Abstract
When 8π appears in a formula of physics, one is tempted to read it as twice 4π. This corpus has so far shown, one case at a time, that the 2 in several of its 8π has separate origins. This paper collects every 8π from all papers up to number 1084, reads them one by one, and counts the origins of the 2. It then sorts the 2 by how it moves when the dimension of space is changed. No new theorem or law is claimed.
First, 8π itself appears in 43 papers. Of the 1083 papers read, the characters 8π appear in 60 papers at 398 places; removing the 17 papers that only have powers such as 8π² or 8π³ leaves 43 papers and 329 places. The paper in which it appears most is number 17, with 47 places.
Second, there were ten origins for reading 8π as 2×4π. The trace reversal of the Einstein equation (16 papers, $\mathbf{0.372093}$ of the 43), the 1/2 of motion (6), the transverse 2/3 (5), the mass staircase (4), the Friedmann equation (3), the number of polarisations (2), and one paper each for identical bosons, the integral of a power, the horizon area of a rotating black hole, and the canonical momentum of a rolling hoop. Ten papers deal with two origins at once.
Third, there were also three kinds of 8π that are not 2×4π. The 8π of the Hawking temperature is 4×2π (4 papers), the 8π in the number of years over which an orbit widens is also 4×2π (1 paper), and the 8π of a charge normalisation is the square root of 16π×4π (1 paper). Reading 8π as 2×4π is itself not always right.
Fourth, and this is the core of the paper: when the dimension of space is changed, the 2 sorts into three ways of moving. The 2 of the number of transverse directions, of the number of polarisations, of the derivative of the area of a sphere and of the integral of a power moves to d−1 and becomes 3 in four dimensions. The 2 of the trace reversal of the Einstein equation moves to (d−1)/(d−2) and becomes $\mathbf{1.500000000}$ in four dimensions. The 2 of the 1/2 of motion and of identical particles stays $\mathbf{2.000000000}$ in every dimension.
Fifth, all three are 2 together only in three dimensions. Checking from two to eleven dimensions, d−1 and (d−1)/(d−2) are both 2 only at d = 3. In three dimensions, twos that arose for separate reasons all wear the same face.
Sixth, two new items were counted. The mass of a non-rotating black hole divided by the product of its temperature and entropy is 2 when space has three dimensions and 1.5 when it has four, the same (d−1)/(d−2) as the factor of the Einstein equation (after checking that the three formulas satisfy the first law). The 2 in the derivative 8πR of the area of a sphere is d−1, the same 2 as in the formula 2/R for the force of a stretched surface.
To be candid, the labels were attached by me after reading the context, not decided by a machine. Twos that move in the same way are not written here as having the same root. What has been shown is that a different way of moving means a different root, and that the difference cannot be seen by looking at three dimensions alone.
Keywords: 8π, census, trace reversal, polarisation, dimension, black-hole thermodynamics, distinct roots.
Scope (what is not claimed)
No new theorem or law is claimed. The trace reversal of the Einstein equation, the higher-dimensional Schwarzschild solution with its temperature and entropy (Tangherlini 1963, Myers and Perry 1986), the number of polarisations, and the symmetrisation for identical particles are all known. The unit of the census is fixed in advance. It counts papers up to number 1084 in whose Japanese text (before the references) an 8π that is not a power of 8π appears. Papers added later will not move the numbers of this census. The labelling is not said to be the only one. Ten papers were given two labels. The table of labels is fixed in the check, and a different labelling would change the numbers. The same way of moving is not written as the same root. Only obeying the same formula in the dimension is counted; nothing beyond that is decided. No claim is made that gravity exists in higher dimensions. The formulas are only listed against the dimension. Relation to earlier papers: Paper 17 showed that the 8π in the kinetic term of the Schrödinger equation and the 8π of the Einstein equation have a different root factor by factor. Here those two, together with eight more origins, are placed in one table. Paper 58 showed that the factor of the Einstein equation is (D−2)/(D−3) in spacetime dimension D and meets the 2 of the mass staircase only in four-dimensional spacetime. Here the same method is applied to the other twos. Paper 52 and Paper 142 showed that the 8π of the Hawking temperature is 4×2π and that what is universal is the 2π. Here that reading is counted as an example of an 8π that is not 2×4π. Paper 907 showed that the 8π/3 of the Friedmann equation is the product of the 1/2 of motion and the volume of a ball, and separates from the transverse 8π/3 in four dimensions. Here that separation is placed in two columns of the table. Paper 1005 showed that the 8π of Planck’s law is the product of the number of polarisations and 4π. Here the number of polarisations is put in the group that moves to d−1. Paper 1081 counted the 8π in the cross section of identical bosons, Paper 1082 the 8π of the healing length, and Paper 1084 the transverse 8π/3. Here those three are added to the census. Paper 01 left open the question whether the 2 of the mass staircase and the 2 of the Einstein equation are the same. Paper 58 has answered that question; here the 2 of the staircase is counted as a 2 that moves not with the dimension but with the choice of the exponent of the density. What is added: reading all 43 papers in which 8π appears and labelling them, counting ten origins of the 2 and three readings that are not 2×4π, sorting them into three by the way they move when the dimension is changed, and adding to that table the ratio of mass to temperature times entropy of a black hole and the derivative of the area of a sphere.
1. The question
$8\pi$ appears all over physics. The equation of gravity $G_{\mu\nu}=8\pi G T_{\mu\nu}$, the formula $8\pi h\nu^3/c^3$ for the light emitted by a hot body, the temperature $1/(8\pi M)$ of a black hole. Knowing that $4\pi$ is the full solid angle, one is tempted to read $8\pi$ as “twice the solid angle”. What, then, is the 2?
This corpus has so far answered the question one case at a time. Paper 17 showed that the $8\pi$ in the Schrödinger equation and the $8\pi$ of the Einstein equation are merely the same number, with both the 2 and the $4\pi$ coming from different places. Paper 58 compared the 2 of the Einstein equation with the 2 of the mass staircase by moving the dimension. This paper does it all together. It collects every $8\pi$ in the corpus, counts the origins of the 2, and sorts them by moving the dimension.
2. How the census is taken
The unit is fixed first. The Japanese text of each paper up to number 1084 is read, up to the references. Powers such as $8\pi^2$ and $8\pi^3$ are counted separately, since the reading “twice $4\pi$” does not apply to them in the first place.
| Papers | Places | |
|---|---|---|
| Papers read | 1083 | |
| Papers in which the characters $8\pi$ appear (powers included) | 60 | 398 |
| Papers in which $8\pi$ itself appears | 43 | 329 |
| Papers with powers only ($8\pi^2$, $8\pi^3$ and the like) | 17 |
For all 43 papers the sentences around each $8\pi$ were read and a label was attached saying where the 2 comes from. Where one paper deals with two origins, two labels were attached (10 papers). The table of labels is written into the check, and it was verified by machine that the set of labelled papers equals the set of papers in which $8\pi$ itself appears. The paper in which $8\pi$ appears most is number 17, with 47 places.
3. Ten origins read as 2×4π
| Origin of the 2 | Papers | Paper numbers |
|---|---|---|
| The $8\pi G$ of the Einstein equation (trace reversal) | 16 | 1, 17, 35, 44, 50, 52, 57, 58, 62, 63, 72, 73, 74, 87, 109, 142 |
| The $1/2$ of motion (integral over speed, quadratic energy) | 6 | 13, 17, 18, 908, 1082, 1083 |
| The transverse $2/3$ ($8\pi/3$) | 5 | 905, 907, 908, 912, 1084 |
| The mass staircase (two sheets of $4\pi$) | 4 | 1, 2, 58, 1081 |
| The $8\pi G/3$ of the Friedmann equation | 3 | 169, 907, 1001 |
| The number of polarisations | 2 | 521, 1005 |
| Identical bosons | 1 | 1081 |
| The integral of a power | 1 | 14 |
| The horizon area of a rotating black hole | 1 | 887 |
| The canonical momentum of a rolling hoop | 1 | 913 |
The most frequent is the Einstein equation, with 16 papers, $0.372093$ of the 43. Each 2 in one sentence:
The 2 of the Einstein equation appears when the right-hand side of the equation is rearranged (trace reversal). The $1/2$ of motion is the $v^2/2$ obtained by accumulating speed from 0 to $v$; it shows up as $8\pi$ in formulas where the mass is set to $4\pi$ and in the formula for an energy proportional to the square of the electric field. The transverse $2/3$ is the fact that there are two directions perpendicular to a given one (Paper 1084). The 2 of the mass staircase is the value when the exponent of the density is chosen as $5/2$ (Paper 01). The $8\pi/3$ of the Friedmann equation can be read as the product of the $1/2$ of the motion of a shell and the $4\pi/3$ of the volume of a ball (Paper 907). The 2 of polarisation is the fact that light can oscillate in two ways (Paper 1005). The 2 of identical bosons comes from adding two waves (Paper 1081). The 2 of the integral of a power is the 2 that appears when $1/r^3$ is integrated to $1/(2r^2)$. The 2 of the horizon area comes from the horizon condition of a rotating black hole. The 2 of a rolling hoop is the fact that translation and rotation carry the same momentum.
4. 8π that is not 2×4π
| Reading | Papers | Paper numbers |
|---|---|---|
| The $1/(8\pi M)$ of the Hawking temperature ($4\times2\pi$) | 4 | 43, 52, 73, 142 |
| Years over which an orbit widens ($4\times2\pi$) | 1 | 909 |
| A charge normalisation (square root of $16\pi\times4\pi$) | 1 | 57 |
| The arithmetic decomposition $(2\pi)\times(8\pi)=16\pi^2$ | 3 | 22, 29, 30 |
| Mention only (citing another paper’s $8\pi$, a list of titles, as a number) | 4 | 125, 200, 1056, 1057 |
Paper 52 and Paper 142 showed that the $8\pi$ of the Hawking temperature is $4\times2\pi$. The $2\pi$ comes from a period in the direction of time, and the 4 from the relation between surface gravity and mass. Paper 142 also showed that giving the black hole charge or rotation makes the coefficient differ from $8\pi$, and that what is universal is the $2\pi$. The $8\pi$ of the paper numbered 909 is four times the angle $2\pi$ of one revolution. The $8\pi$ of the paper numbered 57 is the square root of $16\pi G$ with $G=4\pi$ put in. Seeing the characters $8\pi$ does not mean it may be read as $2\times4\pi$.
5. Changing the dimension: the 2 that moves to $d-1$
Merely saying that the origins differ remains an assertion. The method used in Paper 58 is to move the dimension and see how the 2 changes. Write $d$ for the dimension of space.
| $d$ | Transverse directions | Transverse sum $\div$ longitudinal sum | Derivative of the area of a sphere | Integral of a power |
|---|---|---|---|---|
| 2 | 1 | $0.9992$ | $1.000000000$ | $1.000000000$ |
| 3 | 2 | $2.0003$ | $2.000000000$ | $2.000000000$ |
| 4 | 3 | $2.9901$ | $3.000000000$ | $3.000000000$ |
| 5 | 4 | $4.0023$ | $4.000000000$ | $4.000000000$ |
| 6 | 5 | $4.9779$ | $5.000000000$ | $5.000000000$ |
The number of transverse directions was counted as the rank of the projection that picks out the directions perpendicular to a given one. The transverse sum divided by the longitudinal sum was measured with 400,000 random points on the sphere. Light oscillates perpendicular to its direction of travel, so the number of polarisations is this $d-1$ as well (Paper 1005).
The derivative of the area of a sphere is counted here for the first time. Differentiating the area $4\pi R^2$ of a sphere of radius $R$ with respect to $R$ gives $8\pi R$. In $d$ dimensions the area is proportional to the power $d-1$ of $R$, so differentiation brings $d-1$ out in front. The values in the table are the numerical derivative of the area divided by the area over the radius. This 2 is also the 2 in the formula $2\gamma/R$ for the pressure with which the surface of a bubble or a drop pushes inward ($\gamma$ being the tension of the surface). That is because the derivative of the area divided by the derivative of the volume is $2/R$; the three-dimensional value was $2.000000000/R$.
The integral of a power is $1/r^d$ integrated from $r$ to infinity, which is $1/((d-1) r^{d-1})$. The 2 in the $1/(8\pi r^2)$ of the paper numbered 14 is this $d-1$ in three dimensions.
All four can be read as counting the fact that a sphere inside a $d$-dimensional space is $d-1$-dimensional. This paper does not, however, go on to call the four the same root. What has been counted is only that they move in the same way with the dimension.
6. Changing the dimension: the 2 that moves to $(d-1)/(d-2)$
Rearranging the Einstein equation $G_{\mu\nu}=\kappa T_{\mu\nu}$ for dust at rest gives $R_{00}=\kappa\rho (D-3)/(D-2)$, where $D=d+1$ is the dimension of spacetime. Matching this to Newton’s law of gravity (the Poisson equation) makes $\kappa$ equal to $(D-2)/(D-3)$ times the coefficient of the Poisson equation, or $(d-1)/(d-2)$ times in terms of the dimension of space.
| $d$ | $R_{00}\div(\kappa\rho)$ | $\kappa\div$ (coefficient of the Poisson equation) | Coefficient $(d-1)S_{d-1}/(d-2)$ |
|---|---|---|---|
| 3 | $0.500000000$ | $2.000000000$ | $25.132741229$ |
| 4 | $0.666666667$ | $1.500000000$ | $29.608813203$ |
| 5 | $0.750000000$ | $1.333333333$ | $35.091926759$ |
| 6 | $0.800000000$ | $1.250000000$ |
In three dimensions it is 2, and together with the $4\pi$ of Gauss’s law this makes $8\pi$. In four dimensions it is $1.5$. In two dimensions the denominator is 0 and the formula does not stand. This is the same result as the $(D-2)/(D-3)$ that Paper 58 wrote in terms of the spacetime dimension.
One more number that moves in the same way was found. The mass $M$, entropy $S$ and temperature $T$ of a non-rotating black hole can be written, with the radius $r$ of the horizon, as
$$M=\frac{(D-2)\,\Omega\,r^{D-3}}{16\pi},\qquad S=\frac{\Omega\,r^{D-2}}{4},\qquad T=\frac{D-3}{4\pi r}$$($\Omega$ is the area of the unit sphere of dimension $D-2$; units in which the gravitational constant and the like are 1). It was checked by symbolic computation that the three satisfy $dM=T dS$ (the first law), and then the ratio is
$$\frac{M}{TS}=\frac{D-2}{D-3}=\frac{d-1}{d-2} .$$When space has three dimensions this is $2.000000000$, with four $1.500000000$, with five $1.333333333$ and with six $1.250000000$: the same numbers as the factor of the Einstein equation.
In four-dimensional spacetime $M=r/2$, $S=\pi r^2$ and $T=1/(4\pi r)$, so that $T=1/(8\pi M)$. The $8\pi$ of the Hawking temperature can therefore also be read as $2\times4\pi$, from $T=M/(2S)$ and $S=4\pi M^2$. Together with the $4\times2\pi$ of Paper 52 and Paper 142, the same $8\pi$ has two ways of being split. It is not that one of them is the correct one. $4\times2\pi$ tells what is universal, and $2\times4\pi$ tells how it moves with the dimension.
7. The 2 that does not move with the dimension
The $1/2$ of motion is the result of accumulating speed. Accumulating a velocity vector in $d$ dimensions from 0 along a straight path and dividing $|v|^2$ by the integral gave $2.000000000$ in each of two to six dimensions. The accumulation takes place along a single path, so it does not change with the dimension of space.
This difference separates two numbers that both look like $8\pi/3$. As Paper 907 showed, the $8\pi/3$ of the Friedmann equation is “the 2 of motion” times the volume of a ball, and the transverse $8\pi/3$ is “the number of transverse directions” times the volume of a ball. In three dimensions both are $8.377580$, but in four dimensions they separate into $9.869604$ and $14.804407$.
The 2 of identical bosons does not move either. Paper 1081 counted, in three to six dimensions, that the cross section at long wavelength is $2.000000$ times that of distinguishable particles. The operation of adding two waves does not know the dimension.
8. They agree only in three dimensions
| $d$ | Transverse directions, polarisations, derivative of the area, integral of a power | Trace reversal, $M/(TS)$ | The $1/2$ of motion, identical particles |
|---|---|---|---|
| 2 | 1 | (the formula does not stand) | 2 |
| 3 | 2 | $2.000000$ | 2 |
| 4 | 3 | $1.500000$ | 2 |
| 5 | 4 | $1.333333$ | 2 |
| 6 | 5 | $1.250000$ | 2 |
The three columns are 2 together only in the row $d=3$ (checked from two to eleven dimensions). We live in three dimensions, so twos that arose for separate reasons all look like the same 2. Moving the dimension by one separates them into 3, $1.5$ and 2.
Three of the ten origins do not enter this table. The 2 of the mass staircase is the result of choosing the exponent of the density, and it moves continuously when the exponent is changed (Paper 58 called it “a chosen integer”). The 2 of the horizon area of a rotating black hole is a number attached to one solution in four-dimensional spacetime. The 2 of a rolling hoop is a number of the shape called a hoop; for a disc or a ball it is a different number. These three move not with the dimension but with something else: an exponent, a solution, a shape.
9. The limits of the census, and the relation to earlier papers
The labels were attached after reading the context. The $8\pi G/3$ of the Friedmann equation, for instance, comes from the Einstein equation in general relativity, so it could also be given the label of trace reversal. This paper gave it a separate label, following the form in which Paper 907 read it from the motion of a shell. A different labelling would change the numbers in the tables. What would not change is that the origin is not single, and that there are three ways of moving with the dimension.
Paper 17 compared two $8\pi$ factor by factor and showed that no part of them coincides. Paper 58 separated the 2 of the Einstein equation from the 2 of the staircase by the method of moving the dimension. What this paper has done is to extend that method to every $8\pi$ in the corpus. As a result, numbers that look like the same 2 in three dimensions turned out to have at least three different ways of moving. The “origin of the factor 2” that Paper 01 left as an open question was not a question with one answer, but a question whose answer differs from one 2 to another.
Closing
Closing: the 2 in 8π is not a single thing. Reading the 43 papers of the corpus up to number 1084 in which 8π itself appears gave ten origins read as 2×4π and three readings that are not 2×4π. The most frequent is the trace reversal of the Einstein equation, with 16 papers, $\mathbf{0.372093}$ of the 43. When the dimension of space is changed, the 2 of the number of transverse directions, of polarisation, of the derivative of the area and of the integral of a power moves to d−1; the 2 of trace reversal and of the ratio of the mass of a black hole to its temperature times entropy moves to (d−1)/(d−2); the 2 of the 1/2 of motion and of identical particles does not move. In four dimensions they separate into 3, $\mathbf{1.500000000}$ and 2, and they are 2 together only in three dimensions. To be candid, the labels were attached by reading, and the labelling is not unique. Twos that move in the same way are not written as having the same root. The separator is whether, when the dimension is changed, the 2 moves to the number of transverse directions, moves to the trace-reversal factor, or does not move.
References
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- Yamagishi, Y. (2026). Without Stimulated Emission, Radiation Follows Wien’s Law, and Only the 8π of A over B Carries the Polarization Count [Paper 1005]. YOI: https://www.yamagishiyuuki.com/yoi/1005; DOI: 10.5281/zenodo.22823713
- Yamagishi, Y. (2026). A Hard Sphere Has Four Cross Sections, 1, 2, 4 and 8 Times $\pi a^2$, and the 4 Appears Only in Three Dimensions [Paper 1081]. YOI: https://www.yamagishiyuuki.com/yoi/1081
- Yamagishi, Y. (2026). The $4\pi$ in $g=4\pi\hbar^2a/m$ Is a Solid Angle, and Setting $m=4\pi$ Turns It into $g=\hbar^2a$ [Paper 1082]. YOI: https://www.yamagishiyuuki.com/yoi/1082
- Yamagishi, Y. (2026). $4\pi$ Splits into a Longitudinal $4\pi/3$ and a Transverse $8\pi/3$, and the 21 cm Line of Hydrogen Measures the Transverse Part [Paper 1084]. YOI: https://www.yamagishiyuuki.com/yoi/1084
Reference
Yamagishi, Y. (2026). A Census of the 2 in 8π: It Has Ten Origins, and Changing the Dimension Sorts Them into Three Ways of Moving [Paper 1085]. YOI: https://www.yamagishiyuuki.com/yoi/1085
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 1085. This paper is a preprint and has not been peer reviewed.