Four corners.
Six edges.
What lines up?

The shape supplies a frame.

A tetrahedron has four triangular faces meeting at four corners. It is regular when all six edges have the same length, so every face is an equilateral triangle. Label the corners 0, 1, 2 and 3. Adjacent edges meet at 60°; opposite edges do not meet, but their direction vectors are perpendicular.

We can take the common edge length as one unit: a = 1. This fixes a scale for comparing the geometry; it does not choose a physical length or material.

The six edges are not mutually orthogonal. They form three opposite perpendicular edge-direction pairs. The three lines joining opposite-edge midpoints are mutually orthogonal axes.

Four balanced coordinates. All of 3D space.

Let v₀,…,v₃ be the vectors from the centre to those four corners. Their sum is zero. Assign four real coefficients φᵢ (phi) whose sum is also zero. Three can be chosen freely; the fourth balances them. Collect the corner vectors as the columns of a matrix V.

Here 𝟙 = (1, 1, 1, 1)ᵀ is the all-ones vector, not the unit edge length. 𝟙^⊥ means the lists perpendicular to it: their four entries sum to zero. ℝ⁴ is the space of real four-number lists; ≅ says the two spaces are linearly equivalent. The sum sign Σ adds the four corner contributions; ᵀ writes a row as a column.

This is a one-to-one linear representation of ordinary Cartesian space. Regularity makes it isotropic: every direction receives the same scale. The coefficients may be negative; this represents all of space, not just the tetrahedron’s interior.

The six edges are the six pairwise differences of those same four vectors.

Six edge values split exactly into three plus three.

Orient each edge from its lower to higher corner index. The incidence matrix D adds incoming values and subtracts outgoing ones at each corner. Write the six edge values as a list z. The vector Dz records its four corner balances.

Here ker D means the inputs D sends to zero; im Dᵀ means the outputs of its transpose. The symbol ⊕ means a unique sum, and P names a projector—an operation that extracts one component. I₆ leaves a six-number list unchanged.

The cycle part circulates with zero corner balance. The cut part consists of differences of corner values. Every edge participates in both. This splits any oriented edge vector z; it does not yet say which values a physical circuit can sustain.

Symmetry limits the form of a relation.

Rotations and reflections rearrange the corner labels and their edge values. The cycle and cut spaces each stay within their own pattern family, conventionally called T₁ and T₂. These are irreducible representations: families that cannot be split further while preserving every symmetry. Any linear edge-to-edge operator X respecting all these symmetries has just two gains, α and β:

Under rotations alone (A₄), the two three-dimensional copies are equivalent: the generic commuting operator has four parameters. A symmetric operator has three; reciprocity gives this reduction only when expressed in the appropriate matched coordinates.

A measured departure from the appropriate fit is a symmetry/model residual. Geometry, leads, materials, sensor error, calibration or omitted physics can contribute. The residual alone does not identify the cause.

What lives on an edge matters: reversing a directed quantity changes its sign; reversing the label of an edge temperature does not. The mathematical companion follows both cases →

Now choose your medium.

The same structure can organize conductors, a conducting fluid, or another physical system. In control language, the medium, apparatus and their dynamics form the plant. Here we choose current-carrying conductors as the worked example.

Let iⱼ be the current in channel j and bⱼ(x) its magnetic field per ampere at position x. For fixed complete paths in air and a suitable slow-variation regime, Biot–Savart supplies these columns. Their sum is the magnetic field B:

In a closed six-edge network with no external injection, steady currents satisfy Di = 0. Independent edge drives need complete return paths; those paths contribute to the field. Four geometric corners are always present. Four corner actuators are an optional, separately modeled addition.

A pattern becomes a waveform.

Choose a basis matrix Q whose columns are admissible channel patterns. The coefficients q(t) say how much of each pattern to use at time t. Their sum gives the current vector i(t). In a fixed linear circuit, voltage v depends on resistance R and inductance L:

Resistance sets losses. Inductance sets the voltage required to change current. Faraday’s law couples changing magnetic fields to electric fields. A current waveform is not a voltage waveform.

Measure the map before inverting it.

Declare the observation: a field vector at one point, gradients, or samples over a region. Record the channel inputs, sensor pose, background and uncertainty. Fit y = Âi + b and test predictions on held-out commands.

The physical bench prototype named V2 has historical command-to-field and placement-dependent motion evidence. This apparatus name is separate from the lowercase corner vector v₂. Those observations remain separate from calibrated per-edge current authority, a whole-volume field map, and closed-loop mechanical control. Read the evidence boundary →

Measurement redraws possibility.

Let y* be a requested output, b the measured background, and A the response map. After scaling output units and input budgets, singular value decomposition (SVD) separates reachable output directions from the residual. Small singular values mean weak authority. Current, voltage, slew and thermal limits bound the reachable set further.

U and W give perpendicular output and input directions; Σ contains their gains. Uᵣ keeps the directions resolved above uncertainty. The dagger † denotes the pseudoinverse, which inverts supported gains. These formulas use the chosen normalized coordinates.

The implemented inverse must respect those limits. Feedback compares requested and measured outputs, corrects within the available authority, and updates the map when supported by new measurements. Static rank alone does not establish dynamic controllability or loop stability.

Holding one task leaves only the authority of A₂ ker A₁ for a second. Changing coordinates or scheduling signals does not add simultaneous physical channels.