What this tool does
Regular pentagons cannot tile a plane, so pentagon graph paper means the Cairo tiling: a pattern of identical five-sided cells that meet in the basket-weave you see on Cairo's pavements. Choose the size, the line colour and weight, and an optional border.
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Octagon Graph Paper
Regular octagons with small squares at the corners, the truncated square tiling, in five sizes and three colours.
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Why it is not made of regular pentagons
A regular pentagon has 108 degree corners, and three of them around a point leave a gap while four overlap; that is why no page of regular pentagons exists. The Cairo tiling uses a pentagon with two right angles and three larger ones, and it tiles perfectly. Every cell is the same shape, they meet in rows that run at right angles to each other, and the effect is the paving that gives the tiling its name.
How it is drawn
The generator builds the tiling from its skeleton: a short bar at every lattice point, horizontal and vertical on a checkerboard, and four spokes from each cell centre to the nearest ends of the four bars around it. Every face that produces is a pentagon, with the two right angles at the spoke ends. The size setting is half the lattice spacing, so a 12 mm sheet has pentagons about 12 mm tall.
Uses
Tessellation and geometry lessons, where the point is to see that the shape works and to work out its angles. Quilting and mosaic patterns. Games that want a cell with five neighbours. And drawing: the Cairo tiling shades beautifully in two or three colours along its rows.
Settings shared with the octagon page
Five sizes from 10 to 25 mm, grey, blue or green lines, three weights and an optional border. The same generator draws octagon graph paper; switch between the two with the Tiling dropdown.
FAQs
Quick answers
Are these regular pentagons?
No; regular pentagons cannot tile a plane. These are Cairo pentagons, with two right angles, which do.
What does the size mean?
Roughly the height of one pentagon: 10, 12, 15, 20 or 25 mm.
What is it used for?
Tessellation lessons, pattern and quilt design, game maps that want five-sided cells, and colouring. It is also the quickest way to explain why the tiling works: count the angles at each corner.
Can I make the lines fainter?
Light weight in grey is the faintest; it is the default.
Is there an octagon version?
Yes, the octagon graph paper page uses the same generator with the truncated square tiling.
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