Every map has a domain.

The regular simplex as a coordinate frame

A regular tetrahedron has four equilateral triangular faces and six equal edges. Let their common length be a > 0; a = 1 means one chosen length unit. Set c = a/(2√2). The columns of V are the centroid-to-corner vectors. Corner radius ρ = √3c. Coordinates have the length units of a; φ (phi) is the dimensionless list of four balanced coefficients.

A vector is an ordered list; a matrix combines lists by weighted addition. Here 𝟙 = (1,1,1,1)ᵀ is the all-ones vector, Iₙ is the n-by-n identity, and ᵀ transposes rows and columns. The outer product 𝟙𝟙ᵀ is the four-by-four all-ones matrix. P removes the mean; 𝟙⊥ is the zero-sum subspace. Vertical bars denote Euclidean length.

V = c [ 1   1  −1  −1 ]
      [ 1  −1   1  −1 ]
      [ 1  −1  −1   1 ]
P = I₄ − 𝟙𝟙ᵀ/4
V𝟙 = 0          VVᵀ = 4c²I₃          VᵀV = 4c²P
x = Vφ          φ = Vᵀx/(4c²),  with 𝟙ᵀφ = 0
|x|² = 4c²|φ|²  on 𝟙⊥

The restriction V:𝟙⊥→ℝ³ is invertible and isotropic, with length scale 2c. Signed balanced coefficients cover all Cartesian space. They are not nonnegative barycentric weights confined to the solid.

For a length-preserving version, define T = V/(2c) and the normalized Cartesian position ξ = x/(2c). Then ξ = Tφ, φ = Tᵀξ, TTᵀ = I₃ and TᵀT = P. For example ξ = (1,0,0)ᵀ corresponds to φ = (½,½,−½,−½)ᵀ. This is the normalized frame used in the foundations document.

The six vectors vⱼ−vᵢ are the edges. For example (v₁−v₀)·(v₃−v₂) = 0. The opposite-edge midpoint differences are (2c,0,0), (0,2c,0) and (0,0,2c), up to direction. Adjacent edges are not perpendicular.

x(t) = Vφ(t)       ẋ(t) = Vφ̇(t)
∫[t₀,t₁] x(t)dt = V ∫[t₀,t₁] φ(t)dt
∫γ F(x)·dx = ∫γφ [VᵀF(Vφ)]·dφ
∇𝟙⊥(f∘V) = Vᵀ∇ₓf

Incidence makes the split constructive

Use edge order (01,02,03,12,13,23), oriented i→j for i<j. D is incidence, never the magnetic field B. For an arbitrary oriented edge vector z, s = Dz is net incoming amount at each corner. The complete graph K₄ means that each pair of corners is connected. Here ker is the set of inputs mapped to zero, im is the set of attainable outputs, rank counts independent outputs, and ⊕ denotes a unique direct sum. A projector extracts a component and leaves it unchanged if applied again.

D = [ −1 −1 −1  0  0  0 ]
    [  1  0  0 −1 −1  0 ]
    [  0  1  0  1  0 −1 ]
    [  0  0  1  0  1  1 ]
DDᵀ = 4I₄ − 𝟙𝟙ᵀ = 4P
Pcut = DᵀD/4         Pcycle = I₆ − Pcut
zcut = Dᵀ(Dz/4)      zcycle = z − zcut
rank D = 3           ℝ⁶ = ker D ⊕ im Dᵀ
Pcut²=Pcut; Pcycle²=Pcycle; Pcut Pcycle=0

ψ = (−3, 1, 1, 1)ᵀ → g = (4, 4, 4, 0, 0, 0)ᵀ
Dg/4 = ψ
loop 0→1→2→0: z = (1, −1, 0, 1, 0, 0)ᵀ; Dz = 0

The edge geometry itself is VD. The projector formula uses complete, equally weighted K4. Reversing edge orientations conjugates its projectors by a diagonal sign matrix.

Allowed operator forms

T_d is the group of all tetrahedral rotations and reflections; it has the same structure as S₄, the permutations of four labels. A₄ contains only the proper rotations. A representation describes how a list changes under these operations; an irreducible representation (irrep) is a pattern family that cannot be split into smaller invariant families. Directed edges carry orientation signs. Their cycle family is called T₁ and their cut family T₂. Under T_d these are distinct three-dimensional types. X maps six oriented inputs to six oriented outputs; α, β, γ and δ below are gains.

Full T_d symmetry:     X = αPcycle + βPcut
Rotations A₄ only:     X ≃ [ αI₃  γI₃ ]
                           [ δI₃  βI₃ ]
Symmetric X:           δ = γ

The block form uses aligned orthonormal bases for the two equivalent A₄ copies. It has four real parameters in general, three if symmetric, and two with full T_d symmetry. Frequency-domain impedances may have complex coefficients; reciprocity is transpose symmetry, not automatically Hermitian symmetry.

A symmetry/model residual can reflect geometry, leads, material properties, sensor error, calibration or omitted physics. It does not uniquely diagnose construction error or handedness.

An edge temperature is unchanged when its label is reversed. Such unoriented scalar lists split differently: A₁ ⊕ E_irrep ⊕ T₂, with dimensions 1 + 2 + 3. A₁ is the common value. E_irrep is the two-dimensional family with equal values on opposite edges and three pair values summing to zero. These are symmetry labels, not new physical quantities.

Choose complete paths before predicting fields

Now let the edge values represent branch currents i, measured in amperes. A steady closed network without external corner injection obeys Kirchhoff’s current law (KCL): Di = 0. Independent edge currents require a larger complete circuit including external returns or injections.

For fixed thin conductors in air or vacuum, a magnetostatic or suitable magnetoquasistatic model constructs the per-amp field from each complete channel path Cⱼ. B is magnetic field (tesla), iⱼ is current (amperes), and bⱼ is field per ampere. Source position is x′, dℓ′ a directed path element, μ₀ vacuum permeability, × the cross product, and ∮ a sum around the closed path:

bⱼ(x) = (μ₀/4π) ∮Cⱼ dℓ′ × (x−x′)/|x−x′|³
B(x,t) = Σⱼ bⱼ(x)iⱼ(t)
Mpartial,ab = (μ₀/4π) ∫a ∫b (dℓ·dℓ′)/|r−r′|

Straight perpendicular filament segments have zero mutual partial inductance. Complete wound circuits include other path elements. A bare segment is a contribution to the integral; arbitrary isolated segment currents are not a complete steady conserved current distribution.

In a magnetostatic current-free region, divergence ∇·B=0 means no net magnetic flux source, and curl ∇×B=0 means no local circulation. These make the field gradient symmetric and trace-free, with five independent components. At arbitrary time-varying frequency, displacement current and regime assumptions matter.

Optional corner dipoles or electrodes require their own geometry and columns. Magnetohydrodynamics further adds fluid state, conductivity, boundaries and force coupling; these are not supplied by the graph.

From coordinates to signals

Q collects admissible current patterns as columns; q(t) gives their time-dependent weights and i(t) the resulting currents. Voltage v is measured in volts, resistance R in ohms, and inductance L in henries, including mutual coupling. A dot or d/dt gives rate of change.

i(t) = Qq(t)
v(t) = Ri(t) + L di(t)/dt

Q spans physically admissible current combinations. Amplitude, frequency, phase and offset choose q(t). The RL equation assumes a fixed lumped linear circuit; nonlinear cores, motion, parasitics and frequency-dependent losses require additional terms.

A changing magnetic field also produces a circulating electric field, E_field, measured in volts per metre. Faraday’s law is ∇×E_field = −∂B/∂t; ∂ denotes a partial derivative. This physical field is separate from the symmetry label E_irrep introduced for scalar edge patterns.

Two complementary readings of the six edges

For each directed edge, let ê be its unit direction and r̂ the unit vector from its midpoint to the centre. Form two three-by-six matrices: W₀ has columns ê; F₀ has columns ê × r̂. For the regular tetrahedron these give a second, constructive view of the cycle/cut split.

F₀ᵀF₀ = 2Pcycle       W₀ᵀW₀ = 2Pcut       W₀F₀ᵀ = 0
β = F₀z               ε = W₀z
z = ½(F₀ᵀβ + W₀ᵀε)
|β|² + |ε|² = 2|z|²

Here β and ε are normalized geometric readings, each a three-component list. Together they recover every oriented edge list z. In the equal straight-filament model, F₀ also gives the directions of the centre magnetic contributions. Turning either reading into a physical field requires the source law, units, complete paths and boundary conditions.

A snapshot and a dynamic response answer different questions.

The browser evaluates a fixed Biot–Savart source map at the current values on the playhead. Its edge sources are straight, finite-length, zero-thickness filaments; optional corner sources are point dipoles. Finite length is part of the defined geometry. Thin filaments and point dipoles simplify the calculation; they are not a claim of better physical performance.

Magnetostatics concerns steady sources. A magnetoquasistatic model allows time variation while neglecting displacement current in Ampère’s law and electromagnetic propagation effects; magnetic induction remains. The simple snapshot map additionally assumes a fixed spatial current distribution and a linear, effectively memoryless material response.

One necessary scale check compares propagation time with the fastest significant variation: ωℓ/v_prop ≪ 1, where ω = 2πf is angular frequency, ℓ is the largest relevant physical dimension and v_prop the propagation speed in the medium. This alone does not establish every quasistatic or circuit approximation. Skin and proximity effects, eddy currents, capacitance and driver bandwidth can matter before wave propagation does. Sharp waveform edges contain harmonics above their fundamental frequency.

Hysteresis is a material-history effect. In a hysteretic magnetic medium, field response depends on prior magnetization and may do so even under slow driving. It is not automatically required merely because frequency increases. A material with memory needs a state-dependent response model; an air-core model does not acquire ferromagnetic hysteresis by turning up its frequency.

A linear time-invariant dynamic plant can instead use complex amplitudes at each angular frequency:

B_ω(x) = F_B(x, ω) i_ω

F_B carries amplitude and phase response. Nonlinear or hysteretic materials generally require more than this linear frequency map. At scales where propagation matters, use the appropriate full Maxwell model. The normalized browser geometry and frequency sliders do not specify a validated apparatus bandwidth.

Physical basis: MIT: quasistatic laws, time and length scales, and magnetization and hysteresis.

Define possibility at the observer

Choose output features y, background b and a calibrated map Â. If inputs are duty commands, label it a command-response map rather than a per-amp map. Validate against held-out commands and declared sensor poses. A hat marks an estimate; a star marks a target. Sᵧ divides outputs by positive reference scales, and Sᵢ supplies positive current scales. Thus ỹ = Sᵧ(y−b) and normalized current ī = Sᵢ⁻¹i.

y − b = A i                  (or A Q q if constrained)
à = Sy A Si                  output and input scales
à = UΣWᵀ
ỹreachable = Ur Urᵀ ỹtarget
residual = (I − Ur Urᵀ) ỹtarget

SVD is singular value decomposition: U contains perpendicular unit output directions, W contains perpendicular unit input directions, and Σ their nonnegative gains. Uᵣ retains the r directions resolved above uncertainty. Use a noise-informed rank threshold. Static output rank is not dynamic controllability. Singular values depend on units and budgets; field and gradient rows need scaling before their joint condition number is meaningful.

minimize  ||Sy(Ai − (y*−b))||² + λ iᵀRi
subject to current, voltage, slew and thermal constraints

Unconstrained minimum loss for feasible y, R positive definite:
i* = R^(−1/2) (A R^(−1/2))† (y−b)

Holding A₁i fixed leaves secondary authority A₂ ker(A₁).

Here † is the Moore–Penrose pseudoinverse: it inverts supported singular gains and sets unsupported ones to zero. The norm ‖ ‖ measures Euclidean length; λ weights the loss penalty. R⁻¹ᐟ² is the inverse positive square root of R. All weights and units must be chosen consistently.

A current-loop model can add K(i*−i_meas) to feedforward Ri*+L di*/dt, where K converts measured current error into voltage correction. Delays, sampling, saturation and plant dynamics still require stability analysis and experimental validation. New measurements revise  and uncertainty; they do not guarantee a larger reachable set.

Entailed, size untested: a validated map retaining rank and operating headroom permits a constrained command to test. Open: volume control, mechanical trapping and computational advantage need their own plant-specific evidence.

From a point to a region

In a source-free magnetostatic neighbourhood, B is locally the gradient of a harmonic scalar potential. Its value has 3 independent components; its first spatial derivative adds 5, its second adds 7, and its third adds 9. Thus arbitrary local Taylor data through curvature has 15 components; through third derivative, 24. These counts describe compatible field targets, not actuator performance.

Matching every such target requires at least that many independent, effective input directions and a full-row-rank response map. More conductors alone do not guarantee that rank. Splitting an edge into series segments adds no independent current; separately driven complete paths can change the input space. Finite-region accuracy must be checked away from the fitted point.