The Lozenge — Modernist Tilted Geometry

A lozenge is a square turned onto its corner, so that its corners point up, down, left and right. Rosalind Krauss uses Mondrian's lozenge canvases as her clearest example of the centrifugal grid, the grid that presents the picture as a fragment of a field running on past the frame: "the vertical and horizontal grids seen within diamond-shaped canvases—the contrast between frame and grid enforcing the sense of fragmentation, as though we were looking at a landscape through a window" (sources/ingested/krauss-grids-1979.md; → The Grid).

Mondrian and van Doesburg

Piet Mondrian first used the lozenge format in 1918 and returned to it until the 1940s, for instance in Composition with Yellow Lines (1933). Neoplasticism admitted only horizontal and vertical lines. In 1924 Theo van Doesburg brought the diagonal into his counter-compositions, the programme he called Elementarism, and the usual account makes the diagonal the cause of the break between the two painters. Mondrian's neo-plastic lozenges of 1924–26 are read as his answer: the canvas is turned 45°, so the frame supplies the diagonal while every line inside it stays horizontal or vertical. Van Doesburg turned the lines; Mondrian turned the frame.

Jan Dibbets took the turned and distorted square into photography. In the Perspective Corrections (1967–69) shapes laid out on the ground are drawn so that, photographed from one position, they appear as true squares in the picture plane. Collage (1973, Tate) is made of colour photographs on cards with graphite on paper. In van den Dorpel's archive the name appears once, as a file name. Note 1035 lists "UNTITLED (DUTCH FRACTAL), 2005, computer animation (infinite)", and its image, dibbets.png, is a turned square subdivided in the blue, green, magenta and grey gradients of Markov's Window.

Markov's Window (2004)

Markov's Window: a square on its point subdivided into grey gradient fields and green, blue, magenta, black and white squares, many turned 45° — *Markov's Window* (2004), software

The program starts from one square that fills the screen, and the square "can split into multiple segments, it might change its colour, or it might rotate 45°". The work text names Mondriaan as its comparison: "Whereas Mondriaan might have made many studies and versions to reach one single best possible composition, this work instead embraces perpetual change" (harm.work). The 45° turn happens inside the frame, to single squares, which is van Doesburg's device rather than Mondrian's. In the still above, green and white squares stand upright beside squares turned on their points, and gradient squares sit turned inside upright cells.

Markov's Window on a monitor hung on its point, beside a lozenge painting with two black lines, a square canvas with a red square turned 45°, and a small De Stijl composition — *Markov's Window* (2004), installation view, Stedelijk Museum Amsterdam

In the museum the whole work is turned. The Stedelijk hangs the monitor on its point, in the 2020 hang beside works by Malevich (→ The Markov's Dream Palette) and in the hang shown here, which harm.work files with Stedelijk Base and which van den Dorpel put among the first blocks of his Our Inner Child research channel on 21 July 2023 (block). The monitor hangs beside a Mondrian lozenge with two black lines and a square canvas whose red square is turned 45°, van Doesburg's device. Installed, Markov's Window makes both moves: the frame is turned as in Mondrian, the squares inside are turned as in van Doesburg.

Untitled Nested Composition 2 (2018)

Square plotter drawing set on its corner: nested squares, circles and rounded rectangles in black and blue pen, a few cells coloured red, brown and yellow — *Untitled Nested Composition 2* (2018), hand-coloured plotter drawing on paper, 27 × 27 cm — are.na, 19 August 2018

On 19 August 2018 the artist posted the drawing to his are.na channel Drawings, captioned "2018 / hand coloured plotter drawing on paper / 27 × 27 cm". The square sheet is shown on its corner. The plotter divided it into squares parallel to its edges and divided some of them again, down to a cluster of small quartered squares at the left corner. Several cells contain a heavy black square upright to the room, with a smaller tilted square inside it. Upright squares sit inside the lozenge and tilted squares inside the upright ones, so the two arrangements are nested in one drawing. Circles and rounded rectangles are drawn in blue pen, and a few cells are coloured by hand in red, brown and yellow pencil.

The Resolution Paintings (2023)

For Our Inner Child (Upstream Gallery, Amsterdam, 2023) the artist "used the same algorithm that was involved in producing Markov's Window in 2004" (exhibition text). harm.work lists Synthesis Euphoria, Hell and Nacre (150 × 150 cm) and Rite (120 × 120 cm) with the note "Dimensions are measured diagonally": they hang on their points. An installation view the artist filed on 19 November 2023 shows four paintings hung as lozenges along one wall and beside the tall window (block). Here the whole frame is turned 45° on the wall, as Mondrian turned the canvas, while the subdivisions run parallel to each panel's edges and so diagonal to the wall.

His research channel for the show records the turn. From late September 2023 the working files end in _rotated (267509753_rotated, 27 September), each a square output turned 45° on a black ground. A render of 1 October, saved as cherry-blossom, sets an upright square inside the turned frame. On 5 October a mock-up saved as venster-hell-impression puts Hell on its point on a gallery wall beside a tall window.

Four square paintings hung on their corners as lozenges in a white room with a wooden floor and a tall window — Installation view, *Our Inner Child*, Upstream Gallery (2023)

Pink and violet lozenge on black with a white border; an upright square of subdivided pink cells in the centre, smaller lozenges at the four corners — *cherry-blossom*, working render; 1 Oct. 2023

Mock-up of a red, orange and black square work hung on its point on a white wall, a tall street window to the left — *venster-hell-impression*, mock-up of *Hell*; 5 Oct. 2023

Venster (2024)

Venster is "A new version of Markov's Window from 2004, but now fully on-chain": 100 "generative animated recursive SVG files, calculated by Solidity code", released with Bright Moments (harm.work). Venster is Dutch for window, and also the Dutch word for a window on a computer screen.

Two tall leaded-glass windows with red, blue, yellow and white panes and red diagonal bars — Theo van Doesburg, stained-glass windows (building and date not given) — are.na, 24 July 2023

The research channel for Our Inner Child shows which windows he was looking at while the name took shape. It opens on 21 July 2023 with van Doesburg's Composition with window with coloured glass III (block); three days later comes a photograph of a pair of van Doesburg stained-glass windows. On 19 November 2023 three blocks arrive together: the Markov's Window 2023 remake (block), a glass panel after Frank Lloyd Wright's Saguaro design (block) and a file named Venster.gif. The name was in use by November 2023, before the 2024 release.

Venster #1 — seed 78448782, lozenge root, optimistic palette — *Venster #1* (2024), seed 78448782; shape: lozenge, palette: optimistic

In Venster the lozenge is a trait. Every composition grows from one of three roots, lozenge, cover or diagonal split, and on venster.harm.work the viewer can filter the seed space by root, by feel (classic, optimistic, melancholic) and by complexity. In Markov's Window a turned square was a chance event inside the chain; in Venster it is a class the viewer can ask for. Because each composition is computed by Solidity and rendered by the viewer's browser, the same score looks different in Chrome, Firefox and Safari, and the artist decided "to deem the varying SVG interpretations a feature of the artwork" (→ Venster — the writing).

Endless Knot (2025)

Detail of Endless Knot: fields of stepped zigzag lines in pink, blue, maroon, orange and black running on the diagonal — *Endless Knot* (2025), Sakura fineliner plotter drawing on Hahnemühle watercolour paper, 59.4 × 84.1 cm, detail

The traditional endless knot, one of the eight auspicious symbols of Tibetan Buddhism, is drawn as a lattice turned on the diagonal, so the drawing is a lozenge as well. The detail shows fields of stepped zigzag lines in pink, blue, maroon, orange and black, all running on the diagonal of the sheet. The work and its iconography are discussed on Recursion and Self-Reference.

Agent's reading: centrifugal frame, centripetal subdivision

Thesis. An earlier version of this page called the lozenge recursive because its frame implies a composition that continues beyond it.

Antithesis. Note 876 defines recursion as a structure whose definition refers to itself, and adds that "the demand for self referentiality — immediately also forbids reference to anything outside of itself." A frame that points past itself is the opposite.

Synthesis. Krauss's pair covers both. The turned frame is centrifugal: in Mondrian's lozenges it makes the picture "a mere fragment, a tiny piece arbitrarily cropped from an infinitely larger fabric", a view through a window. The subdivision inside Markov's Window, the Resolution Paintings and Venster is centripetal: each square is cut into squares by the same rule, a repetition that refers only to itself, which is the recursion of 876. Hung on its point, Markov's Window is both at once. Krauss notes that artists "from Mondrian, to Albers, to Kelly, to LeWitt" think the grid "in both ways at once". The higher concept is the frame that looks outward around a subdivision that repeats inward.