The Volume of the Unit Ball Is Greatest in Five Dimensions — second edition: that five is a consequence of having set the radius to one.
That the volume of the unit ball in n dimensions, pi to the n over two divided by Gamma of n over two plus one, is greatest at n equal to five among the integers (and at about 5.2569 as a continuous variable) is classically known (Smith and Vamanamurthy 1989). The first edition placed this within the corpus, noting that reading c as a radius makes c to the n the scaling of an n-dimensional ball’s volume, and that the dimension of greatest volume corresponds to the intuition of greatest capacity, a server — while rightly warning that this five must not be confused with the three of stability. The second edition pushes that warning further than the first edition anticipated: it turns it on the five itself. NO NEW PHYSICAL LAW AND NO NEW MATHEMATICAL THEOREM IS CLAIMED.
First, and this is the core of the edition. The answer five is a consequence of having set the radius to one. Comparing balls of radius lambda, what is to be maximised is the volume coefficient times lambda to the n, and differentiating its logarithm gives the condition that the digamma of n over two plus one equals the logarithm of pi lambda squared. At lambda equal to one the right side is log pi and the peak is at about 5.2569; but changing lambda moves the peak (all verified numerically here): 0.48 at one half, 5.26 at one, 24.1 at two, 55.5 at three, and 627 at ten. Measure the same ball with a different ruler and the peak goes wherever you like; comparing balls of radius ten, the special dimension is 627. The statement that the unit ball’s volume is greatest in five dimensions therefore says nothing about five — it says something about one. In the vocabulary of Paper 95 it is a convention, not a fact. The theorem itself is correct; what is corrected is having read its five as a fact about dimension.
Second, the volumes in four and five dimensions cannot be compared at all. The coefficient is dimensionless, while the actual volume of a ball of radius r is the coefficient times r to the n, of dimension L to the n. Saying that the five-dimensional coefficient exceeds the four-dimensional one sets a number attached to L to the fifth beside one attached to L to the fourth. This is the error the corpus has named repeatedly: Paper 28 judged that three halves is a scaling exponent and not a dimension, and Paper 45 that an area is not a solid angle. Quantities of different dimension cannot be compared as they stand, and the comparison looks possible only because the radius was set to one.
Third, this sharpens the first edition’s fourth section. That section said, correctly, that changing the criterion changes the special dimension. What the second edition adds is that the criteria themselves are of two kinds. That all bound orbits close (Bertrand’s theorem, three dimensions), that an attractive potential has a bound ground state (at most three), and that Huygens’ principle holds (odd dimensions) — none of these changes whether length is measured in metres or in feet. Whether an orbit closes is not a question about rulers. Only the peak of the unit ball’s volume changes its answer when the ruler changes. So the first edition was right that the two must not be confused, but the reason is not only that the criteria differ: three dimensions is a fact about the world, and five is a fact about the number one. The two may be listed in one table but cannot be counted with the same weight.
Fourth, reading c as a radius can be put more strongly than the first edition thought. It weakened this to a choice of representation on a par with setting G to four pi, adding that it is not a claim that c is a length. No weakening was needed: in special relativity attainable velocities satisfy the magnitude of v being less than c, so velocity space really contains a ball of radius c. Its volume is the coefficient times c to the n, of dimension length over time to the n, which is consistent; c need not be reinterpreted as a length at all, since it is a radius in velocity space. This object is already in the corpus, in Paper 20 (the volume of the velocity sphere) and Paper 63 (the curvature duality between velocity space and spacetime). It is not a patch but a real geometric object.
Accordingly the first edition’s corollary about a five-dimensional server is withdrawn, for two reasons. First, the comparison itself depends on the ruler: holding the radius at c and varying the dimension compares a quantity of length-over-time to the fourth with one to the fifth, and which is larger depends on the units in which c is measured; five appears only where c equals one. Second, the dimension of velocity space equals that of position space, which is three, so a five-dimensional velocity space does not exist in our world. The first edition rightly limited the correspondence to a suggestive resonance rather than a proof; the second edition goes further and says that the resonance itself was a product of the ruler.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). That the unit ball’s volume peaks at five dimensions is classically known, and no new physical law or mathematical theorem is claimed. What the second edition adds is only a judgement about the range within which that known fact means anything. What is not claimed: any new physical law (there is none) / any new mathematical theorem (there is none) / that five is a fact about dimension (it follows from setting the radius to one and is a convention) / the correspondence with a five-dimensional server (withdrawn) / that comparing the four- and five-dimensional coefficients is meaningful (they are quantities of different dimension) / that five may be counted with the same weight as three (one moves with the ruler, the other does not). What remains: on seeing a number that looks special, ask first what was set equal to one.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Volume of the Unit Ball Is Greatest in Five Dimensions [Paper 59]. Zenodo.
DOI: 10.5281/zenodo.21597865
ORCID: 0009-0006-8587-3450