The Squircle — Impure Geometry and the Ideology of Interface

The form

A superellipse is the curve |x/a|ⁿ + |y/b|ⁿ = 1. With n = 2 and a = b it is a circle. As n grows the curve fills out toward a square or rectangle. The squircle is the case n = 4, a = b, also called Lamé's special quartic: straight-looking sides joined by broad rounded corners.

Gabriel Lamé generalised the equation of the ellipse in this way in the early nineteenth century. Hermann Zapf's typeface Melior (1952) used superellipses for round letters. The Danish poet and scientist Piet Hein gave the curve its name. His winning design for the roundabout at Sergels Torg in Stockholm (1959) used a superellipse with n = 2.5 and an aspect ratio of 6 : 5, an oval with flattened sides rather than a squircle. Apple's app icons approximate a squircle; Wikipedia describes the shape as "actually a quintic superellipse", n = 5 (Superellipse, Squircle).

What the artist says

In the interview "The Squircle of Life" (Spike #70, 2021/22), Tina Rivers Ryan set Malevich's square, a refusal of representation that heralds "art's supposed autonomy", against digital art's "embeddedness … within social and technical systems", and noted that his squares recall "the default format of Instagram posts". He answered with the squircle (sources/ingested/spike-70-harm-van-den-dorpel.md; → Spike #70):

"A very important element is the so-called squircle, which is a square circle. Especially on the Mac, the user interfaces have these rounded edges, which are supposed to give it a human feel. Actually, they're horrible. They look like 60s wallpaper. But anyway, the squircle is considered impure. Mathematicians hate it. And yet, this is the primary shape that defines the aesthetics of the interfaces of our era. For me, to intervene in contemporary life means to operate on the level of user interface design, because that's where we spend most of our time now when we consume culture."

His text for Markov's Dream (2022) gives the history behind the shape (Memories and Memorylessness). Programs used to be tools chained together on the command line; graphical systems turned them into platforms "owned by corporations", which "embrace digital enclosure, rather than interoperability". Then: "To help users feel comfortable in these limiting interfaces, UX designers gradually began to give all square-shaped user interface elements (e.g. windows, buttons, input fields) rounded edges, shadows, and gradient colours. … This is all in an effort to make the UX more metaphorically connected with human bodies, which have no hard edges. Computers become platforms for humans who become tools in the profit sequestration chain."

Mutant Garden and Markov's Dream

Mutant Garden (2019–) is the work the artist was discussing in Spike. Its compositions are layers of rectangles and rounded squares placed inside one another; Rivers Ryan describes them as "capsule-like, semi-transparent squircles that float over each other". The programs that draw them come from Cartesian genetic programming (→ Evolutionary Logic).

Markov's Dream (IV): rounded rectangles in pink, orange and grey nested in a large grey-gradient squircle — *Markov's Dream (IV)* (2022), token #12

Markov's Dream answers the rounding directly. "'Markov's Dream' is a response to this phenomenon. The pure Platonic forms of 'Markov's Window' have been replaced by impure circles or rounded squares in the dream space." Token #12, Markov's Dream (IV), nests rounded rectangles in pink, orange and grey inside a large squircle shaded from white to black. harm.work calls it a remake of Markov's Window "with altered palette"; token #1 keeps the 2004 palette (→ The Markov's Dream Palette). The recursive subdivision of 2004 is still there, now carried out in the shape the about text describes as the platform's way of seeming human.

On photographic aluminium

Blush Array: a large orange rounded rectangle with a smaller rounded square nested in its lower half and rows of small pink and purple rectangles above — *Blush Array* (2022), *Mutant Garden*; light exposure on photographic aluminium paper, CNC routed engraving on top surface, 80 × 112.9 cm

A large orange rounded rectangle contains a smaller rounded square in its lower half and rows of small pink and purple rectangles above. In 2022 the artist exposed Mutant Garden outputs onto photographic aluminium paper: Blush Array, Blush Array II (100 × 130 cm) and Modular Mind (107 × 130 cm), each in an edition of 1 + 1 AP. Blush Array II and Modular Mind were shown in Context Switch, his two-person exhibition with Achraf Touloub at Galerie Noah Klink, Berlin (12 November 2022 – 29 January 2023). In Artforum Adina Glickstein wrote that he "makes software that gives way to colorful sets of nested squircles", with "soft beveled edges recalling the gentle contours of a user interface" (→ Context Switch, Artforum). The exposure fixes a software output on paper.

The wall objects of 2017–2018

Lammer Asbestos: an orange-to-grey rounded square printed on a square acrylic panel bolted to the wall — *Lammer Asbestos* (2018), *Nested Exchange*; wall object, 100 × 100 × 4 cm

Before Mutant Garden, works from Death Imitates Language (2016–) and Nested Exchange (2017–2019) were printed on layered acrylic panels of 100 × 100 cm and bolted to the wall. Lammer Asbestos prints a rounded square, shading from orange to grey, on the square panel; two faint diamond lattices show through the layers. Not every panel shows a squircle. Juancar Zolim Juancar (2017) prints a circle with a cusp at the top, and Vvgdamn Pipikaka Yozdczmi (2017) keeps a sheet of packaging foil between its acrylic layers (harm.work). The artist's talk of the squircle comes later, with Mutant Garden; these panels show the square container and the rounded figure before he named the shape.

White gallery with a square acrylic panel bolted to the right wall and a portrait screen standing on the floor at left — *Juancar Zolim Juancar* (2017), *Death Imitates Language*, installation view

The panel hangs on the right wall of a white gallery with a skylit roof; on the floor at left a portrait screen shows a circle composition.

Agent's reading: presented as natural, derived by rule

Thesis. An earlier version of this page argued that sacred geometry, like the squircle, presents a constructed form as natural and so conceals the interests it serves.

Antithesis. The artist's own Senescenence text places his work in that tradition: "Within traditions of sacred geometry, structure and spirit are intertwined rather than opposed. Van den Dorpel approaches the algorithm with this mindset" (harm.work; → Sacred Geometry).

Synthesis. The objection in the about text is to a form presented as natural: the rounded corner as "metaphorically connected with human bodies", hiding the platform behind it. What the practice takes from sacred geometry, from the loom and from Kunz's pendulum points is a form derived by rules that stay visible in the result. The difference lies in whether the rule shows. Markov's Dream takes the interface's rounded square and puts it back under a visible rule, the recursive subdivision of 2004 (→ Process Legibility).