From the load-sharing hexagon to a house you can build from the dirt — and how it holds when the ground shakes
The last two chapters built lattices to look at. This one has to stand up. Chapter 16 proved the hexagon wins on packing; Chapter 17 put spin on the same frame. Here the frame carries weight — the weight of a roof, and the far more dangerous, sideways, cyclic weight of an earthquake — and the question is no longer aesthetic. It is: when the ground shakes, does the wall hold? The answer the operator chain gives turns out to be buildable by hand, from the soil under your feet, and it is already a published plan.
This is the chapter where the mathematics leaves the page. The same hexagonal geometry that governs the G6 Crystal lunar base governs a compressed-earth-block house you can raise on land you own. The bridge is a single structural fact: a hexagonal block passes an applied load to six neighbours, so no block bears more than one sixth of it. That fact — and the crack-arresting geometry that comes with it — is what makes the wall seismic. Part of it is proved in SeismicLattice.lean; the hardest part, formal progressive-collapse superiority, is the same open obligation S2 we first met in Chapter 16, and it is still open.
sorry; the full collapse-resistance claim (S2) is named and left open.
Begin with a fact old as tiling itself: only three regular polygons cover the plane with no gaps — the equilateral triangle, the square, and the hexagon. The reason is pure divisibility. A regular $n$-gon tiles only if a whole number of copies closes the $360^\circ$ around each vertex, which happens exactly when $n-2$ divides $4$. The divisors of $4$ are $1, 2, 4$, giving $n = 3, 4, 6$ and nothing else — the pentagon fails because $3 \nmid 10$.
tiling_reduces_to_four proves $(n-2)\mid 2n \Leftrightarrow (n-2)\mid 4$, and only_regular_tilings concludes $n \in \{3,4,6\}$. The three positive cases (triangle_tiles, square_tiles, hexagon_tiles) and the pentagon's failure (pentagon_no_tile) are proved outright.
Among the three survivors the hexagon is special in a way that matters under load. Count the edge-neighbours of one tile — the tiles it shares a full face with, and therefore the tiles it can hand a force to. A triangle has three, a square four, a hexagon six. The hexagon touches the most neighbours of any tile that fills the plane, which means when you push on it, it has the most ways to pass the push along.
Here is the structural heart of the whole architecture programme, and it is one line. If a loaded block splits its burden equally among its contact faces, each face carries $1/(\text{neighbours})$. So:
$\displaystyle \underbrace{\tfrac{1}{6}}_{\text{hexagon}} \;<\; \underbrace{\tfrac{1}{4}}_{\text{square}} \;<\; \underbrace{\tfrac{1}{3}}_{\text{triangle}}.$
The hexagonal block puts the least load on any single face of any tiling polygon — and the six shares still add back to the whole, $6 \times \tfrac16 = 1$, so nothing is lost, only distributed. A square wall concentrates a quarter of every load onto one neighbour; a hexagonal wall never lets any neighbour feel more than a sixth. Under the steady weight of a roof this is a nicety. Under the reversing, hammering load of an earthquake — where failure begins wherever stress concentrates — it is the difference between a wall that cracks and a wall that spreads.
hexagon_most_neighbors: $3 < 6$ and $4 < 6$ — the hexagon has the most edge-neighbours. Proved.hex_load_share_min: $\tfrac16 < \tfrac14 < \tfrac13$ — least load per face. Proved.hex_load_conserved: $6 \cdot \tfrac16 = 1$ — the shares conserve the load. Proved.seismic_bridge, proved without sorry.
There is a second, subtler gift. A crack is a straight-through path looking for the easiest route across a wall. In a square grid the mortar lines run straight, and a crack can travel a full course without turning — a ready-made fault. In a hexagonal tiling there are no straight-through lines: every path must turn at a vertex, and to cross the wall a crack has to zig-zag around six-fold junctions, spending energy at each turn. The geometry itself arrests the crack. That claim — call it Q1, crack-path tortuosity — is stated in the Lean file and left open, because a faithful proof is a topological statement about the tiling graph.
The interactive wall below lets you drop a load on a block and watch it distribute. Toggle between square and hexagonal bond and compare the peak face-load; send a crack and watch it run straight through the square and stall in the hexagon.
hex_load_share_min), and every path turns at a vertex — the crack stalls. The 1/6 figure is the mathematical minimum for a convex tile that fills the plane.
The G6 Crystal paper derived this geometry for lunar regolith — a place with no lumber, no cement plant, no supply chain, only the material underfoot and the mathematics of how to build with it. But, as the Earth House plan puts it, the physics does not care where you are: dirt is dirt, compression is compression, the hexagon holds. The operator chain that governs the Moon base governs a house in rural New Mexico or in Newark, and it runs in the same four steps:
| Operator | On the Moon (G6 Crystal) | On Earth (G6 Earth House) |
|---|---|---|
| C · Compress | Regolith is the material universe | Test and screen your soil — the jar test, the ribbon test |
| K · Curvature | Sinter / press blocks | The CEB press: the pressure at which loose earth becomes a structural block |
| F · Fold / test | Load the vault against vacuum and quakes | The wall tests the block; hexagonal tiling spreads the force so no block bears it alone |
| U · Unfold | An inhabited pressurised module | A room, a wall, a roof — a home, from what was already underfoot |
Compressed earth block is not new — adobe and rammed earth are ten thousand years old. What the framework adds is the formal justification for the hexagonal interlocking profile: with the G6 interlocking block, load crosses six contact faces instead of two, and the wall needs no mortar because geometry replaces adhesive. This is the same load-sharing that beehives, basalt columns, and the lunar vault use. The Earth House plan estimates a $192\text{ ft}^2$ starter shell in materials for roughly the price of a used car, against tens of thousands for conventional construction, because the primary material — the soil — costs nothing.
The full step-by-step plan — soil testing, the CEB press, foundation and bond beam, the hexagonal interlocking courses, roof, plaster, off-grid systems, and a real material-cost table — is published as an open guide:
→ G6 Earth House · Build with Dirt · part of the Hour House programme, Newark NJ. The plan is motivated by the housing crisis it cites (HUD's 2024 figure of roughly 771,000 people without shelter; the NLIHC shortage of affordable homes) — architecture as a humanitarian instrument, not only a mathematical one.
Spreading load is necessary but not sufficient. Earthquakes attack in ways a static roof never does — lateral, reversing, resonant — and earth construction has real, known failure modes that the hexagon alone does not fix. The plan and the standard practice address them directly, and it is worth naming them so the claim stays honest:
Tie the top together. A reinforced concrete bond beam around the full top of the wall turns a stack of independent blocks into a single ring that moves as one — the most important seismic detail in earth-block building. Keep it off the water. A rubble-trench foundation capped with brick lifts the earth wall above the moisture line; water, not shaking, is the usual killer of earth walls. Give it thickness or a frame. In earthquake zones the plan calls for doubled wall thickness or a bond beam and, for two storeys, real engineering. And detune from the ground. A building has a natural period; a quake has a dominant period; safety means keeping them apart so the structure does not resonate with the shaking — the exact inverse of the Schumann lock we wanted in Chapter 16. Here we want a mismatch.
Chapter 16 tuned the G6 Crystal onto a resonance (the Schumann $n{=}4$ lock, obligation S1). Seismic design is the same physics run backwards: keep the structure's fundamental period $T$ away from the ground's dominant period $T_g$, so the forcing never builds. In the framework both are Arnold-tongue questions about the dm³ contact flow — one asks to enter a tongue, the other to stay out of it. The proposition detune_from_ground_period states the mismatch condition; like S1 it needs the ODE response model, and is left open (Q2).
Eleven facts are formalized without sorry in SeismicLattice.lean; three hard obligations are named and open. As with the other Architecture files these are elementary Mathlib proofs (divisibility, arithmetic inequalities), stated below and slated for the repository's kernel CI. None depends on the Structural Hypothesis.
| Claim | Lean name | Status |
|---|---|---|
| Tiling condition reduces to $(n-2)\mid 4$ | tiling_reduces_to_four | ✓ Formalized |
| Only $n\in\{3,4,6\}$ regular polygons tile the plane | only_regular_tilings | ✓ Formalized |
| Triangle/square/hexagon tile; pentagon does not | triangle_tiles … pentagon_no_tile | ✓ Formalized |
| Hexagon has the most edge-neighbours (6) | hexagon_most_neighbors | ✓ Formalized |
| Load-share minimum $1/6 < 1/4 < 1/3$ | hex_load_share_min | ✓ Formalized |
| Load is conserved across the six faces ($6\cdot\tfrac16=1$) | hex_load_conserved | ✓ Formalized |
| Hexagonal face-load below both square and triangle | hex_face_load_below_all | ✓ Formalized |
| Fewest polygons meet at a hexagon vertex (3) | hexagon_fewest_at_vertex | ✓ Formalized |
| dm³ invariant signs $\mu_{\max}<0$, $T^\ast>0$ | mu_max_neg, T_star_pos | ✓ Formalized |
| Bundled proved core | seismic_bridge | ✓ Formalized |
| S2 · hexagrid progressive-collapse superiority — needs FEM formalisation (same obligation as Ch 16) | hexagrid_collapse_superior_placeholder | ○ OPEN |
| Q1 · crack-path tortuosity — topological claim on the tiling graph | crack_tortuosity_placeholder | ○ OPEN |
| Q2 · seismic response-spectrum detuning — needs a structural-dynamics response model | detune_from_ground_period | ○ OPEN (sorry) |
Chapters 16, 17 and 18 are one arc seen three ways: the crystal's shape, its spin, and now its strength. Chapter 16 proved the hexagon packs best and left S2 — hexagrid collapse resistance — open; this chapter is where S2 becomes a house, and the obligation is carried forward unchanged and honest. Upstream it rests on Chapter 16's packing optimum and Chapter 10's dm³ dynamics (the period $T^\ast$ that seismic design must detune from). Downstream it leaves the book entirely: the G6 Earth House plan is the operator chain poured into real walls. The formal content lives in SeismicLattice.lean, beside G6Crystal.lean, DM3Bridge.lean and MagneticLattice.lean.
tiling_reduces_to_four, show by hand that a regular octagon ($n=8$) cannot tile the plane, and identify the one Archimedean (semi-regular) tiling that does use octagons. What extra freedom does mixing two polygon types buy, and why does it not change the load-share argument for a single-block wall?
detune_from_ground_period (open) asks that $|T - T_g|$ be bounded below. For a single-storey CEB house with fundamental period $T \approx 0.1\text{ s}$ and a quake with dominant period $T_g \approx 0.3\text{ s}$, argue qualitatively whether the house is in danger of resonance, and write the Lean theorem signature you would add to state "a stiffer, lighter wall raises $|T - T_g|$."