Fairy Tale Architecture: Flatland

The brother-sister duo of writer Kate Bernheimer and architect Andrew Bernheimer curate a series in which diverse architects explore the intimate relationship between the domestic structure of fairy tales and the imaginative realm of architecture.

Flatland, as designed by Ultramoderne
The characters of Flatland in the act of extrusion. [All images by Ultramoderne]

Fairy tales have transfixed readers for thousands of years, and for many reasons; one of the most compelling is the promise of a magical home. How many architects, young and old, have been inspired by a hero or heroine who must imagine new realms and new spaces — new ways of being in this strange world? Houses in fairy tales are never just houses; they always contain secrets and dreams. This project presents a new path of inquiry, a new line of flight into architecture as a fantastic, literary realm of becoming.

— Kate Bernheimer & Andrew Bernheimer


The 19th century novel Flatland: A Romance of Many Dimensions, by Edwin A. Abbott, is a science-fiction satire set in a hierarchical society. Widely taught in math classes, it is a fairy tale about reality as much as it is a lesson in geometry and physics.

The main character is a Square living in a two-dimensional world, where the more lines you have and the larger your angles, the more powerful your social position. Hexagons are nobles, and Circles are priests. Lowly Isosceles Triangles aspire to be Equilaterals. Women, at the bottom of the caste system, have only one dimension. They are sharp line segments, treacherous and stuck swishing about to make themselves known. But Flatland is not the only universe; there’s also Spaceland, where the Spheres live, and Pointland and Lineland, which may exist only in Square’s dreams. When I try to summarize the plot, I get it mixed up with Philip K. Dick’s Do Androids Dream of Electric Sheep? — so many wheels within wheels turning around the question of what it is to be human, what it is to experience time. I remember how I felt as a young child in math class. (“But why can’t seven go into two? Maybe seven is mean, and wasn’t invited…”)

Flatland, as designed by Ultramoderne
Basic forms of Flatland characters.

The author addresses the book to those readers “who decline to say on the one hand, ‘This can never be,’ and on the other hand, ‘It must needs be precisely thus, and we know all about it.’” Flatland is about what is impossibly true. Curiosity about the paradox of impossible truth has inspired science, philosophy, literature, art, and design across many ages.

Aaron Forrest and Yasmin Vobis of the architecture firm Ultramoderne initially fretted that their choice of Flatland for this series was “nerdy,” and if “nerdy” means marvelous and intelligent we will grant them that here. They are drawn especially to Abbott’s admission, in the preface to the second edition, that the inhabitants of Flatland do have height but simply cannot conceive of the third dimension (as we cannot imagine more than three). To possess a quality one cannot perceive — that is a real conundrum. Likewise, the architects are fascinated that the Polygons are unable to distinguish between buildings and people, for houses in this world look exactly the same as a certain category of person. Ideas about perception and identity are central both to the novel and to Ultramoderne’s spacious, angular, atmospheric designs.

Flatland, as designed by Ultramoderne
The well-tempered bases of the extruded characters.

Swiss scholar Max Lüthi identifies flächenhaftigkeit as one of the most important techniques in European fairy tales, whose interchangeable plots, characters, and objects exist in a depthless, two-dimensional space. Flächenhaftigkeit is difficult to translate, but the word implies a space of diffuse flatness. It is a kind of narrative grammar, a poetics, which gives fairy tales their mysterious aura. Think of a Mark Rothko painting, its frame filled with abstraction. Picture the sentences that make up a book — their font, their progression — being as depthless as the canvas in Michael Martone’s classic study of the American Midwest, The Flatness and Other Landscapes. Think of Kara Walker’s cutouts, Hans Christian Andersen’s silhouette art. Flatness is a portal to the uncanny. (Even the three-dimensional figures in Madame Tussauds may be described as having flat affect.)

Flatness is a rich concept in a lot of disciplines: quantum physics, geometry, architecture, modern art, and more. When hurled by contemporary book reviewers, the word “flat” alleges a lack of imagination. But when a word can be understood as an insult, it’s a good time to be contrary. Ultramoderne’s homage to Flatland is a beautiful, multidimensional architectural project: an homage to the nerds.

— Kate Bernheimer

Flatland, as designed by Ultramoderne
Aerial view. Click here to see elevation mugshots (20 MB animation).

Three Questions for Ultramoderne from Kate Bernheimer

As you began to brainstorm ideas for designing Flatland, did you draw on any childhood memories, dream images, or particular buildings or people? How was the process of working from fiction different from working with a client?

Flatland is an unusual fairy tale because it is written by a math teacher, and so much of the story is dedicated to explaining the idiosyncrasies of the two-dimensional world in which the story takes place. While you are reading these descriptions, of course, your mind is drifting elsewhere and imagining the infinite strangeness of this world that Abbott invented. We couldn’t help but be reminded of high school math classes where the half of your brain that isn’t busy learning new mathematical concepts begins to busy itself imagining the new worlds made possible by these concepts and their sheer abstraction.

You mentioned Flatland’s idiosyncratic notions of dimensionality and identity. Can you speak to these idiosyncrasies from a design perspective? What problems did they present?

There are so many interesting unanswered questions about Flatland, which exists entirely on a very large sheet of paper. For example, gravity in Flatland works very differently from our own world. There is a constant pull towards the bottom of the sheet of paper, or across the geometrical figures. It is never explained whether this gravity would be strong enough, even at the very bottom of the page, where its pull is strongest, to distort the figures, or even whether shifts in the gravitational field would entail shifts in perception as they do in our own universe. And in Flatland, any deviation from regular geometry is taken as a sign of deviancy and can result in execution of the figure in question, so all of a sudden the question of gravity isn’t just one of abstraction but a very pressing social matter!

Flatland, as designed by Ultramoderne
Character line-up.

We became fascinated by the problems of materiality in a two-dimensional world. Abbott, in his foreword to the second edition, discusses the problem of the “height” of a line drawn on a sheet of paper, and how, even if they do have some thickness, it would be impossible for the inhabitants of Flatland to perceive it. So, in theory, the figures could be infinitely tall and never know it. We decided somewhat arbitrarily that in our version of Flatland the characters would all be 6 feet tall, and we built models of them out of aluminum mesh. The mesh naturally began to fight back. Creating perfect folds of equal angle in them was quite difficult, and even if we managed to get the folds right at one end the other end would always look a bit off. And this doesn’t even broach the issue of creating closed figures out of an open-weave material.

In the end, the imperfection of the models turned out to be an asset. We found the way they shifted in the light and began to lean up against one another in clusters to be beautiful. What would be considered a serious liability in Flatland was of great benefit in our world. So we considered ourselves lucky to be inhabitants of Spaceland where we didn’t have to worry so much about whether each side is formed equally or not.

If your project could be built, where would it be sited?

There is no question that Flatland would be most interesting sited in a metropolis of tall buildings, like New York, Chicago, or Hong Kong. If simple planar triangles, squares, and septagons can have human characteristics and play out human dramas on the surface of a page, why can’t office and apartment towers be characters interacting in the space of the city?

About the Series: Fairy Tale Architecture

Places’ ongoing series on fairy tale architecture is curated by writer Kate Bernheimer and architect Andrew Bernheimer.

Kate and Andrew Bernheimer, “Fairy Tale Architecture: Flatland,” Places Journal, December 2016. Accessed 20 Jan 2017. <>

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