The pi that carries no dimension – the pi on Ramanujan’s page did not come from a circle.
The corpus has followed 4pi for a long time. This paper turns the same test on pi itself: when the dimension d is moved, does that pi move? NO NEW MATHEMATICAL THEOREM IS CLAIMED. Ramanujan’s series, Chudnovsky’s series, the Gauss-Legendre algorithm, the BBP formula, and the imaginary quadratic fields of class number one with their j-invariants are all standard.
First, this is one of the rare papers that confirms a shared root. The pi of a solid angle and the pi of a Gaussian have the same root. The area of the unit sphere is twice pi to the d over two divided by the gamma function at d over two, and the Gaussian integral over d-dimensional space is pi to the d over two. Integrating the ball volumes from a recursion that assumes no pi to the d over two anywhere – starting from a length of two in one dimension and integrating cross-sections – reproduces the closed form for d from two through eight, with differences of zero or of order ten to the minus one hundred and twenty. The d-th power of the Gaussian integral was likewise checked against pi to the d over two at d of two, three and six. Both carry the exponent d over two and move together when d moves – indeed the Gaussian integral is the very instrument by which one computes the area of a sphere. It is rare for the corpus to confirm a shared root; this is the second time, after Paper 142. Confirming sameness is as much a part of a census as counting differences.
Second, and this is the core. There is no d in the pi of Ramanujan’s series. Two root two over nine thousand eight hundred and one, times the sum over k of the factorial of four k times one thousand one hundred and three plus twenty-six thousand three hundred and ninety k, divided by the fourth power of k factorial times three hundred and ninety-six to the four k – nowhere in this formula does a dimension appear, so there is nothing to move. By the corpus’s own test (dimension-independence, Paper 129) this pi is not a solid angle. Here it appears as a period of a modular form, with no sphere involved. This does not mean there are two pis: as a number it is one. What differs is the reason it has to be there, and since Paper 95 the corpus has called that difference an address.
Third, a census of digits. The Leibniz series gives one digit in ten terms, three in a thousand, five in a hundred thousand and six in a million – about ten to the k terms for k digits. Ramanujan gives seven digits in one term, fifteen in two and forty-seven in six, that is about eight per term. Chudnovsky gives thirteen in one term and eighty-four in six, about fourteen per term (computed here). Yet all three are of one type – a fixed number of digits per term – and only the constant differs.
Fourth, fast has three roots too. Gauss-Legendre, the arithmetic-geometric mean, reaches three, eight, nineteen, forty, eighty-four, one hundred and seventy-one, and three hundred and forty-five digits on successive iterations: the digits double (computed here at five hundred digit precision). That is not a larger constant but a different law of growth – linear and quadratic are distinct roots. BBP, by contrast, is fast in neither sense: it gives few digits per term and doubles nothing. What it gives is random access, the n-th hexadecimal digit of pi without the preceding n minus one (positions one, two, ten, fifty, one hundred and two hundred all verified here). Among the three good algorithms there is no ordering; which is best depends on what one wants.
Fifth, the brake. e to the pi root one hundred and sixty-three is neither a coincidence nor an integer. Its value is 262537412640768743.9999999999992500725972, and it misses 640320 cubed plus 744 by 7.49927 times ten to the minus thirteen (computed here). Both circulating claims are wrong. It is just a coincidence is wrong, because 640320 cubed is the value of the j-invariant for the class-number-one imaginary quadratic field and the q-expansion of j forces the agreement. It is really an integer is wrong too, because it misses by ten to the minus thirteen. And the same 640320 appears in Chudnovsky’s series – the source of the fourteen digits per term sits where the source of the pi sits.
The census. The pi that appears as a solid angle moves when d moves. The pi of the Gaussian integral shares that root and moves with it. The pi of Ramanujan’s series does not move, there being no d at all. The addresses are geometry for the first two and a period of a modular form for the third. As for fast, the three senses are many digits per term (linear), the digits double (quadratic), and the n-th alone (random access), and they are unordered.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: turning the corpus’s dimension test on pi itself, confirming that the solid-angle pi and the Gaussian pi share a root, writing addresses for three senses of fast, and fencing both circulating claims about e to the pi root one hundred and sixty-three. What remains: the habit of seeing a circle behind every pi is a strong one. But move d, and some pis move while others do not. The pi on Ramanujan’s page did not come from a circle.
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 $\pi$ That Carries No Dimension [Paper 31]. Zenodo.
DOI: 10.5281/zenodo.22090430
ORCID: 0009-0006-8587-3450