K4 · GEOMETRY INTO POSSIBILITY
Four corners.
Six edges.
What follows?
Start with a regular tetrahedron. Follow its geometry into coordinates, its connections into algebra, and only then choose what to build.
Each step supplies something the next can use. Physical capability enters through a specified plant and its measurements.
Dashed midpoint axes form a Cartesian frame.
Dimensionless regular simplex · oblique projection
The shape supplies a frame.
Every corner connects to the other three. All six edges have the same length. Adjacent edges meet at 60°. Opposite edges do not meet, but their direction vectors are perpendicular.
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.
OF THE MIRROR
Four balanced coordinates. All of 3D space.
Let v₀,…,v₃ run from the centre to the four corners. Their sum is zero. Assign four real coefficients φᵢ whose sum is also zero. These have exactly three independent degrees of freedom.
x = Σ φᵢvᵢ = Vφ
1⊥ ⊂ ℝ⁴, with 1⊥ ≅ ℝ³
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.
The inverse, and a useful calculus bridge
For corner radius ρ, VVᵀ = (4ρ²/3)I₃, so φ = 3Vᵀx/(4ρ²). A fixed frame commutes with differentiation and integration:
For a path integral, dx = V dφ and F(x)·dx = [VᵀF(Vφ)]·dφ. This is ordinary coordinate calculus; the metric and the integration measure must transform too. See the exact frame relations →
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. The vector Dz records the four corner balances.
cycle (3) ⊥ cut (3)
Pcut = DᵀD/4 · Pcycle = I₆ − Pcut
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.
Under all tetrahedral rotations and reflections, oriented edge space splits into the distinct representations T₁ and T₂. Any linear edge-to-edge operator respecting that action has just two coefficients:
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.
Two meanings of E, kept distinct
Unoriented scalar edge values transform as A₁ ⊕ Eirrep ⊕ T₂. Here Eirrep is a two-dimensional symmetry representation. The electric field is written Efield; it is a different object. Quadratic observables require their own map, including cross terms.
Now choose a physical plant.
The graph can organize computation and control before it represents conductors, a coupled fluid, or another physical system. A plant supplies units, dynamics, constraints and an observer. Here the worked example is electromagnetism.
For thin complete current paths in a linear magnetoquasistatic regime, Biot–Savart supplies one field column per current channel. Those columns add:
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.
What perpendicularity and winding actually imply
Ideal straight perpendicular filaments have zero mutual partial inductance because dℓ·dℓ′ = 0. Complete circuits also contain turns, feeds and returns; their mutual inductance need not vanish.
Endpoint magnetic poles describe an equivalent long-solenoid model, not magnetic monopoles. Coincident, matched equivalent poles can cancel at graph corners; that does not prove cancellation for arbitrary finite helices or complete windings.
A pattern becomes a waveform.
Choose a basis Q for the admissible channel space and time-dependent coordinates q(t). The composition is exact; delivering it is a plant problem.
v(t) = Ri(t) + L di(t)/dt
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, and PWM duty is not a calibrated current measurement.
See what the six signals compose.
Start with one edge. Add a second and compare the field slices. Then try a coordinated pattern and follow its six waveforms. The scope and all four views use the same instantaneous currents.
The retained simulator uses ideal finite edge segments and optional ideal corner dipoles. Its colours are relative. It shows source superposition and local cancellation; omitted returns prevent treating arbitrary edge patterns as complete-circuit predictions.
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.
V2 has historical command-to-field and placement-dependent motion evidence. 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.
After scaling output units and input budgets, singular value decomposition separates reachable output directions from the residual. Small singular values mean weak authority. Current, voltage, slew and thermal limits bound the reachable set further.
yreachable = UᵣUᵣᵀ(y* − b)
i* = A†(y* − b) · unconstrained reference
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.
Read every claim at its own level.
Exact geometry / algebra [A]Identities under stated mathematical assumptions.
Established law / regimePhysical laws with their applicability named.
Model [M] / numericalA specified approximation and its computed output.
MeasuredA dated observation with input, pose and uncertainty.
Entailed · size untestedA conditional consequence; practical magnitude still needs a test.
Open hypothesis [C]A proposed mechanism or capability awaiting evidence.
These labels describe evidence, not milestones that automatically promote one another. The existing tags are retained: [G] exact for a named model, [G*] regime-limited exact, and [H] engineering guidance. “Measured” and “entailed, size untested” are evidence descriptors, not replacements for these grades.