The one idea behind all of them: reaching the unreachable
Every system here obeys the same kind of rule: some directions of motion are allowed, others are forbidden, and the forbidden ones can only be reached indirectly. A car cannot slide sideways — yet it parks sideways, by a back-and-forth wiggle. That wiggle is the geometric heart of the whole subject: combine “drive forward” and “steer” in the right order and you get net sideways motion that neither move alone provides. Mathematicians call that combination the bracket of the two moves; engineers call the whole situation nonholonomic.
Now, one system at a time.
Heisenberg $H^3$ — a charged particle, and the shortest fence
The system. Picture a charged particle moving in a plane with a magnetic field pointing straight up out of it. The Lorentz force curves its path into circles — Larmor orbits. Track a third number alongside its position: the magnetic flux swept out by its trajectory, which is just the signed area it encloses. Position plus that area is the Heisenberg group, and the particle’s circular orbits are exactly its sub-Riemannian geodesics.
The everyday version. A boat that can go forward/back and left/right, with a meter that ticks up by the signed area it encloses. “Get home having enclosed exactly this much area, by the shortest possible route” is the ancient isoperimetric (Dido) problem — and its answer is a circular arc. The area is the “vertical” coordinate, and you can only change it by going around — never directly.
Where it also lives. Heisenberg is literally the algebra of quantum mechanics: position and momentum with $[x, p] = i\hbar$, the center being phase. It underlies the uncertainty principle, the Gabor transform, and time–frequency analysis.
Growth vector $(2,3)$; caustic. Two directly-drivable directions, one area coordinate reached by a bracket (weight 2). As the series proves, the caustic collapses to a line: every orbit of a given curvature refocuses at the same phase point after one full turn, at time $2\pi/|w|$. It is the flat reference — the simplest possible version of “reaching the unreachable.”
SE(2) — the parking car and the visual cortex
The system. A car (or bicycle, or unicycle) that drives forward and steers but cannot slip sideways: the Dubins/Reeds–Shepp car of robot motion planning. Its configuration is position plus heading, the group $\mathrm{SE}(2)$, and its length-optimal paths are the workhorses of autonomous-vehicle planners.
The surprising twin. Your visual cortex runs the same geometry. Area V1 lifts each edge to a (position, orientation) pair and completes broken contours along $\mathrm{SE}(2)$ geodesics — the “association field” of Petitot and Citti–Sarti that explains illusory contours like the Kanizsa triangle. The Geometry of Seeing series is entirely about this group.
The shape of its geodesics. They are Euler’s elastica — the curve a thin springy rod bends into. So the optimal path of a parking car, the contour your brain hallucinates, and the shape of a bent leaf-spring are the same curve.
Growth vector $(2,3)$; the aliasing. Same as Heisenberg — which is the whole point of the series’ hardest case: zoomed in, a parking car and a magnetic orbit are indistinguishable. Only the finite-scale curvature (the deviation $\delta$) tells SE(2) from Heisenberg (Part 3).
Engel — the truck with one trailer
The system. A car towing a single trailer. Its state is (position, cab heading, trailer heading) — four numbers. You steer the cab directly, but the trailer angle you can only change indirectly, by driving while turning. Engel is the nilpotent model of this generic car-with-trailer kinematics.
Why backing up a trailer is hard. The trailer heading is a weight-3 coordinate: reaching it needs a bracket of a bracket — a maneuver nested two deep. That is the precise mathematical reason a trailer is so much harder to reverse into a spot than a car: its key coordinate is buried one indirection deeper than the car’s sideways slide.
Growth vector $(2,3,4)$; abnormals. And Engel is the first group in the series with abnormal geodesics — special optimal motions (here, driving dead straight) that owe their optimality to the shape of the constraints, not the metric. That yes/no bit is a robust fingerprint (Part 3, Appendix C5).
Cartan $(2,3,5)$ — rolling spheres
Not a second trailer. A car with two trailers has growth vector $(2,3,4,5)$ — each trailer buries one more coordinate one level deeper. The Cartan case is different: it has two coordinates at weight 3, growth vector $(2,3,5)$, and its natural machine is a rolling one.
The beautiful version. Take two spheres, one rolling on the other without slipping and without twisting. The allowed motions (two independent rolling directions) generate a $(2,3,5)$ distribution — and when the radius ratio is exactly 1 : 3, this humble system has the (split real form of the) exceptional Lie group $G_2$ as its local symmetry. This is the system Élie Cartan singled out in his 1910 “five variables” paper; the rolling-sphere realization is one of the few places the smallest of the exceptional groups shows up in something you could build on a desk.
Growth vector $(2,3,5)$; abnormals present. Its deepest coordinate is the most noise-fragile to recover — which is exactly why the detector finds Cartan the hardest of the four to pin down under noise (Part 3).
SE(3) — the MRI scanner and the drone
The system. Move through 3D space and carry an orientation: position in $\mathbb{R}^3$ plus a full orientation (a pointing direction on the sphere $S^2$ together with the roll about it), the six-dimensional group $\mathrm{SE}(3)$. You may go forward along your axis and reorient that axis two ways, but not slide sideways — a rank-three structure, the first in the series with three drivable directions.
Where it lives.
- Diffusion MRI. Each voxel of brain white matter records the local nerve-fibre direction; tracing a tract means following $\mathrm{SE}(3)$ geodesics through position + orientation. This is Duits’ fibre-tracking geometry, and the real-world target of the series’ final post.
- Drones and aircraft. A fixed-wing craft flies forward and can pitch and yaw (reorienting its axis) but cannot translate sideways — the same rank-3 structure. Roll is generated by the bracket of the two reorientations.
- Surgical needle steering. A bevel-tipped needle pushed through tissue turns as it advances: forward + reorient, nonholonomic 3D steering.
Growth vector $(3,6)$. Three drivable directions; the three brackets fill all six dimensions at once (reorienting two ways makes a roll; reorienting while moving makes the two sideways translations). Rank three is loud — no rank-two group can imitate it — so the detector separates SE(3) from all the earlier groups the instant it measures the growth vector (Part 4).
Summary: the groups as machines
| Group | Real-world system | Allowed moves | Hidden coordinate (weight) | Growth vector |
|---|---|---|---|---|
| Heisenberg | charged particle in a magnetic field; the shortest-fence (Dido) problem; quantum phase space | move in the plane | enclosed area / flux (2) | $(2,3)$ |
| SE(2) | parking car (Dubins); visual-cortex contour completion | drive + steer | sideways slip (2) | $(2,3)$ |
| Engel | car with one trailer | drive + steer | trailer angle (3) | $(2,3,4)$ |
| Cartan | two spheres rolling without slip or twist ($G_2$ at 1:3) | two rolling directions | two deepest contact/orientation coordinates (3) | $(2,3,5)$ |
| SE(3) | diffusion-MRI fibre tracking; drone/aircraft; steerable needle | drive forward + reorient axis (×2) | sideways + roll (2) | $(3,6)$ |
The pattern to carry into the rest of the appendices: the growth vector is nonholonomic parking difficulty, made into a number. Weight 1 is a direction you drive; weight 2 needs one wiggle (a bracket); weight 3 needs a wiggle of wiggles. That number is the coarse fingerprint — the first thing the inverse detector reads off a field of caustics, and the thing that sorts a magnetic orbit, a trailer truck, and an MRI fibre field into three different geometric species.
Glossary
- Nonholonomic — a system whose allowed velocities are restricted, but which can still reach any configuration by combining them (a car reaches any parking spot despite not sliding sideways).
- Bracket — the net motion from combining two allowed moves in a back-and-forth wiggle; reaches directions neither move gives alone.
- Growth vector — how many coordinates are reached at each “bracket depth”: directly (weight 1), by one bracket (weight 2), by a nested bracket (weight 3), …
- Larmor orbit — the circular path of a charged particle in a magnetic field; the Heisenberg geodesic.
- Dubins / Reeds–Shepp car — a vehicle that drives and steers without side-slip; the $\mathrm{SE}(2)$ model.
- Euler elastica — the shape of a bent elastic rod; the $\mathrm{SE}(2)$ geodesic.
- Abnormal geodesic — an optimal motion forced by the shape of the constraints rather than the metric (e.g. driving a trailer dead straight); present for Engel, Cartan, SE(3), absent for Heisenberg, SE(2).
References
- A. M. Vershik & V. Ya. Gershkovich (1994). “Nonholonomic dynamical systems, geometry of distributions and variational problems.” Dynamical Systems VII, Springer.
- R. Montgomery (2002). A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys and Monographs 91. (Rolling spheres, $G_2$, abnormal geodesics.)
- J.-P. Laumond, ed. (1998). Robot Motion Planning and Control. Springer LNCIS 229. (Car-with-trailers nonholonomic control.)
- J. Petitot (2003). “The neurogeometry of pinwheels as a sub-Riemannian contact structure.” J. Physiol. Paris 97, 265–309.
- R. Duits, A. Ghosh, T. C. J. Dela Haije & A. Mashtakov (2013). “On sub-Riemannian geodesics in $\mathrm{SE}(3)$ whose spatial projections do not have cusps.” arXiv:1305.6061.
- G. Bor & R. Montgomery (2009). “$G_2$ and the rolling distribution.” Enseign. Math. 55, 157–196.