Avalon Hill hex wargames
The hex-and-counter format that made the hexagon a game surface. Every counter moves a fixed number of equidistant cells, so terrain becomes arithmetic and the diagonal argument disappears.
Specialty grids · honeycomb tiling · tabletop & nature
The hexagonal grid
A hexagonal grid tessellates the plane with regular six-sided cells in a honeycomb pattern, so every interior cell shares an edge with six equidistant neighbours — no separate, longer diagonal the way a square grid has. That single uniform adjacency is why tabletop maps abandoned squares for hexes, and the same shape's least-wall efficiency is why bees, foams, and graphene settle into it. Here is what the overlay lays down, the packing math behind it, the verified history from the three regular tilings to the hex wargame, and when a hexagonal grid earns its place over a square one.
- Type
- Regular tiling
- Built from
- Six-sided cells
- Difficulty
- Intermediate
- Orientations
- Pointy-top / flat-top
- Packing
- Densest equal-circle
- Also known as
- Honeycomb grid
See the hexagonal grid on five subjects
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A hex map turns terrain into countable steps: each cell touches six others at the same distance, so a unit's move range is a clean ring rather than a square with cheaper edges and pricier corners. Drag the handle to see the lattice settle over the scene.
What the overlay shows
The hexagonal overlay lays a tessellation of regular hexagons over the frame — pointy-top by default, with flat-top available. Each interior cell shares an edge with six others, and because the hexagons are regular every one of those neighbours sits at the same centre-to-centre distance. There is no second class of neighbour. Cell size, line weight, opacity, and orientation are all adjustable, so the grid reads over a dark battle map as clearly as a blank quilting sheet.
The defining property is that uniform adjacency. On a square grid a cell touches four others edge to edge and four more only at the corners, further away — the source of the awkward "is a diagonal one step or one-and-a-half" argument that every square-based game has to legislate around. The hexagon dissolves the problem: every direction is the same direction. That is why it became the surface for movement-based games, and why the same shape, read as a packing rather than a map, shows up wherever a material is minimising boundary.
The math, briefly
Start with one regular hexagon. Its interior angle is fixed:
interior angle = 120° · three meet at every vertex (3 × 120° = 360°)
Three facts follow from that single number:
- Only three shapes tile alone. A regular polygon tiles the plane edge to edge by itself only if its interior angle divides 360° — which leaves the equilateral triangle (60°), the square (90°), and the regular hexagon (120°). These are the three regular tilings, a result laid out in Grünbaum and Shephard's Tilings and Patterns.1 The hexagon and triangle are duals: join the centres of the hexagons and you get triangles, and vice versa.
- Least wall, densest circles. Among all ways to divide the plane into equal-area cells, the regular hexagonal honeycomb has the least total perimeter — the honeycomb conjecture, proven by Thomas Hales in 2001.2 Relatedly, packing equal circles so each touches six others is the densest circle packing in the plane, the structural fact behind foams and graphene.56
- Coordinates need a system. Unlike a square grid, a hex grid has no obvious (x, y). Three schemes are standard: offset (rows and columns with an alternating half-cell shift), axial (two axes that make movement clean), and cube (three redundant axes summing to zero that make distance a single formula). The geometry sits comfortably in Coxeter's treatment of plane lattices.7
The overlay keeps the angles and spacing exact so you can count cells instead of measuring them. Try it in the live tool and one hexagon becomes your working unit.
History — what is real and what is myth
Verified history (with primary sources)
Antiquity — the three regular tilings. That only the triangle, square, and hexagon tile the plane on their own has been known since classical geometry; Grünbaum and Shephard's Tilings and Patterns remains the modern reference that collected and systematised this body of knowledge.1 The hexagon was the one of the three that nature seemed to have chosen first.
1859 — Darwin and the bee's cell. In On the Origin of Species, Darwin devoted a long passage of the chapter on instinct to the hive-bee, arguing that the hexagonal comb is the product of natural selection economising wax rather than innate geometry. He called the comb-building instinct "the most wonderful of all known instincts," and concluded that the bee had "practically solved a recondite problem" of enclosing the most honey with the least wax.4
1917 — D'Arcy Thompson and packing. Thompson's On Growth and Form set the hexagon in a wider physical frame, treating the honeycomb and many biological patterns as outcomes of surface tension and packing under pressure rather than design — the line of argument that anticipates the foam physics later formalised by Weaire and Hutzler.35 Peter Pearce's Structure in Nature Is a Strategy for Design carried the same principle forward into architecture and product design.6
1950s — the hex wargame. The hexagonal grid became a game surface through Charles S. Roberts and the founding of Avalon Hill, whose early titles established the hex-and-counter format and made the hex map the default for conflict simulation. James Dunnigan, who designed and published hundreds of such games, documents this lineage in The Complete Wargames Handbook, explaining that hexes were adopted precisely to remove the diagonal-movement ambiguity of square boards.8 The 1995 hit Settlers of Catan later carried the hex tile into mainstream board gaming.
Honest caveats
"The honeycomb conjecture is old folklore." The intuition is ancient and the conjecture was stated long ago, but a complete, peer-reviewed proof did not exist until Hales published it in Discrete & Computational Geometry in 2001.2 Citations that date the proof to 1999 are pointing at the arXiv preprint, not the published result.
"Hexes have no straight rows." They have clean rows in one direction only. Pointy-top hexes form tidy horizontal rows but no vertical columns; flat-top hexes form tidy columns but no horizontal rows. There is no orientation that gives both, which is the price of uniform adjacency.
"Hex coordinates are as simple as squares." They are not. The absence of a single natural (x, y) means you must choose offset, axial, or cube coordinates and convert between them, and beginners routinely mix schemes mid-map and break their own distance maths.
When to use it (and when not)
| If you want to... | Use hexagonal | Don't use it for... | Difficulty |
|---|---|---|---|
| Build an overland or operational game map | Six equidistant neighbours make movement and range uniform in every direction | Indoor tactical maps where walls and doors run square (use a square grid) | Beginner |
| Design a honeycomb decorative pattern or tile layout | Cells pack with no gaps and the least grout line for a given area | Rectilinear layouts that must align to a square room or page margin | Beginner |
| Plan English paper-piecing or hexie quilting | One hexagon equals one fabric piece — count cells, not millimetres | Strip-pieced or block quilts built on a square grid | Intermediate |
| Illustrate a molecular or crystalline lattice | Matches graphene, beryl, and basalt-column geometry directly | Cubic or tetragonal crystals that are not hexagonal at all | Intermediate |
| Teach packing and tiling to students | The clearest physical case of a least-perimeter, densest-circle arrangement | Lessons about diagonal symmetry, where the square reads more plainly | Advanced |
Famous examples of the grid at work
Six documented places where the hexagonal grid is demonstrably the chosen structure — in games, in nature, and in materials.
Settlers of Catan (1995)
The hex tile reaching the mainstream. Resource hexes meet three to a vertex, so a single settlement collects from three terrains at once — the corner-and-edge economy is the whole game.
Beehive honeycomb
Darwin's case study. Bees building in wax converge on hexagons because that shape holds the most honey for the least wall — the least-perimeter result later proven as the honeycomb conjecture.
Graphene lattice
A single sheet of carbon atoms arranged in a hexagonal lattice. The same packing logic that minimises foam-wall energy gives graphene its strength-to-weight ratio.
Foam and bubble packing
A two-dimensional foam relaxes toward hexagonal cells because surface tension minimises wall length — the physical demonstration of the honeycomb result, slightly distorted by real-world disorder.
Hex graph paper for game design
Before a map is digital it is sketched on hex paper. Designers block movement zones and terrain by counting cells, which is exactly what the overlay reproduces over a reference photo or scan.
Common mistakes
1
Mixing coordinate systems mid-map
Labelling part of a map with offset coordinates and then switching to axial for movement maths is the fastest way to compute the wrong distance between two cells, because the conversions are not interchangeable by eye.
Fix: pick one scheme for the whole project. Use offset for printed legends, axial or cube for any code, and convert once at the boundary rather than per cell.
2
Choosing the wrong orientation for the movement axis
Pointy-top gives clean horizontal rows; flat-top gives clean vertical columns. Pick the one that fights your map's main travel direction and every route description becomes a diagonal description.
Fix: decide orientation from the dominant axis of play. East–west overland travel suits pointy-top; a tall, scrolling battlefield suits flat-top.
3
Forcing hexes onto a square subject
Interiors, buildings, and city blocks are rectilinear. Laid over a square-walled dungeon, a hex grid leaves doorways and corridors slicing awkwardly across cells and helps nobody.
Fix: reserve hexes for open terrain and packing problems. Use a square grid where the world itself is built on right angles.
4
Treating the cell size as cosmetic
In quilting, tiling, and tabletop play the hexagon equals a physical thing — a fabric piece, a ceramic tile, a movement step. Setting the spacing to whatever looks nice means the grid no longer measures what you are building.
Fix: set one hexagon to one real unit before drawing anything, so counting cells gives you the true count of pieces, tiles, or steps.
How different disciplines use it
For game designers
The hexagon is the working surface of conflict simulation and overland exploration. Block terrain, zones of control, and movement allowances by counting cells, and the uniform adjacency keeps balance honest — a unit with three movement points reaches the same ring of cells whichever way it faces. Decide pointy-top versus flat-top and your coordinate scheme before laying a single tile, because both choices propagate through every rule that mentions an "adjacent hex."
For quilters
English paper piecing is built on the hexagon — the grandmother's-flower-garden block is rosettes of hexies sewn edge to edge. Set one grid cell to your finished piece size so counting cells across a layout tells you exactly how many papers to cut. The overlay also lets you audition a colour arrangement over a photographed fabric pull before committing a single stitch.
For pattern designers
Honeycomb is a perennial motif in tile, textile, and surface graphics because it packs with no gaps and reads as both organic and engineered. Use the grid to keep a repeat truly seamless — every cell edge has to meet its neighbour — and to scale a hexagonal subway-tile layout against a real wall dimension, grout line included, before ordering material.
For illustrators and educators
Graphene, beryl, basalt columns, foam, and honeycomb are all the same lattice seen at different scales, which makes the hexagon a teaching gift. Overlay it on a micrograph or a photograph of the Giant's Causeway to show students that the pattern is a least-energy outcome rather than a coincidence, and pair it with the packing math to connect the picture to the proof.
"the bee... has practically solved a recondite problem, and has made her cells of the proper shape to hold the greatest possible amount of honey, with the least possible consumption of precious wax."
Charles Darwin, On the Origin of Species, John Murray (1859)4
Frequently asked questions
What is a hexagonal grid?
A hexagonal grid tessellates the plane with regular six-sided cells in a honeycomb pattern, so every interior cell shares an edge with exactly six neighbours, all at the same centre-to-centre distance. That single uniform adjacency — no separate diagonal — is why it became the standard surface for movement-based tabletop games and a natural fit for packing problems.
Why use hexagons instead of squares for game maps?
A square grid has two kinds of neighbour at two different distances: four edge-sharing cells and four corner-sharing diagonals further away. That mismatch makes movement and range ambiguous. A hexagonal grid has one neighbour type — six cells, all equidistant — so a step in any direction costs the same. Hex-and-counter wargames adopted hexes for exactly this reason.
Why do bees build hexagonal honeycomb?
Of the shapes that tile the plane with no gaps, the regular hexagon encloses a given area with the least total wall. Bees building in wax are under direct pressure to minimise material, so they converge on hexagons. Darwin discussed this in On the Origin of Species; the underlying optimality was proven as the honeycomb conjecture by Thomas Hales in 2001.
What is the difference between pointy-top and flat-top hexagons?
Pointy-top hexagons have a vertex at top and bottom and flat left and right edges; they tile into horizontal rows. Flat-top hexagons have flat top and bottom edges and a vertex left and right; they tile into vertical columns. The geometry is identical — only the rotation and the resulting row-or-column reading differ — but coordinate maths and movement axes change with it.
How are hexagonal grids given coordinates?
There is no single obvious scheme as there is for squares. Three are common: offset coordinates (rows and columns with an alternating half-cell shift), axial coordinates (two axes that simplify movement), and cube coordinates (three redundant axes summing to zero that make distance a clean formula). Printed maps usually use offset; software usually uses axial or cube internally.
Are hexagons the densest packing?
For equal circles, yes: the hexagonal arrangement, where each circle touches six others, is the densest packing of equal circles in the plane. Hexagonal cells are also the least-perimeter partition of the plane into equal areas — the honeycomb conjecture. Foams and graphene settle into hexagonal arrangements for the same energy-minimising reason.
What are the regular tilings of the plane?
Only three regular polygons tile the plane edge to edge using a single shape: the equilateral triangle, the square, and the regular hexagon. This follows because the interior angle has to divide 360° evenly at each vertex. The hexagon and triangle are duals — joining the centres of one produces the other.
What software supports hexagonal grids?
Map-making tools (Hex Kit, Worldographer, Inkarnate), virtual tabletops (Foundry, Roll20), and game engines (Unity, Godot) all support hex layouts, as do quilting and pattern apps. Grid Maker Pro overlays a configurable pointy-top or flat-top hexagonal grid over any reference image in the browser, with PNG, SVG, and PDF export.
References
- Grünbaum, B. & Shephard, G.C. Tilings and Patterns. W.H. Freeman, New York (1987). ISBN 0-7167-1193-1.
- Hales, T.C. "The Honeycomb Conjecture." Discrete & Computational Geometry 25, 1–22 (2001). DOI: 10.1007/s004540010071.
- Thompson, D'Arcy W. On Growth and Form. Cambridge University Press (1917; abridged ed., J.T. Bonner, 1961).
- Darwin, C. On the Origin of Species. John Murray, London (1859), Ch. VII ("Instinct"), on the hive-bee's cell.
- Weaire, D. & Hutzler, S. The Physics of Foams. Oxford University Press (1999). ISBN 0-19-850551-5.
- Pearce, P. Structure in Nature Is a Strategy for Design. MIT Press, Cambridge MA (1978). ISBN 0-262-16064-1.
- Coxeter, H.S.M. Introduction to Geometry. Wiley, New York (1961).
- Dunnigan, J.F. The Complete Wargames Handbook. William Morrow, New York (1992). ISBN 0-688-10368-5.
Other specialty overlays
Notes from the studio · Three practitioners on the hexagonal grid
Illustrative composites of how the tool gets used in practice — not quotes from named individuals.
When I prototype a campaign map I drop the scan onto a hex overlay and count movement ranges straight off the cells. The deep-link reopens with the orientation and spacing already set, which is the whole battle.
For teaching close-packing I overlay the hex lattice on a micrograph of foam. Students see that the cells are not drawn that way — they relax into it to minimise wall, which is the same reason graphene holds together.
A honeycomb repeat only reads as seamless if every edge meets its neighbour exactly. Setting one cell to the real tile size and checking the grout line over a photo of the wall saves a reorder every time.
Open the hexagonal grid overlay
Drop a reference image or start from a blank canvas. The hexagonal grid overlay applies in one click, pointy-top or flat-top. Free, in your browser.
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