Free Root 3 Rectangle Grid Generator
Draw the root 3 rectangle instantly with our 100% free grid generator. Overlay the proportion the hexagon and the vesica piscis both produce, with its full armature and quarter divisions, on any image, then export SVG or 600 DPI in seconds. No signup, no downloads needed. Trusted by 60,000+ artists worldwide.
The 1:1.732 rectangle and its armature
The root-3 rectangle with its dynamic-symmetry lines. It is the proportion that arises from the hexagon and the vesica, which is why it recurs in work built on six-fold geometry.
4 controls, not a fixed picture
Toggle rectangle outline, reciprocals, eyes (intersection points) and thirds. 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.
Exact at any enlargement
The proportion is irrational and defined geometrically, so SVG export keeps it exact where a pixel grid would round it. PNG, JPG and PDF up to 600 DPI.
| 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 : 1.73205 is the number the hexagon and the vesica piscis both produce, and it is the rectangle whose armature lands on the quarters rather than the thirds.
The root 3 rectangle generator draws one of the 82 composition overlays in Grid Maker Pro, and the rest stack on the same image.
- 1 : 1.732
- Eyes at 1/4
- SVG + 600 DPI export
- Nothing uploaded
What is the root 3 rectangle?
The root 3 rectangle generator is a free browser tool that draws the 1 : √3 rectangle and its armature over any image or blank canvas. Both diagonals, four reciprocals and the four eyes where those land — which on this proportion fall at exactly a quarter of the width and height, because a rectangle of ratio r puts its eyes at 1/(1+r²) and r² here is exactly 3. Two subdivision lines show the cuts that divide it into three smaller root 3 rectangles.
How to use the root 3 rectangle
A 1:1.732 rectangle divided into its own quarters, with the diagonals and their crossings drawn, so the placements are visible before anything is committed.
Open the rectangle
The largest 1 : 1.73205 rectangle that fits your canvas is drawn and centred with both diagonals. On a canvas that is not root 3 the shape is inset, and the inset is visible.
JPG · PNG · GIF · WEBP
Turn on what you need
Reciprocals for the four eyes at the quarters, the thirds subdivision to see the self-similar cut, or the diagonals alone when you only want direction.
Rectangle outline · Reciprocals
Place, then export
Subject on an eye, 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 3 Rectangle Generator does
Thirds into itself
Cut a root 3 rectangle into three across its long side and every piece is another root 3 rectangle. Where root 2 halves into itself, this one thirds — which is the property that separates the two.
Eyes exactly on the quarters
The reciprocals land at a quarter of each side. Compared with the thirds those points sit noticeably further toward the edges, which is what gives compositions on this rectangle their characteristic openness at the centre.
The hexagon's number
√3 is the ratio between a regular hexagon's width across the flats and its circumradius, and it is the proportion of the vesica piscis lens. Anything built on six-fold symmetry produces this rectangle sooner or later.
Letterboxed, and visibly so
On a canvas of another proportion the overlay draws the largest root 3 rectangle that fits and centres it. Seeing the inset tells you how much of your frame the armature does not govern.
Over your own work
Its own transform, so the armature can be rotated onto a subject that is not square to the frame with the right angles intact.
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.
Root 3 rectangle examples: four ways the armature reads
Root 3 rectangle vs drawing it by hand vs a downloaded template
| Option | Proportion exact | Reciprocals at true right angles | Vector export | Cost |
|---|---|---|---|---|
| Grid Maker Prothis tool | Yes1 : 1.73205 | Yescomputed feet | SVG + raster to 600 DPI | Free |
| Drawing it by hand | Yesthe diagonal of a root 2 rectangle | By construction | No | Time |
| A downloaded template | Usually | Rarely | Sometimes | Free or paid |
| Guides in a layout program | Yesif you compute it | Only if you draw them | Yes | Subscription |
Who the root 3 rectangle is for: work with six-fold structure already in it
How the root 3 rectangle works — The number six-fold geometry keeps producing, and the quarters it puts on a rectangle
Root 3 is the one of the 82 composition overlays you arrive at rather than choose. Every construction built on equilateral triangles, hexagons or the vesica produces √3 somewhere, and this is what that number looks like when you make a rectangle of it.
A root 3 rectangle has sides in the ratio 1 : 1.73205. Its self-similar property is that cutting it into three across the long side gives three rectangles of exactly the same proportion — where root 2 halves into itself, root 3 thirds into itself, and the pattern continues up the family.
The number arrives from six-fold geometry in several independent ways. A regular hexagon's width across the flats is √3 times its circumradius. An equilateral triangle's height is √3/2 of its side. The vesica piscis — two circles each through the other's centre — encloses a lens exactly √3 times taller than it is wide. All the same constant, reached from different directions.
You do not usually decide to use root 3. You notice that the thing you are drawing has already produced it.
The eyes land on the quarters
Both diagonals, four reciprocals dropped onto them at right angles, and the four feet are the eyes. On this rectangle they sit at exactly one quarter of the width and one quarter of the height, because the eyes of a rectangle of ratio r fall at 1/(1 + r²) and r² is exactly 3.
That is a visibly different composition from the thirds. Quarter points sit further out toward the edges, leaving the centre of the frame more open and pushing the subject closer to the boundary. On a long rectangle that reads as spaciousness; used carelessly it reads as a subject falling out of the frame.
The family sequence is worth carrying: root N puts its eyes at 1/(N+1). The root 2 armature generator gives thirds, root 3 quarters, and the root 5 armature generator sixths, root 4 fifths, root 5 sixths. Choosing a root rectangle is really choosing which division you want your placements to sit on.
Where it turns up in real work
Gothic tracery is the obvious case. Window heads built on the vesica inherit √3, and the panels beneath them frequently follow. The same proportion governs a great deal of Islamic geometric ornament for the same reason — six-fold constructions are what a compass produces, and √3 is what six-fold constructions produce.
It also appears in equilateral-triangle-based layouts of any kind: hexagonal tiling, triangular lattices, and the packing arrangements that follow from them. If your subject already has six-fold or three-fold structure, root 3 is the rectangle that agrees with it rather than fighting it.
The three-way subdivision, and using it
The overlay draws two lines across the long axis, at one third and two thirds. Those are the cuts that produce three root 3 rectangles, and they are what makes the self-similarity visible.
In layout that gives you a three-panel division where every panel repeats the whole. A root 3 spread divided into thirds gives three columns each carrying the same armature at the same proportions — the analogue of what root 2 does with halves, and useful in exactly the places a three-column structure is wanted.
It also produces a pleasant coincidence worth noticing: the subdivision is at thirds while the eyes are at quarters, so the two sets of lines never coincide. On root 2 the eyes are at thirds and the subdivision at a half, and there the two also stay clear of each other. That separation is what stops these armatures reading as a grid.
Hambidge, and the honest position
The root 3 overlay page covers where the proportion turns up, beside the rest of the dynamic symmetry category. The root rectangles as a compositional family come from Jay Hambidge's The Elements of Dynamic Symmetry (1920). His historical argument — that Greek vases and temples were designed on these proportions — was contested when he made it and remains contested.
What survived the argument is the method rather than the archaeology. Choosing a rectangle first and composing on its own diagonals and reciprocals produces a coherence that is genuinely hard to reach otherwise, and it is still taught for that reason. The geometry here is exact regardless of what the Greeks were doing.
Practical notes
Root 3 is long — 1.732 to 1 is longer than a 35mm frame and longer than the golden rectangle. On a standard photograph or a standard sheet the letterboxed inset will be substantial, and composing on the armature means composing on a band across the middle. That is fine if it is deliberate.
Where it fits without inset is panel work, banner formats and anything cut to proportion rather than bought in a standard size. If you are making the surface as well as the image, root 3 is available in a way it is not when the surface is A4.
And if you are working from a hexagonal or triangular construction, set the rectangle to match rather than the other way round — the √3 is already in your drawing, and aligning the frame to it costs nothing and removes a source of disagreement you would otherwise spend the whole composition negotiating.
Getting to it from a root 2
There is a compass construction that steps through the whole family and it is worth knowing, because it means you never need to measure any of these proportions. Start with a square. Its diagonal, swung down, gives you root 2. The diagonal of that root 2 rectangle, swung down again, gives you root 3. The next gives root 4, then root 5.
Each step is the same move: the diagonal of the rectangle you have becomes the long side of the next. That is why the family is named the way it is — a root N rectangle's diagonal is √(N+1) of the short side, so the sequence generates itself and no arithmetic is ever involved.
For root 3 specifically that means two swings from a square, which is quick enough to do on the work itself. If you are laying out a panel and want the proportion exactly, drawing it is faster than measuring it and rather more accurate.
One consequence of the length worth planning for: at 1.732 to 1, a root 3 field is long enough that a single subject sitting on one eye leaves a great deal of unoccupied rectangle. That is a feature on a frieze and a problem on a portrait, and the usual remedy is to treat the three-way subdivision as three fields rather than the whole as one — compose each third, and let the armature relate them.
Where the root 3 rectangle gets used
Tracery panels. The vesica gives me √3 whether I want it or not, so the panel proportion may as well agree with the window above it.
For triangular-lattice pattern work the frame at root 3 means the repeat meets the edges cleanly instead of being cropped mid-cell.
Quarters rather than thirds changes the feel of a long panel more than I expected — the centre stays open and the eye travels.
Root 3 rectangle: frequently asked questions
The geometry
Where do the eyes fall on a root 3 rectangle?
At exactly one quarter of the width and one quarter of the height. The eyes of a rectangle of ratio r sit at 1/(1 + r²), and root 3 has r² = 3, so the fraction is exactly 1/4.
How does it subdivide?
Into three. Cutting it across the long side at one third and two thirds produces three rectangles of exactly the same proportion — where root 2 halves into itself, root 3 thirds into itself.
Why does √3 keep appearing in six-fold geometry?
Because it is the number six-fold constructions produce. A hexagon's width across the flats is √3 times its circumradius, an equilateral triangle's height is √3/2 of its side, and the vesica piscis lens is √3 times taller than it is wide. Same constant, reached independently.
How does composing on quarters differ from thirds?
Quarter points sit further toward the edges, leaving the centre more open and pushing the subject nearer the boundary. On a long rectangle that reads as spaciousness; overdone it reads as a subject falling out of the frame.
Does the family follow a pattern?
Yes, and it is worth carrying: a root N rectangle puts its eyes at 1/(N+1). Root 2 gives thirds, root 3 quarters, root 4 fifths, root 5 sixths. Choosing a root rectangle is choosing which division your placements sit on.
Do the eyes and the subdivision ever coincide?
No, and for a viewer that is part of why these armatures do not read as grids. Here the eyes are at quarters and the subdivision at thirds; on root 2 the eyes are at thirds and the subdivision at a half. The two sets always stay clear of each other.
Why is my rectangle inset so much?
Because root 3 is long — 1.732 to 1 is longer than a 35mm frame or a golden rectangle. On a standard sheet or photograph the fitted rectangle is a band across the middle, which is fine deliberately and confusing accidentally.
Is the Hambidge history reliable?
The method is; the archaeology is contested. His 1920 claim that Greek design was built on these proportions was disputed at the time and still is. Composing on a chosen rectangle's own diagonals works regardless of what the Greeks did.
Using it
What does turning the thirds on tell me here?
How far the eyes are from the obvious division. On root 3 the eyes sit at quarters, so the thirds run visibly inside them — the gap is much wider than on root 2, which is the clearest demonstration in the family that the armature is not just thirds under another name.
How do I fit it to a photograph?
Drag and scale so the armature covers the part of the picture you are composing, and accept the letterbox. Root 3 is a long rectangle and almost no camera shoots it, so the overlay will normally sit as a band inside your frame rather than filling it.
Export and print
What should I export to set out a panel at this proportion?
SVG, so the diagonals reach exact corners at whatever size you open it. Root 3 panels come up in six-fold and hexagonal work, where the proportion has to be right for the geometry above it to close — and a raster's rounding is enough to lose that.
Can I print the armature alone?
Yes. Leave the image out and export the grid only: you get the rectangle, both diagonals, the four reciprocals and the four eyes on a transparent ground, at up to 600 DPI on paper from A5 to A2.
Choosing a grid
When should I choose it?
When the surface is yours to cut — panel work, banners, anything made to proportion — or when your subject already has six-fold or three-fold structure. In the second case the √3 is already in your drawing and matching the frame to it removes a disagreement.
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.
Where does root 3 actually turn up?
In hexagonal geometry, mostly — the long diagonal of a regular hexagon is √3 times its short one, which is why the proportion follows the hex grid maker and the vesica piscis generator around. It is much rarer as a picture format than as a construction result.
Can I use it in a class on dynamic symmetry?
Yes, and running root 2, 3, 4 and 5 in sequence is the lesson — the eyes move from thirds to quarters to fifths to sixths in a pattern nobody believes until they see it. Nothing is metered and no account is needed.
Open the tool Root 3 rectangle generator
The hexagon's number, as a rectangle
Quarters rather than thirds, three panels rather than two. Export vector or 600 DPI. No signup, nothing uploaded.
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