Free Root 2 Rectangle Grid Generator
Draw the root 2 rectangle instantly with our 100% free grid generator. Overlay the proportion of every A-series paper size, with diagonals, reciprocals and eyes that land exactly on the thirds, on any image, then export SVG or 600 DPI in seconds. No signup, no downloads needed. Trusted by 60,000+ artists worldwide.
The 1:1.414 rectangle and its armature
The root-2 rectangle — the proportion of every A-series paper size, and the only one that halves into two copies of itself — with its diagonals and reciprocals drawn in.
4 controls, not a fixed picture
Toggle rectangle outline, reciprocals, eyes (intersection points) and halving midline. Every one is live on the canvas.
Stacks with composition & photography
Overlays combine, each carrying its own transform, so this one can sit under or over rule of thirds, golden spiral and armature (14-line), and 15 more in the same set.
Matches A-series stock exactly
Because A4, A3 and A2 are all root-2, the construction and the paper share a proportion: export as PDF at true size and the armature fits the sheet edge to edge.
| Ctrl + Z | Undo |
| Ctrl + Y | Redo |
| Esc | Deselect overlay |
| Alt + drag | Free positioning (no snap) |
| Ctrl + scroll | Zoom canvas |
| Ctrl + P | |
| ? | Show this help |
1 : 1.41421 is the proportion of every A-series sheet, and it is the one rectangle whose dynamic-symmetry eyes land exactly on the rule-of-thirds intersections.
The root 2 rectangle generator draws one of the 82 composition overlays in Grid Maker Pro, and the rest stack on the same image.
- 1 : 1.414
- Eyes at 1/3
- SVG + 600 DPI export
- Nothing uploaded
What is the root 2 rectangle?
The root 2 rectangle generator is a free browser tool that draws the 1 : √2 rectangle and its armature over any image or blank canvas. It letterboxes the largest 1 : 1.41421 rectangle that fits, then draws both diagonals, four reciprocals dropped onto them at right angles, and the four eyes where those land. On this rectangle the eyes fall at exactly one third of the width and height — the rule-of-thirds intersections — because a rectangle of ratio r puts its eyes at 1/(1+r²), and for √2 that is 1/3. A halving midline shows the cut that produces two smaller root 2 rectangles.
How to use the root 2 rectangle
A 1:1.414 rectangle with its reciprocals and eyes marked — the proportion that halves into itself, which is why every sheet of A-series paper has it.
Open the rectangle
The largest 1 : 1.41421 rectangle that fits your canvas is drawn and centred, with both diagonals. If your canvas is A-series the rectangle fills it exactly; on anything else it is inset, and you can see by how much.
JPG · PNG · GIF · WEBP
Turn on what you are working to
Reciprocals to get the four eyes, the halving midline to see the self-similar cut, or the bare diagonals when you only want direction.
Rectangle outline · Reciprocals
Place, then export
Put your subject on an eye and a counterweight on the diagonal running away from it. Export SVG for vector, or a raster up to 600 DPI sized to your paper.
PNG · JPG · PDF · SVG · up to 600 DPI
Three steps, about a minute. No signup, no upload.
What the Root 2 Rectangle Generator does
Halves into itself
Cut a root 2 rectangle across its long side and both halves are root 2 rectangles again. That is the defining property, and it is the entire reason the A-series exists — A4 is half of A3 with the same shape.
Eyes exactly on the thirds
The reciprocals land at one third of each side, which are precisely the rule-of-thirds intersections. Every other root rectangle puts them somewhere else — this is the one where the two systems agree exactly.
Letterboxed, and honest about it
On a canvas that is not root 2 the overlay draws the largest one that fits and centres it. The visible inset tells you how much of your frame sits outside the proportion, which is information rather than an inconvenience.
Both diagonals, four reciprocals
The full dynamic-symmetry armature: baroque and sinister diagonals, and a perpendicular dropped from each corner onto the diagonal it misses. The right angles are computed, not sketched.
Over your own work
Its own transform, so the armature can be rotated onto a subject that is not square to the frame and the right angles travel with it.
Vector or print size
SVG export keeps every line vector. Raster export is sized from the paper format and DPI you choose, up to 600 — and if that format is A-series, the rectangle and the sheet coincide.
Root 2 rectangle examples: four ways the armature reads
Root 2 rectangle vs an A-series page in a layout program vs drawing it by hand
| Option | Proportion exact | Reciprocals at true right angles | Vector export | Cost |
|---|---|---|---|---|
| Grid Maker Prothis tool | Yes1 : 1.41421 | Yescomputed feet | SVG + raster to 600 DPI | Free |
| An A-series page in a layout program | Yesthe page is the rectangle | Only if you draw them | Yes | Subscription |
| Drawing it by hand | Yesthe diagonal of a square | By construction | No | Time |
| A downloaded template | Usually | Rarely | Sometimes | Free or paid |
Who the root 2 rectangle is for: work that ends up on a sheet
How the root 2 rectangle works — The proportion that halves into itself, and the reason thirds work on A4
Root 2 is the least glamorous of the 82 composition overlays and the one you handle every day. It is also, for a reason that is calculable rather than mystical, the rectangle on which the rule of thirds is exactly right.
A root 2 rectangle has sides in the ratio 1 : √2 — one to 1.41421. Its defining property is that halving it across the long side produces two rectangles of exactly the same proportion. Do it again and you get four. The shape is preserved at every halving, indefinitely.
That is the whole basis of the ISO 216 paper series. A0 is a root 2 rectangle with an area of one square metre; A1 is A0 halved, A2 is A1 halved, and so on down. Every sheet in the series has the same shape, which is why enlarging A4 to A3 needs no cropping and why a photocopier's 141% button exists.
It is the only rectangle you can cut in half forever without changing its mind about what shape it is.
Where the eyes fall, and why it matters
Draw both diagonals, then drop a perpendicular from each corner onto whichever diagonal does not pass through it. Those four feet are the eyes, and on a root 2 rectangle they sit at exactly one third of the width and one third of the height.
That is not a coincidence and it is not approximate. For a rectangle of ratio r the eyes land at 1/(1 + r²) of each side. For root 2, r² is exactly 2, so the fraction is exactly 1/3. The rule-of-thirds intersections and the root 2 armature are the same four points.
Which explains something that usually gets asserted rather than derived, and is worth checking against the thirds grid generator on the same sheet: the rule of thirds fits A-series paper particularly well because A-series paper is root 2, and root 2's own structural points are the thirds. On a 3:2 photograph, by contrast, the same construction puts the eyes at 0.30769 — close to the thirds, visibly not on them.
The family, and the pattern it follows
The formula generalises across the whole root family and produces a sequence worth carrying: a root N rectangle puts its eyes at exactly 1/(N+1). Root 2 gives thirds, root 3 gives quarters, root 4 gives fifths, root 5 gives sixths. A square — root 1 — gives halves, which is the degenerate case where all four eyes collapse to the centre.
That single line is the most useful thing to know about dynamic symmetry, and it means you can choose a rectangle by the divisions you want rather than by its reputation. If your composition wants quarters, work in root 3; if it wants fifths, work in a double square.
The halving line, and what to do with it
The overlay draws one subdivision: the midline across the long axis. That is the cut that produces the two smaller root 2 rectangles, and it is the line that makes the self-similarity visible rather than merely stated.
In layout it is more useful than it looks. A root 2 page divided at its midline gives you two root 2 panels, each of which can carry its own armature at the same proportions — which is how a spread, a two-up print or a folded card stays coherent with the sheet it came from. Nothing else in the root family has that property so cleanly, because nothing else halves into itself.
Where it comes from, and what is claimed
The root 2 overlay page covers the paper standard, beside the rest of the dynamic symmetry category. The root rectangles as a compositional system come from Jay Hambidge's The Elements of Dynamic Symmetry (1920), out of lectures at the Yale School of Fine Arts. Hambidge argued that Greek design was built on these proportions and their reciprocal diagonals; that historical claim was contested at the time and remains so.
The paper standard is separate and much better documented. The proportion was proposed by Georg Lichtenberg in 1786, standardised as DIN 476 in Germany in 1922, and became ISO 216. There is no dispute about that lineage, and it has nothing to do with Hambidge — two independent arrivals at the same number, for entirely different reasons.
Using it
The practical case is any work printed on A-series paper, which in most of the world is all of it. Set the paper format to your sheet and the rectangle coincides with it exactly, so the armature you compose against is the armature of the actual page rather than of a region inside it.
For photographs the 3:2 rectangle generator is the one to reach for, because cameras do not shoot root 2 — 3:2 and 4:3 are the common sensor proportions and neither is 1.414. Use it when the output is the page rather than the frame, and use the 3:2 or phi rectangle when the frame is what you are composing.
And use the letterbox as information. If the inset is large, your canvas proportion is a long way from root 2, and composing on this armature means composing on a region rather than on the sheet. That is a legitimate thing to do deliberately and a confusing thing to do accidentally.
Checking a sheet you already have
The quickest way to confirm a surface is root 2 is not to measure it. Fold a corner so a short edge lies along a long one: if the diagonal of the square you have just folded reaches exactly to the far corner, the sheet is root 2. That is the construction restated as a test, and it takes about four seconds on any piece of paper.
It is worth doing on stock you have been given rather than bought. Trimmed sheets, off-cuts and anything cut down by hand drift from the standard surprisingly often, and a sheet that is two millimetres out on A4 will put the armature's eyes visibly off the thirds by the time you are working near the edges.
The overlay gives you the other half of the check. Set the paper format to the size you believe you have, load a photograph of the sheet, and see whether the drawn rectangle matches its edges. Where it does not, you now know which dimension is wrong and by how much.
Where the root 2 rectangle gets used
Everything I set goes onto A-series eventually, so composing on the sheet's own armature rather than an imported grid is the obvious move once you have seen it.
The eyes being exactly the thirds settled an argument I had been having with myself for years about which grid to use on printed work.
Halving into itself is the clearest demonstration of self-similarity I can put in front of a class, because they are all holding a sheet of it.
Root 2 rectangle: frequently asked questions
The geometry
What makes root 2 special?
Halving it across the long side produces two rectangles of exactly the same proportion, indefinitely. That is the property the entire A-series paper standard is built on — A4 is A3 halved, with the same shape, which is why enlarging between sizes needs no cropping.
Where do the eyes fall?
At exactly one third of the width and one third of the height — the rule-of-thirds intersections. For a rectangle of ratio r the eyes sit at 1/(1 + r²), and root 2 has r² = 2, so the fraction is exactly 1/3.
So the rule of thirds is the root 2 armature?
On a root 2 rectangle, yes, exactly. That is why thirds fits A-series paper so well and why it fits a 3:2 photograph less well — on 3:2 the same construction puts the eyes at 0.30769, close to the thirds but visibly not on them.
Does the pattern continue through the family?
It does, and it is the most useful thing to know about dynamic symmetry: a root N rectangle puts its eyes at exactly 1/(N+1). Root 2 gives thirds, root 3 quarters, root 4 fifths, root 5 sixths. You can choose a rectangle by the divisions you want.
What is the halving line for?
It marks the cut that produces the two smaller root 2 rectangles. In layout that matters practically — a root 2 page divided at its midline gives two root 2 panels, each able to carry the same armature, which is how a spread stays coherent with its sheet.
Why is the rectangle inset on my canvas?
Because your canvas is not root 2, so the overlay draws the largest one that fits rather than stretching. The visible inset tells you how much of the frame sits outside the proportion — useful information rather than an inconvenience.
Is it good for photographs?
Not on most canvases. Cameras shoot 3:2 or 4:3 and neither is 1.414, so the armature governs a region rather than the frame. Use root 2 when the output is a page, and the 3:2 or phi rectangle when the frame itself is what you are composing.
Where does the proportion come from historically?
Two independent places. As a compositional system it comes from Hambidge's dynamic symmetry (1920), whose historical claims about Greek design are contested. As a paper standard it was proposed by Georg Lichtenberg in 1786, standardised as DIN 476 in 1922 and later ISO 216 — well documented and unrelated to Hambidge.
Using it
Should I leave the halving midline on?
While you are working out the structure, yes — it is the property that makes root 2 what it is, and seeing the two half-pages inside the whole explains the proportion faster than the number does. Once the eyes are placed it is one more line across the picture.
How do I use it on an image that is not A-series?
Let it letterbox. The overlay fits the largest 1 : 1.41421 rectangle inside your canvas and leaves the remainder outside the armature, which is honest — a 3:2 photograph is not a root 2 rectangle, and stretching the armature to fill it would move the eyes off their real positions.
Export and print
Can I export it as an SVG?
Yes. Ten straight lines and four dots export cleanly as vector, which is what you want for plotting a page layout or scaling the armature to a large surface. Raster export goes to 600 DPI at your chosen paper format.
Can I print it at true A-series size?
Yes, and this is the one armature where that is exactly right: choose A4 or A3 as the paper format and the printed rectangle IS the proportion, with no letterboxing at all. At 600 DPI the reciprocals stay fine enough to rule against.
Choosing a grid
If the eyes are so close to thirds, why not use thirds?
Practical matters
Does my image get uploaded?
No. Everything happens in your browser — the image is drawn to a canvas on your own device and never sent anywhere.
Is this useful for laying out a document?
Very — every ISO paper size is a root 2 rectangle, so the armature is the page you are actually designing. Putting a headline on an eye or a column edge on a reciprocal is a real layout decision, not an analogy.
Can I use the exports in a book about proportion?
Yes, with no watermark and no attribution required. The construction is Hambidge's and is long out of copyright; the drawing you export is yours under the terms.
Open the tool Root 2 rectangle generator
The paper proportion, with its own thirds
A root 2 rectangle is the shape of your sheet. Compose on the armature it actually has, and export vector or 600 DPI. No signup, nothing uploaded.
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