Here’s Looking at Euclid: Helen Friel Folds Byrne’s Proofs into Paper
Paper Engineering · Geometry · Object study
In 1847 Oliver Byrne freed Euclid's proofs from letters and turned them into coloured figures. A hundred and sixty-five years later, the London paper engineer Helen Friel lifted those same figures off the page and folded them into three dimensions.
Ali Kaya
Here’s Looking at Euclid — Helen Friel’s paper models of Oliver Byrne’s coloured proofs.
There is almost no lettering in a Euclid printed in London in 1847. Instead of labelling the angles and edges A, B, C, Oliver Byrne painted every one of them red, blue, and yellow and translated the proof, from start to finish, into a visual language. Open a page and what you see looks less like a textbook than like a Mondrian printed seventy years before Mondrian.
01
Byrne freed the proof from letters
Byrne's claim was not a modest one either. He wrote that Euclid's Elements could be learned this way in less than a third of the usual time, and added that it would stay in the memory far longer. His publisher, William Pickering, paid dearly for that claim. Each colour was printed from a separate plate, so the cost soared; the volume sold for twenty-five shillings while a black-and-white equivalent went for two, and of the thousand copies printed, seventy-five percent stayed on the printer's hands. When Pickering went bankrupt a few years later, this book carried part of the blame.
The book historian Ruari McLean called the same volume one of the oddest and most beautiful books of the whole century. That is often what a commercial disaster looks like from the other side. The full edition — reissued as a facsimile you can still buy — is worth seeing in colour: Oliver Byrne's The First Six Books of the Elements of Euclid. We wrote about the book itself in our note on Byrne's coloured Euclid.
Byrne’s flat planes, folded up: colour that used to mean area now has weight.
02
Friel picked the proof up off the ground
What Byrne did has a curious side effect. The moment you drop the letters and move to coloured planes, the parts of the proof stop reading as abstract labels and start looking like things you could cut out and lift away. Looking at the page, you half feel you could take hold of the red square and turn it. In 2012 the London paper engineer Helen Friel did exactly that.
Here's Looking at Euclid is a set of three-dimensional paper models of Byrne's diagrams. Rather than enlarge and repaint the figures, Friel turned them into templates, worked out the fold allowances, and converted a proof that lay flat into an object that stands on a desk. What comes out is neither a model nor an illustration so much as a physical copy of the proof itself.
Byrne flattened the proof. Friel picked the same proof up off the ground.
A proof that lay flat for two thousand years, now standing on a desk.
03
Why paper works where a diagram doesn’t
It is not hard to see why this works. The classic figure for the Pythagorean theorem is three squares set on the sides of a right triangle, and those squares already mean area. Folded in paper, area stops being a symbol to be read and becomes a surface with weight in your hand. The most maddening thing about the figure drawn on a classroom board is that students look at the squares as pictures. In the paper model there is nothing left to look at, only something to hold.
The Pythagorean figure drawn in Byrne’s colour logic. In the template view the edges become fold lines and glue tabs push out along the sides — the exact move that turns a proof into a paper object.
This is the same instinct behind the very first English Euclid. Three centuries before Friel, Henry Billingsley glued folding paper solids into the pages of his 1570 translation — the story we tell in Paper that stands up. The tool changes; the idea does not. When you hold a shape, you stop translating.
04
A late repair from Byrne’s side
Friel's work counts as a late repair from Byrne's side too. When the book first appeared it was largely ignored and even criticised, and it took until the twentieth century to draw fresh interest — helped in no small part by the room Edward Tufte gave it in _Envisioning Information_and by Taschen's facsimile. Byrne's page turns up in a different medium almost every generation: first in an exhibition case, then in design books, then in a browser, and finally as a template to be cut out and folded.
Every generation gets Byrne back in a new medium. This one you can fold.
05
The name, and the joke inside it
The name is its own matter. Here's Looking at Euclid is a play on that very well-known line from Casablanca, and for exactly that reason the project reads less like a self-serious study of geometry than like something that looks at mathematics with a smile. Given what happened to Byrne's book, a little humour from someone working geometry in paper seems fair enough.
Friel shares her Byrne-inspired Pythagorean theorem template free on her site, cleared for personal use and for use in schools. Printing it on coloured paper and cutting it out is all it takes. Seeing a figure that has been drawn on classroom boards for years finally stand on its own on a desk is another thing entirely.
Printed on coloured paper, cut, and folded — the proof, standing up.
Where to go nextYou can page through the whole of Byrne's 1847 edition in
Nicholas Rougeux's digital version, clicking the shapes in each proof to follow which colour answers to which proposition. The rest of Friel's paper work lives on
her own site, and the colour edition that started it all is
readers loved this