r$ | $K=+1$ over $K=0$ | $K=0$ over $K=-1$ |
|---|---|---|
| $0.5$ | $0.951174091153$ | $0.951287278359$ |
| $1$ | $0.818026929881$ | $0.819574517212$ |
| $1.5$ | $0.635306664876$ | $0.641219749076$ |
| $2$ | $0.445950233935$ | $0.457995044653$ |
| $2.5$ | $0.286028365184$ | $0.301045763042$ |
| $3$ | $0.174428208283$ | $0.183942666317$ |
Every step is smaller than the last. The computation carries an assertion that stops it if a ratio rises, and it never stopped.
Observation 6.2 The content of the comparison theorem is not its value at one radius — it is that it never reverses. An inequality at one point could hold by accident; being monotone could not. What was checked here is only between models, and says nothing about a general manifold with a lower bound on its Ricci curvature.
7. A Hyperbolic Ball Never Becomes Mostly Interior
Check 7.1: divide the surface of the ball by its volume.
| $r$ | $K=-1$ | $K=0$ |
|---|---|---|
| $1$ | $3.3957378$ | $3.0$ |
| $2$ | $2.259195051$ | $1.5$ |
| $5$ | $2.001635891$ | $0.6$ |
| $10$ | $2.000000157$ | $0.3$ |
| $20$ | $\mathbf{2.0}$ | $0.15$ |
Observation 7.2 In flat space it goes to zero as $3/r$ — the larger the ball, the more it becomes “interior”. In hyperbolic space it settles on $2$ — however large it is made, the proportion of skin to interior does not change. It is the eighty-six per cent of Section 4, said another way.
8. The Separator and the Fence
Observation 8.1 (the separator) The same observation, “the volume is gathered toward the outside”, has two separate causes — dimension (Paper 182) and curvature (this paper). To separate them, vary the dimension: crowding from dimension goes away when the dimension is lowered, and crowding from curvature stays put in three dimensions.
Observation 8.2 (the fence) The fence around “the volume of a ball grows like the $n$-th power of the radius” is that the curvature is zero. Outside the fence the form of the growth itself changes — exponential if negative, and finishing partway if positive. And this is not a difference of degree between faster and slower growth: the $K=+1$ column has nothing at all beyond radius $\pi$.
Observation 8.3 In Paper 206 the dimension of a thin cylinder had no answer until a scale was named — here the growth of a ball's volume is not settled until a curvature is named. Both have the same shape: the name of the quantity is not enough.
9. The Census and the Close
What this paper measured:
- Three kinds of growth (Check 2.1): exponential, cubic, and finishing partway.
- The tool was checked (Check 3.1): quadrature and closed form agree to fifteen digits.
- The hyperbolic shell keeps $86\%$ (Check 4.1): ten digits of $1-e^{-2}$.
- Imitating it needs twice the radius in dimensions (Check 5.1): $n=98.997$ at $r=50$.
- The ratio never turns back (Check 6.1): confirmed with an assertion.
- Surface over volume settles on $2$ (Check 7.1).
- The fence is that the curvature be zero (Observation 8.2).
References
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- Gromov, M. (1981). Structures métriques pour les variétés riemanniennes. Paris: CEDIC.
- Cheeger, J., Ebin, D. G. (1975). Comparison Theorems in Riemannian Geometry. Amsterdam: North-Holland.
- Ledoux, M. (2001). The Concentration of Measure Phenomenon. Providence: American Mathematical Society.
- Chavel, I. (2006). Riemannian Geometry: A Modern Introduction. Second edition. Cambridge: Cambridge University Press.
- Yamagishi, Y. (2026). The Wigner Angle Is an Area [Paper 33]. Zenodo. 10.5281/zenodo.22090872
- Yamagishi, Y. (2026). $c$ as the Ball of Velocity Space [Paper 60]. Zenodo. 10.5281/zenodo.21598012
- Yamagishi, Y. (2026). The Velocity–Spacetime Curvature Duality [Paper 63]. Zenodo. 10.5281/zenodo.21606250
- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. Zenodo. 10.5281/zenodo.22104170
- Yamagishi, Y. (2026). The Power in the Eigenvalue Count Is Half the Dimension [Paper 204]. Zenodo. 10.5281/zenodo.22110032
- Yamagishi, Y. (2026). The Spaces Converge and the Dimension Does Not [Paper 206]. Zenodo. 10.5281/zenodo.22110637
- Yamagishi, Y. (2026). There Are Three Tools for Writing Polarization [Paper 207]. Zenodo. 10.5281/zenodo.22110902
Reference
Yamagishi, Y. (2026). Curvature Does the Same Work as Dimension [Paper 208]. YOI: https://www.yamagishiyuuki.com/yoi/208
DOI: 10.5281/zenodo.22111071
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 208. This paper is a preprint and has not been peer reviewed.