Farish's plates (1822)
The founding document. Farish's engraved machine drawings show gears and frames in equal-axis projection so a workshop could read dimensions directly off the page.
Perspective · three axes 120° apart · Farish, 1822
Isometric grid for technical drawing
An isometric grid draws three axes 120° apart — two climbing at 30° from horizontal, one vertical — with no vanishing point, so parallel edges stay parallel and one cell measures the same unit on every axis. It is the measurable cousin of linear perspective: less atmosphere, more dimension. Here is what the overlay does, the foreshortening math nobody mentions, why most "isometric" video games are technically dimetric, and when the grid earns its place over a true vanishing-point drawing.
- First documented
- 1822 (Farish)
- Popularised in
- 19th-c engineering
- Origin culture
- British (Cambridge)
- Difficulty
- Intermediate
- Axis angles
- 30° / 150° / 90°
- Also known as
- 30° axonometric
See the isometric grid on five built subjects
‹›
Building edges that recede along two horizontal directions map onto the 30° isometric axes. If the structure is genuinely rectilinear, every cornice and sill should ride a grid line — drag the handle to check.
What the overlay shows
The isometric overlay lays down a triangular lattice: a family of lines climbing left-to-right at 30° above horizontal, a mirrored family at 150°, and a set of true verticals. Those three directions are the projected images of the object's three principal axes — width, depth, and height — and in the drawing they sit exactly 120° apart. Cell size, line weight, and opacity are all adjustable so the grid reads over a dark reference as easily as a blank page.
The defining property is measurability. Because there is no vanishing point, a unit cube becomes one grid cell on each axis, with no further foreshortening calculation. You can take a measurement off the drawing and trust it back in the scene — the reason isometric became the engineer's projection. The cost is depth: near and far objects render the same size, so the grid trades atmosphere for dimension.
If you are wondering what angle an isometric grid uses, the answer is fixed: the two ground axes always sit 30 degrees from horizontal, and the three axes are 120 degrees apart. To draw on an isometric grid, trace every edge along one of those three directions and let the cell size set your unit — that is the whole technique.
The math, briefly
Project a cube so that all three axes make equal angles with the picture plane. Each axis then foreshortens to the same factor:
cos(35.26°) = √(2/3) ≈ 0.8165 · drawing axes at 30° from horizontal
Three facts fall out of that single constraint:
- The 30° angle. When the foreshortening on all three axes is equal, the two ground axes project to exactly 30° above the horizon and the three axes are 120° apart — the angle you set on the grid.
- The 0.816 scale. True isometric shrinks every edge to ≈81.6% of life. Draughtsmen usually skip this by drawing full length along each axis — an "isometric scale" that enlarges the whole drawing by ≈1.22× but keeps the three axes consistent. Both are valid; mixing them is not.
- Circles become ellipses. A circle on any isometric face projects to an ellipse with a major-to-minor ratio of √3 : 1 ≈ 1.732, tilted to its face. There are exactly three ellipse orientations, one per face.
The grid does the angle-keeping for you — every cell is pre-aligned to the three axes. Try it in the live tool and the cell size sets your working unit.
History — what is real and what is myth
Verified history (with primary sources)
1822 — William Farish. Farish, a professor at Cambridge, read his paper On Isometrical Perspective to the Cambridge Philosophical Society and published it in the society's Transactions. He coined "isometric" (Greek: equal measure) for a system that let engineers draw machines so that measurements survived on every axis — a genuine problem in an era of accelerating mechanisation.1
1853 — Karl Pohlke. The German geometer stated what is now Pohlke's theorem, the fundamental theorem of axonometry: any three line segments from a point can serve as the projected image of three equal, mutually perpendicular axes. This put Farish's practical trick on rigorous footing and generalised it to dimetric and trimetric projection.7
19th–20th century — the engineer's standard. Through the industrial era isometric became the dominant convention for machine drawing, exploded assembly diagrams, and patent illustration, a lineage traced in Peter Booker's A History of Engineering Drawing.2 It was eventually codified internationally as one of the axonometric methods in ISO 5456-3, and remains the architect's quick three-dimensional notation in references such as Francis Ching's Architectural Graphics.56
Confusions that won't die
"Video-game isometric is isometric." Most of it isn't. The 2:1 pixel ratio of SimCity 2000, Diablo II, and the pixel-art revival produces axis angles of arctan(0.5) ≈ 26.57°, not 30° — which makes it technically dimetric. The 2:1 ratio was an engineering choice: integer pixel steps rendered faster and tiled without seams on 1990s hardware.3
"Isometric has no distortion." It removes perspective convergence, not distortion. Every axis is foreshortened to 0.816, and the projection visibly skews squares into rhombi and circles into ellipses. What it preserves is ratio along each axis, not appearance.
"It's a kind of perspective." Farish's own title says "isometrical perspective," but in modern terms isometric is a parallel (axonometric) projection, the opposite limit of linear perspective. Carlbom and Paciorek's classic survey places both on one continuum: perspective has a finite centre of projection, axonometric pushes it to infinity.4
When to use it (and when not)
| If you want to... | Use isometric | Don't use it for... | Difficulty |
|---|---|---|---|
| Draw a machine or part that must stay measurable | Every axis reads to scale — dimensions survive straight off the drawing | Hero renders where depth and drama matter (use 2-point perspective) | Beginner |
| Build tile-based game art | Identical tiles tessellate with no perspective math per cell | First-person or cinematic camera views (use a perspective rig) | Intermediate |
| Show a product from a single neutral angle | Width and depth share one scale — clean for spec sheets and packaging | Lifestyle product shots needing a sense of place | Beginner |
| Explain a system or assembly (infographic, exploded view) | Parallel axes keep parts aligned as they separate in the diagram | Organic or curved subjects with no rectilinear structure | Intermediate |
| Teach axonometric construction to students | Equal foreshortening makes it the simplest axonometric to grid by hand | Quick atmospheric thumbnails (use a horizon + vanishing point) | Advanced |
Famous examples of the projection at work
Six works and products where isometric (or its 2:1 dimetric cousin) is demonstrably the chosen system.
Q*bert & Zaxxon (1982)
Among the first games to read as 3D on 2D hardware. The stacked-cube pyramid of Q*bert is pure axonometric — no camera, just parallel axes.
SimCity 2000 (1993)
The template for the dimetric city-builder. Tiles repeat at a 2:1 ratio (≈26.57°), so every zone block measures the same regardless of where it sits on the map.
Monument Valley (2014)
Built on true isometric precisely so impossible Escher geometry reads as solid. With no perspective convergence, two paths at different depths can be made to "touch."
IKEA assembly instructions
Wordless, language-independent, and isometric by design. Exploded parts drift apart along parallel axes so a panel's edge always lines up with its socket.
M.C. Escher's parallel worlds
Works like Cube with Magic Ribbons exploit axonometric projection: with no vanishing point, near and far can be deliberately, beautifully confused.
Common mistakes
1
Drawing true circles on isometric faces
A wheel, a dial, a coin — drawn as a perfect circle on an isometric face, it instantly looks pasted-on and breaks the projection. Circles must become ellipses at the √3:1 ratio, tilted to the face they lie on.
Fix: use an isometric ellipse guide. There are only three orientations — top, left, and right faces — so learn all three once.
2
Mixing true 30° with the 2:1 game ratio
Authoring some assets at true 30° isometric and others at the 26.57° 2:1 pixel ratio produces a subtly inconsistent rhythm — tiles that won't quite line up and shadows that point two ways.
Fix: pick one standard per project. Use 2:1 only when committed to pixel-perfect retro art; use true 30° everywhere vector coordinates allow.
3
Reaching for isometric when you need depth
Isometric flattens distance — far objects are as large as near ones. Used for a landscape or an interior meant to feel deep, it reads as airless and toy-like.
Fix: if the goal is atmosphere and recession, switch to 1- or 2-point perspective. Reserve isometric for subjects where measurement beats mood.
4
Adding a vanishing point "to help"
Letting the parallel axes converge even slightly defeats the entire purpose: the drawing stops being measurable and becomes a weak perspective instead.
Fix: keep every axis strictly parallel. If you find yourself wanting convergence, you actually want perspective — change tools, don't bend the grid.
How different disciplines use it
For illustrators
Isometric is the backbone of the modern "tech illustration" look — flat-colour cities, server diagrams, editorial infographics. Block the scene on the grid first, commit every edge to one of the three axes, then add lighting that respects a single light direction. Because there is no convergence, you can extend the composition in any direction without re-solving a horizon, which is why isometric sets tessellate so happily into patterns.
For game artists
Decide true 30° versus 2:1 dimetric before drawing a single tile — the choice dictates your whole asset pipeline. Tile-based engines (Unity Tilemap, Godot, Phaser) expect a consistent ratio so sprites stack without gaps. The payoff is that a tile drawn once reads correctly anywhere on the map, and depth-sorting reduces to a simple back-to-front cell order rather than a z-buffer.
For product designers
Isometric is the neutral "three-quarter" view for spec sheets, packaging nets, and exploded assembly diagrams. Width and depth share one scale, so a caliper measurement maps straight to the drawing. Pair it with a consistent ellipse guide for fillets and bores, and keep one face flat to the reader when a feature needs emphasis — at which point you may actually prefer oblique projection.
For architects
The axonometric (often a 30°/30° or plan-oblique variant) is the architect's fast 3D notation, covered as a core skill in Ching's Architectural Graphics. It communicates massing and spatial relationships while staying measurable — useful for early massing studies, worm's-eye ceiling plans, and construction details where a perspective would hide dimensions behind foreshortening.
"Axonometry is a matter of perspective — or rather, the deliberate refusal of it. By sending the viewpoint to infinity, the draughtsman trades the illusion of depth for the certainty of measure."
Jan Krikke, Axonometry: A Matter of Perspective, IEEE Computer Graphics & Applications (2000)3
Frequently asked questions
What is an isometric grid?
A parallel-projection drawing guide whose three principal axes sit 120° apart in the drawing — the two ground axes at 30° above horizontal, plus a vertical. There are no vanishing points, so parallel edges stay parallel and one cell measures the same unit on every axis. That measurability is why engineers and game artists adopted it.
Is video-game "isometric" really isometric?
Usually not. The 2:1 pixel ratio in SimCity 2000, Diablo II, and most pixel-art games yields axis angles of about 26.57°, not the true 30° — making it technically dimetric. The 2:1 ratio was chosen because integer pixel steps rendered faster on 1990s hardware and the visual difference is small.
How much does isometric projection foreshorten?
True isometric foreshortens every axis to √(2/3) ≈ 0.816 of true length. In practice most draughtsmen ignore this and draw full length using an "isometric scale," which enlarges the whole drawing by about 1.22× but keeps all three axes consistent.
Why do circles look like ellipses on an isometric grid?
A circle on a tilted isometric face projects to an ellipse with a major-to-minor ratio of √3:1 ≈ 1.732, aligned to that face. Drawing a true circle on an isometric face is the most common beginner error — use the correct ellipse for each of the three face orientations.
Who invented isometric projection?
William Farish, a Cambridge professor, formalised it in his 1822 paper "On Isometrical Perspective." The broader mathematics of axonometric projection was generalised by Karl Pohlke's 1853 theorem.
What's the difference between isometric, dimetric, and trimetric?
They are the three axonometric families. Isometric foreshortens all three axes equally (one scale); dimetric has two axes equal and one different (two scales); trimetric has all three different (three scales). Isometric is the most regular and the easiest to grid.
Does isometric show real depth?
No. Because parallel lines never converge, isometric has no perspective depth cue — distant objects are the same size as near ones. That is a feature for measurement and a limitation for atmosphere; use linear perspective when you need a sense of distance.
What is the difference between true isometric and 2:1 isometric?
True isometric draws the two ground axes at exactly 30° from horizontal, with all three axes 120° apart and foreshortened equally. The "2:1 isometric" common in game art steps two pixels across for every one down, which lands at arctan(0.5) ≈ 26.57° — close to 30° but technically dimetric. Choose true 30° when vector coordinates allow it, and 2:1 only when you need integer-pixel tiles that tessellate without seams.
What software supports isometric grids?
Vector tools (Illustrator, Affinity Designer, Inkscape, Figma plugins) and 2D game engines (Unity Tilemap, Godot, Phaser) all support isometric or near-isometric grids. Grid Maker Pro overlays a true 30° isometric grid over any reference image in the browser, with PNG, SVG, and PDF export.
References
- Farish, W. "On Isometrical Perspective." Transactions of the Cambridge Philosophical Society, Vol. 1, pp. 1–20 (1822).
- Booker, P.J. A History of Engineering Drawing. Chatto & Windus, London (1963).
- Krikke, J. "Axonometry: A Matter of Perspective." IEEE Computer Graphics and Applications 20(4), 7–11 (2000). DOI: 10.1109/38.851742.
- Carlbom, I. & Paciorek, J. "Planar Geometric Projections and Viewing Transformations." ACM Computing Surveys 10(4), 465–502 (1978). DOI: 10.1145/356744.356750.
- International Organization for Standardization. ISO 5456-3:1996 — Technical drawings — Projection methods — Part 3: Axonometric representations. Geneva (1996).
- Ching, F.D.K. Architectural Graphics (6th ed.). Wiley (2015). ISBN 978-1-118-73948-1.
- Pohlke, K. Fundamental theorem of axonometry (1853). See Monge, G. Géométrie descriptive, Baudouin, Paris (1799), for the descriptive-geometry foundation it generalises.
- Maynard, P. Drawing Distinctions: The Varieties of Graphic Expression. Cornell University Press (2005). ISBN 978-0-8014-4263-6.
Other Perspective overlays
Notes from the studio · Three practitioners on the isometric grid
Illustrative composites of how the tool gets used in practice — not quotes from named individuals.
For isometric tech illustration I block the whole scene on the grid first. The deep-link reopens with the exact overlay configured — no clicking through menus mid-session.
I keep three Grid Maker Pro tabs open when prototyping a tile set — true 30° in one, the 2:1 ratio in another, to compare. The bookmarkable URLs make this workflow possible.
Free and browser-only is the right shape for a measuring tool. Lower friction means I actually drop a part photo onto the grid instead of saving it for special occasions.
Open the Isometric grid overlay
Drop a reference image. The Isometric grid overlay applies in one click. Free, in your browser.
Launch Grid Maker Pro →