Abstract
The numbers 4π/3 and 8π/3 appear often in the formulas of physics. Their sum is 4π, the full solid angle. This paper counts that the two are the longitudinal and transverse shares of one solid angle, and looks at which formula takes which share and what fixes the split. No new theorem or law is claimed.
First, summing a quantity that has a direction over all directions gives 1/3 longitudinally and 2/3 transversely. When a directed quantity p is split into the component along the line of sight (longitudinal) and the component perpendicular to it (transverse) and each is summed over all directions, the longitudinal sum is $\mathbf{4.188790205}$ and the transverse sum $\mathbf{8.377580410}$, that is 4π/3 and 8π/3. Added together they return 4π.
Second, electricity takes the longitudinal share. Place positive and negative charges inside a sphere and sum the electric field they make over the inside of the sphere: the result is −(4π/3)p whatever the placement and whatever the size of the sphere (p being the strength and direction of the set of charges). For five placements the ratio of the summed value to −(4π/3)p was $\mathbf{1.000000000}$ in every case. In a control with one charge moved outside the sphere it was $\mathbf{0.395312500}$, which does not agree.
Third, magnetism takes the transverse share. Place a loop of current inside a sphere and sum the magnetic field it makes over the inside: the result is +(8π/3)m (m being the strength and direction of the loop). For four placements the ratio was $\mathbf{1.000000000}$. The electric field comes out like the slope of a height, so the longitudinal part remains; the magnetic field comes out like a whirl, so the transverse part remains. The signs are opposite, and the difference of the two coefficients is 4π.
Fourth, the split is 1/3 and 2/3 only when the shape is round. For a uniformly magnetised spheroid the share of the field inside changes with the shape. The longitudinal share is $\mathbf{0.000429899}$ for an elongated shape whose axis is 100 times its width, $\mathbf{0.333333333}$ for a sphere, and $\mathbf{0.984489707}$ for a flat shape whose axis is one hundredth of its width.
Fifth, and this is the core of the paper: the 21 cm line of hydrogen measures the transverse 8π/3. The electron of a hydrogen atom surrounds the position of the proton as a round cloud. The magnetic field of the proton felt by the electron is the +(8π/3)m summed inside a sphere. Computing the frequency from this coefficient and the values of the table of constants alone gives $\mathbf{1420.485452}$ MHz, against the measured 1420.405751768 MHz (Hellwig et al. 1970), a relative difference of 5.611×10⁻⁵. Replacing the coefficient by the longitudinal 4π/3 gives $\mathbf{710.242726}$ MHz and by the full 4π gives $\mathbf{2130.728178}$ MHz, far from the measurement.
Sixth, when the dimension is changed the shares become 1/d and (d−1)/d. In two dimensions the longitudinal and transverse shares are a half each; in three they are 1/3 and 2/3, in four 1/4 and 3/4. A dielectric sphere placed in a uniform electric field has an inside field d/(ε+d−1) times the outside one. The 2 in the ε+2 of the three-dimensional formula is d−1.
To be candid, the remaining difference of 5.611×10⁻⁵ in the 21 cm line comes from higher-order corrections and the size of the proton, which are not included here, and this paper does not explain it. What could be shown is that the measurement supports, up to just before the fifth digit, that the coefficient is 2/3 and neither 1/3 nor 1.
Keywords: solid angle, electric dipole, magnetic dipole, demagnetising factor, hyperfine splitting, 21 cm line, dimension.
Scope (what is not claimed)
No new theorem or law is claimed. The theorems on the electric and magnetic fields summed inside a sphere (Jackson 1998), the contact interaction between point-like magnets (Fermi 1930), the lowest-order formula for the hyperfine splitting of hydrogen (Griffiths 1982), the demagnetising factors of a spheroid (Osborn 1945), and the cross section of Thomson scattering are all known. The 21 cm line is not computed precisely. Only the g factor of the electron is put into the lowest-order formula; higher-order corrections of quantum electrodynamics and the size of the proton are not included. The difference from the measurement is not explained. Only a round electron cloud is considered. States in which the electron is not at the position of the proton (states with orbital angular momentum) are not treated. Mass is not discussed. The 4π of this paper is a solid angle; the 4π that this corpus places as a choice of the unit of mass does not appear. No claim is made that electromagnetism exists in higher dimensions. The directional average is only listed against the dimension. Relation to earlier papers: Paper 912 showed that a coefficient taking the average over all directions is the volume of the unit ball and one taking the transverse average is d−1 times that, and placed 4π/3 and 8π/3 in two different families. Here it is counted that the two appear, with signs, as the sums of the electric and magnetic fields inside a sphere, and that they add to 4π. Paper 905 showed that the 2/3 of Larmor’s formula is the spherical average 1 − 1/d of the angular distribution of radiation. Here it is seen that the same transverse 2/3 appears in the average of a static magnetic field and in a measured frequency. Paper 908 showed that the 4/3 of the electromagnetic mass of the electron is twice the average 2/3 over field directions. Here the longitudinal 1/3 that pairs with that 2/3 is placed beside it in a table. Paper 665 showed that the demagnetising factor of a sphere is 1/3 in all three directions, so that the effect of shape vanishes for a sphere. Here that 1/3 is read as the longitudinal share, and it is counted that the share moves when the shape is changed. Paper 37 counted that the viscous drag on a slipping sphere splits into 4/3 from the pressure and 8/3 from the normal viscous stress. These are the same 1/3 and 2/3, but whether they come from the same root has not been measured in this paper. What is added: listing 4π/3 and 8π/3 as the longitudinal and transverse shares of one solid angle, checking numerically, with a set of charges and a loop of current of finite size, the electric and magnetic fields summed inside a sphere, comparing, in the frequency of the 21 cm line, the transverse coefficient with the longitudinal and the full one, and changing the shape and the dimension to count that 1/3 and 2/3 are consequences of roundness and of three dimensions.
1. The question
The electric field inside a uniformly polarised sphere is $-(4\pi/3)P$, and the magnetic field inside a uniformly magnetised sphere is $(8\pi/3)M$ (Gaussian units). The cross section with which an electron scatters light is $(8\pi/3)r_e^2$, and $8\pi/3$ also enters the formula for the hyperfine splitting of the hydrogen atom. $4\pi/3$ is easy to remember as the coefficient of the volume of a ball, but where $8\pi/3$ comes from is harder to see.
The two add up to $4\pi$. This is not a coincidence: the two are shares of the full solid angle $4\pi$. This paper counts what splits $4\pi$ into $1/3$ and $2/3$, and which formula takes which share. At the end, as a case in which the $2/3$ side is actually measured, the 21 cm line of hydrogen is computed.
2. Summing a directed quantity over all directions
Take a quantity $\mathbf{p}$ that has a direction, and a unit vector $\mathbf{n}$ for the line of sight. $\mathbf{p}$ splits into the component along $\mathbf{n}$, $\mathbf{n}(\mathbf{n}\cdot\mathbf{p})$, and the component perpendicular to it, $\mathbf{p}-\mathbf{n}(\mathbf{n}\cdot\mathbf{p})$. Call the former longitudinal and the latter transverse. Summing each over all directions gives
$$\oint \mathbf{n}(\mathbf{n}\cdot\mathbf{p})\,d\Omega = \frac{4\pi}{3}\,\mathbf{p},\qquad \oint \bigl[\mathbf{p}-\mathbf{n}(\mathbf{n}\cdot\mathbf{p})\bigr]\,d\Omega = \frac{8\pi}{3}\,\mathbf{p} .$$Summed over 80,000 points on the sphere, the longitudinal sum was $4.188790205$ and the transverse sum $8.377580410$, together $12.566370614$, with fractions $0.333333333$ and $0.666666667$. Space has three directions, one along $\mathbf{n}$ and two perpendicular to it.
3. Electricity takes the longitudinal share
Place several charges inside a sphere. The sum of charge times position, $\mathbf{p}$, gives the strength and direction of the set of charges. The electric field made by the charges, summed over the whole inside of the sphere, is then
$$\int_{\text{sphere}}\mathbf{E}\,dV = -\frac{4\pi}{3}\,\mathbf{p} .$$The electric field is the gradient of the potential $\Phi$, so the sum over the inside can be turned into a sum over the surface, $-\oint\Phi \mathbf{n} dS$. Seen from far away the potential has the form $\mathbf{p}\cdot\mathbf{n}/R^2$, so the surface sum is the longitudinal sum of the previous section. This was checked without the formula, by summing the potential directly over the surface.
| Placement | Summed value $\div (-4\pi/3)\mathbf{p}$ |
|---|---|
| Small pair (separation $0.02$), centred, sphere radius 1 | $1.000000000$ |
| Large pair (separation $0.8$), centred, sphere radius 1 | $1.000000000$ |
| Oblique pair, off centre, sphere radius 1 | $1.000000000$ |
| The same oblique pair, sphere radius 3 | $1.000000000$ |
| Three charges ($+2$, $-1$, $-1$), sphere radius 1 | $1.000000000$ |
The largest departure of the ratio from 1 was $1.613\times10^{-13}$. Whether the set is small or large, off centre or not, and whatever the size of the sphere, the value does not change. There is one condition: all the charges must be inside the sphere. In a control with one charge moved outside the sphere, the ratio was $0.395312500$.
4. Magnetism takes the transverse share
Now place a loop of current inside the sphere. The area of the loop times the current gives its strength and direction, $\mathbf{m}$. The magnetic field made by the loop, summed over the whole inside of the sphere, is
$$\int_{\text{sphere}}\mathbf{B}\,dV = +\frac{8\pi}{3}\,\mathbf{m} .$$The magnetic field is the curl of the vector potential $\mathbf{A}$, so the sum over the inside can be turned into the surface sum $\oint\mathbf{n}\times\mathbf{A} dS$. Seen from far away $\mathbf{A}$ has the form $\mathbf{m}\times\mathbf{n}/R^2$, and since $\mathbf{n}\times(\mathbf{m}\times\mathbf{n})=\mathbf{m}-\mathbf{n}(\mathbf{n}\cdot\mathbf{m})$, the surface sum is the transverse sum. The loop was divided into 2000 short segments, $\mathbf{A}$ was summed, and the result was integrated over the surface.
| Placement | Summed value $\div (8\pi/3)\mathbf{m}$ |
|---|---|
| Small loop (radius $0.01$), centred, sphere radius 1 | $1.000000000$ |
| Large loop (radius $0.6$), centred, sphere radius 1 | $1.000000000$ |
| Tilted loop (radius $0.3$), off centre, sphere radius 1 | $1.000000000$ |
| The same tilted loop, sphere radius 3 | $1.000000000$ |
The largest departure of the ratio from 1 was $3.624\times10^{-13}$. As with electricity, the value does not depend on the size of the loop, its placement or the size of the sphere.
The reason the shares differ between electricity and magnetism lies in how the fields come out. The electric field is the gradient of a potential and comes out like the slope of a hill. The slope points towards the centre of the hill, so the longitudinal part remains. The magnetic field is a curl and comes out like a whirl. A whirl goes around its centre, so the transverse part remains. Far away, the electric field of a set of charges and the magnetic field of a loop of current have the same shape. They differ only near the source, and that difference summed up is the $4\pi$ between $-4\pi/3$ and $+8\pi/3$.
5. The split changes with the enclosing shape
$1/3$ and $2/3$ came out because the enclosing shape was a sphere. What happens for another shape is counted with a uniformly magnetised spheroid. With a magnetisation $M$ along the axis, the field inside is uniform, with $H=-4\pi NM$ and $B=4\pi(1-N)M$. $N$ is the demagnetising factor, fixed by the shape alone. The aspect ratio $r$ is the length of the axis divided by the width.
| $r$ | $N$ (longitudinal share) | $1-N$ (transverse share) | $H\div M$ inside | $B\div M$ inside |
|---|---|---|---|---|
| $0.01$ | $0.984489707$ | $0.015510293$ | $-12.371462523$ | $0.194908091$ |
| $0.1$ | $0.860804277$ | $0.139195723$ | $-10.817185565$ | $1.749185049$ |
| $0.5$ | $0.527200283$ | $0.472799717$ | $-6.624994139$ | $5.941376476$ |
| $1$ | $0.333333333$ | $0.666666667$ | $-4.188790205$ | $8.377580410$ |
| $2$ | $0.173563998$ | $0.826436002$ | $-2.181069518$ | $10.385301096$ |
| $10$ | $0.020285880$ | $0.979714120$ | $-0.254919890$ | $12.311450724$ |
| $100$ | $0.000429899$ | $0.999570101$ | $-0.005402267$ | $12.560968348$ |
$N$ was obtained in two ways, from the closed formula (Osborn 1945) and numerically by summing the field at the centre from the magnetic charge that appears on the surface; for all seven shapes the difference was at most $6.019\times10^{-13}$. Only for the sphere ($r=1$) are the values $-4\pi/3$ and $+8\pi/3$. For a thin needle almost all of $4\pi$ goes to the transverse side, and for a flat disc almost all goes to the longitudinal side. For every shape the two add up to $4\pi$. Paper 665 showed that the demagnetising factor of a sphere is $1/3$ in all three directions, so that the effect of shape vanishes for a sphere. $1/3$ and $2/3$ are the shares for a shape that treats every direction alike, that is, a round shape.
6. The 21 cm line of hydrogen measures the transverse side
A hydrogen atom consists of a proton and an electron, and both are small magnets. The energy differs slightly according to whether the two magnets point the same way or opposite ways. The radio wave corresponding to that difference arrives from all over the universe as a line of wavelength 21 cm.
In the state of lowest energy the electron cloud surrounds the proton as a round cloud, and is present at the position of the proton as well. For a round cloud, the outer part of the magnetic field of the proton averages to zero over directions, and what remains is only the part $+(8\pi/3)\mathbf{m}$ summed inside a sphere. The difference in energy can be written
$$\Delta E = \frac{8\pi}{3}\,g_e\mu_{\rm B}\,g_p\mu_{\rm N}\,|\psi(0)|^2,\qquad |\psi(0)|^2=\frac{1}{\pi a_0^3}$$(Gaussian units; $g_e$ and $g_p$ are the g factors of the electron and the proton, $\mu_{\rm B}$ and $\mu_{\rm N}$ the Bohr and nuclear magnetons, $a_0$ the Bohr radius, and $|\psi(0)|^2$ the density of the electron at the position of the proton). The $\pi$ of $8\pi/3$ cancels against the $1/\pi$ of the density, giving $\Delta E=(8/3) g_eg_p\mu_{\rm B}\mu_{\rm N}/a_0^3$. The $\pi$ goes, but the $8/3$ remains.
The frequency was computed with the values of the table of constants (CODATA 2022). The correction for the proton not being infinitely heavy compared with the electron (a factor $0.998367927$) was included.
| Coefficient | Frequency (MHz) | Ratio to the measurement |
|---|---|---|
| Transverse $8\pi/3$ (the actual formula) | $1420.485452$ | $1.000056111$ |
| Control: replaced by the longitudinal $4\pi/3$ | $710.242726$ | $0.500028$ |
| Control: replaced by the full $4\pi$ | $2130.728178$ | $1.500084$ |
| Measurement (Hellwig et al. 1970) | $1420.405751768$ |
The value computed with the transverse coefficient agrees with the measurement to a relative difference of $5.611\times10^{-5}$ (a difference of $0.079700$ MHz). The same frequency was also obtained from another formula using the Rydberg constant and the fine-structure constant; the relative difference between the two routes was $2.502\times10^{-11}$. Setting the g factor of the electron to exactly 2 gives $1418.840091$ MHz. The measured frequency converted to a wavelength is $21.106114$ cm.
The remaining difference of $5.611\times10^{-5}$ comes from higher-order corrections not included in this calculation and from the proton not being a point. This paper does not treat it. What can be said is that mistaking the coefficient for $1/3$ or $1$ would halve the frequency or multiply it by 1.5, whereas $2/3$ agrees up to just before the fifth digit. The measurement picks the transverse share of $4\pi$.
7. Other occurrences of $2/3$ and $1/3$
The cross section with which an electron scatters light (Thomson scattering) is $(8\pi/3)r_e^2$. Light is a transverse wave, and the strength of the scattered light is fixed by the component perpendicular to the line of sight alone, so the transverse sum appears here as well. Summing the angular distribution of unpolarised light, $(1+\cos^2\theta)/2$, over all directions gave $8.377580410$. The Thomson cross section of the table of constants divided by the square of the classical electron radius $r_e$ of the same table is $8.377580409$, a relative difference of $-2.070\times10^{-11}$ from $8\pi/3$.
The same shares appear in earlier papers of this corpus. Paper 905 showed that the $2/3$ in the formula for the radiation of an accelerated charge (Larmor’s formula) is the spherical average $1-1/d$ of the angular distribution. Paper 908 showed that the $4/3$ of the electromagnetic mass of the electron is twice the average $2/3$ over field directions. Both are the transverse share. Paper 912 separated coefficients that take the average over all directions and those that take the transverse average into two families, and placed $4\pi/3$ and $8\pi/3$ in different families. The longitudinal and transverse of this paper correspond to those two families.
Paper 37 counted that the drag $4\pi\mu Ua$ on a slipping sphere in a viscous fluid splits into $4/3$ from the pressure and $8/3$ from the normal viscous stress. The numbers are the same $1/3$ and $2/3$. Whether they come from the same root as the longitudinal and transverse of this paper has not been measured here.
8. Changing the dimension
The longitudinal share is $1/3$ and the transverse $2/3$ because space has three directions. In $d$ dimensions the longitudinal share is $1/d$ and the transverse $(d-1)/d$.
| $d$ | All directions $S_{d-1}$ | Longitudinal $S_{d-1}/d$ | Transverse $S_{d-1}(d-1)/d$ | Longitudinal fraction | Transverse fraction |
|---|---|---|---|---|---|
| 2 | $6.283185307$ | $3.141592654$ | $3.141592654$ | $0.500000$ | $0.500000$ |
| 3 | $12.566370614$ | $4.188790205$ | $8.377580410$ | $0.333333$ | $0.666667$ |
| 4 | $19.739208802$ | $4.934802201$ | $14.804406602$ | $0.250000$ | $0.750000$ |
| 5 | $26.318945070$ | $5.263789014$ | $21.055156056$ | $0.200000$ | $0.800000$ |
| 6 | $31.006276680$ | $5.167712780$ | $25.838563900$ | $0.166667$ | $0.833333$ |
Placing random points on the sphere and measuring the mean square of one coordinate gave $0.499784$, $0.333427$, $0.249817$, $0.200080$ and $0.166833$ in two to six dimensions, each within $1.52$ standard errors of $1/d$. The longitudinal sum $S_{d-1}/d$ is the same number as the volume of the unit ball (Paper 912). In two dimensions the longitudinal and transverse sums are the same $\pi$ and cannot be told apart. Two different numbers such as $4\pi/3$ and $8\pi/3$ stand side by side only from three dimensions upward.
The shares also enter the formulas for matter. Place a sphere of dielectric constant $\varepsilon$ in a uniform electric field $E_0$; the field inside the sphere is uniform. Solving the boundary conditions in $d$ dimensions gives
$$\frac{E_{\rm in}}{E_0} = \frac{d}{\varepsilon+d-1} .$$In three dimensions this is $3/(\varepsilon+2)$, the well-known formula. The 2 in the denominator $\varepsilon+2$ is the number of transverse directions, $d-1$. For $\varepsilon=4$ the field inside a three-dimensional sphere is $0.500000$ times the outside field.
Closing
Closing: 4π/3 and 8π/3 are the longitudinal and transverse shares of the full solid angle 4π. Summing a directed quantity over all directions gives $\mathbf{4.188790205}$ longitudinally and $\mathbf{8.377580410}$ transversely. The electric field summed inside a sphere is −(4π/3)p and the magnetic field +(8π/3)m; this was checked with a set of charges and a loop of current of finite size, with a ratio of $\mathbf{1.000000000}$. The split is 1/3 and 2/3 only when the enclosure is round; for an elongated shape the transverse side, and for a flat shape the longitudinal side, takes almost all of 4π. The frequency of the 21 cm line of hydrogen computed with the transverse coefficient is $\mathbf{1420.485452}$ MHz and agrees with the measurement to a relative difference of 5.611×10⁻⁵. With the longitudinal or the full coefficient it would be a half or 1.5 times that. When the dimension is changed the shares are 1/d and (d−1)/d, and the 2 in the ε+2 of the formula for a dielectric sphere is d−1. To be candid, the remaining difference in the 21 cm line has not been explained. What could be said is that, of the ways 4π splits, the transverse 2/3 is what enters the measured frequency. The separator is whether the field comes out longitudinally or transversely, and whether the enclosing shape is round.
References
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Reference
Yamagishi, Y. (2026). 4π Splits into a Longitudinal 4π/3 and a Transverse 8π/3, and the 21 cm Line of Hydrogen Measures the Transverse Part [Paper 1084]. YOI: https://www.yamagishiyuuki.com/yoi/1084
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1084. This paper is a preprint and has not been peer reviewed.