Chapter 4. Multi-Ring Orthogonal Geometry: The Spherical Octahedron and the Superposition of Fields

One ring has a plane and an axis. Three rings set at right angles to one another, like the bands of an armillary sphere, have no preferred direction at all. This chapter describes the geometry of that arrangement, calculates how the magnetic fields of the three rings add, and shows why rings at right angles do not disturb one another, a fact engineers rely on in three-axis coils and antennas.

Three great circles

Three rings of one cubit circumference at right angles. Each lies in a plane through the centre; the arrows are the rings’ normals. The six dots, where the rings cross, are the vertices of an octahedron.

Three rings of equal size with a common centre, each perpendicular to the other two, lie on the surface of one sphere as three great circles. They cross at six points, at the ends of the three axes, and these points are the vertices of a regular octahedron. The rings divide the sphere into eight equal spherical triangles, each with three right angles: a spherical octahedron.

Calculation

For rings one cubit in circumference:

import math

C = 523.5e-3                      # circumference of each ring, m
R = C / (2 * math.pi)             # radius of the sphere the rings lie on
mu0 = 4e-7 * math.pi
print(f"sphere radius {R*1000:.1f} mm, diameter {2*R*1000:.1f} mm")
print(f"12 arcs of 90 degrees, each {C/4*1000:.1f} mm; total wire {3*C*1000:.1f} mm")
print(f"8 spherical triangles, each {4*math.pi*R**2/8*1e4:.1f} cm^2")

I = 1.0                           # ampere
B1 = mu0 * I / (2 * R)            # field at the centre of one ring
print(f"one ring, 1 A:    B = {B1*1e6:.2f} uT at the centre")
print(f"three in phase:   B = {math.sqrt(3)*B1*1e6:.2f} uT, along the (1,1,1) diagonal")
for z_mm in (0, 40, 83.3, 166.6):  # on the axis of one ring
    z = z_mm / 1000
    Bz = mu0 * I * R**2 / (2 * (R**2 + z**2) ** 1.5)
    print(f"  on axis, z = {z_mm:5.1f} mm: B = {Bz*1e6:.2f} uT")
# sphere radius 83.3 mm, diameter 166.6 mm
# 12 arcs of 90 degrees, each 130.9 mm; total wire 1570.5 mm
# 8 spherical triangles, each 109.0 cm^2
# one ring, 1 A:    B = 7.54 uT at the centre
# three in phase:   B = 13.06 uT, along the (1,1,1) diagonal
#   on axis, z =   0.0 mm: B = 7.54 uT
#   on axis, z =  40.0 mm: B = 5.52 uT
#   on axis, z =  83.3 mm: B = 2.67 uT
#   on axis, z = 166.6 mm: B = 0.67 uT

The sphere is 166.6 mm across. Each arc between two crossings is a quarter cubit, seven fingers long (130.9 mm). For scale, the Earth’s magnetic field is a few tens of microtesla, so one ampere in a cubit ring produces a field at its centre about a fifth of the Earth’s, and the field falls off steeply with distance.

Fields add as vectors

Magnetic fields obey the principle of superposition: the field of several currents is the vector sum of the fields each would produce alone (Grover 1946). For three orthogonal rings this has simple consequences.

Equal currents in phase in all three rings produce, at the centre, a field $\sqrt{3}$ times that of one ring, pointing along the diagonal between the three axes. The combination is equivalent to a single larger ring tilted to face that diagonal. Nothing new appears; the field simply points in a new direction.
Currents in quadrature in two rings (equal in size, a quarter cycle apart) produce a field of constant size that rotates, completing one turn per cycle. This is how a rotating magnetic field is made in an electric motor, and how a pair of crossed antennas produces circular polarisation (Balanis 2016).
No current, no field. A passive ring that is not connected to any source carries current only when a changing external field induces it. The rings do not produce, store or concentrate any field of their own.

Rings at right angles do not couple

Calculation

The mutual inductance $M$ between two circuits measures how much magnetic flux from one passes through the other; it is given by Neumann’s formula, $M = \frac{\mu_0}{4\pi} \oint\!\!\oint \frac{d\mathbf{l}_1 \cdot d\mathbf{l}_2}{|\mathbf{r}_1 – \mathbf{r}_2|}.$ Evaluating it numerically for two concentric rings, one 2% larger so that the wires do not touch:

import numpy as np

mu0 = 4e-7 * np.pi
C = 0.5235                        # circumference of each ring, m
R = C / (2 * np.pi)

def ring(normal, radius, n=720):
    """Points and segment vectors of a circular ring centred at the origin."""
    normal = np.asarray(normal, float) / np.linalg.norm(normal)
    a = np.cross(normal, [1, 0, 0] if abs(normal[0]) < 0.9 else [0, 1, 0])
    a /= np.linalg.norm(a)
    b = np.cross(normal, a)
    t = np.linspace(0, 2 * np.pi, n, endpoint=False)
    pts = radius * (np.outer(np.cos(t), a) + np.outer(np.sin(t), b))
    return pts, np.roll(pts, -1, axis=0) - pts

def mutual(r1, r2):
    """Neumann formula: M = mu0/4pi * sum(dl1 . dl2 / |r1 - r2|)."""
    (p1, d1), (p2, d2) = r1, r2
    dist = np.linalg.norm(p1[:, None, :] - p2[None, :, :], axis=2)
    return mu0 / (4 * np.pi) * np.sum((d1 @ d2.T) / dist)

x = ring([1, 0, 0], R)
y_orth = ring([0, 1, 0], R * 1.02)       # 2% larger so the wires do not touch
x_par = ring([1, 0, 0], R * 1.02)        # same plane, for comparison
print(f"coplanar rings:   M = {mutual(x, x_par)*1e9:8.2f} nH")
print(f"orthogonal rings: M = {mutual(x, y_orth)*1e9:8.2f} nH")

# isotropy: induced EMF in each ring is proportional to the field component along its normal
rng = np.random.default_rng(1)
for _ in range(3):
    u = rng.normal(size=3); u /= np.linalg.norm(u)
    single = abs(u[0])                  # one ring with normal along x
    triad = np.sqrt(np.sum(u ** 2))     # three rings, combined in quadrature
    print(f"field direction {np.round(u, 2)}: one ring {single:.2f}, three rings {triad:.2f}")
# coplanar rings:   M =   423.15 nH
# orthogonal rings: M =     0.00 nH
# field direction [0.36 0.86 0.35]: one ring 0.36, three rings 1.00
# field direction [-0.79  0.55  0.27]: one ring 0.79, three rings 1.00
# field direction [-0.62  0.67  0.42]: one ring 0.62, three rings 1.00

Two concentric rings in the same plane are tightly coupled (423 nH). Rotate one by a right angle and the coupling vanishes: by symmetry, every bit of flux that one ring sends through one half of the other is cancelled by flux through the other half. Three mutually orthogonal rings are therefore three independent circuits sharing a centre.

The second part of the program shows what the three rings gain together. One ring responds only to the component of a field along its own axis; a field lying in its plane induces nothing, so a single ring has “blind” directions. Three orthogonal rings, with their responses combined in quadrature, respond equally to a field from any direction. This is the principle of three-axis search-coil magnetometers and of three-dimensional coil antennas.

A practical complication: the crossings

Note

The calculation assumes the rings are insulated from one another where they cross. If bare rings touch, or are soldered together, at the six crossings, they become one electrical network of twelve arcs rather than three separate rings. Currents can then take shortcuts through the crossings, the rings are no longer independent, and the zero mutual inductance no longer applies to every way the network can carry current. Builders who want three independent rings should keep the crossings insulated, for example with a small sleeve or a non-conducting frame; builders who solder the crossings are making a different object, which has to be measured on its own terms.

Measurement plan 1.0, continued

Hypothesis · not established

The author proposes the spherical octahedron of three cubit rings as an object worth measuring precisely because its electrical behaviour can be predicted in advance. Two further objects are added to the plan in Chapter 3:

three single rings of 1.6 mm bare wire, one cubit in circumference, held at right angles in a non-conducting frame with insulated crossings;
the same three rings with all six crossings soldered.

Predictions, written down before measuring.

Object 6: with a probe loop aligned in turn with each axis, the coupling change is the same within 1 dB for all three axes; the coupling between any two of the rings, measured by driving one through a probe and sensing the other, is at least 20 dB weaker than for two coplanar rings at the same spacing.
Object 7: the coupling between rings is no longer weak, and new resonances appear below the single-ring loop resonance, corresponding to current paths around shorter loops of the network.

Summary

Three orthogonal rings of one cubit lie on a sphere 166.6 mm across, cross at the six vertices of an octahedron and divide the sphere into eight right-angled triangles.
Their fields add as vectors: in phase they act as one tilted ring; in quadrature they make a rotating field. Without a current they make no field.
Rings at right angles have zero mutual inductance; together they respond equally to fields from every direction.
Soldering the crossings turns three rings into one network with different behaviour; the measurement plan tests both.

References

Balanis, Constantine A. 2016. Antenna Theory: Analysis and Design. 4th ed. Wiley.

Grover, Frederick W. 1946. Inductance Calculations: Working Formulas and Tables. Van Nostrand.

Fan, Y. (2026). The Sacred Cubit and Resonant Geometry: Measuring the Egyptian Royal Cubit, the Quarter Wave and the Bifilar Ring [神圣肘尺与谐振几何] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23666560 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).