Rotae

The work and its context

Rotae is a 529-edition generative art series by Nadieh Bremer, released on Art Blocks on Ethereum mainnet, contract 0xa7d8d9ef8d8ce8992df33d8b8cf4aebabd5bd270, project ID 194.¹ Each output is the closed path traced by the tip of a chain of wheels rotating on wheels: the registry description says the series "explores the vast array of beautiful and unexpected shapes that can appear from the simple process of letting wheels rotate on (many more) wheels", chaining "anywhere from three to eight wheels", and it promises that a few seconds after the static shape appears "you can watch how the line gets drawn, possibly revealing unexpected shapes hidden within."¹ The title is Latin for "wheels", and on her own project page Bremer explains it as a nod to the paper that describes the math of achieving a chosen m-fold symmetry no matter how many circles are chained together.²

Bremer is a Dutch data visualization designer and artist, trained in astronomy, founder and owner of the studio Visual Cinnamon, whose practice the AB5D dossier summarizes as bringing "the investigative habits of data visualization into generative art."³⁴ Rotae is her only project in AB[500]; the dossier reads it as a place where "circular systems become animated fields whose clarity never suppresses their capacity for surprise."³ Smashing Magazine called the collection "the grown-up sibling of the good ol' Spirograph fascination."⁵

The series runs as library-free JavaScript, script type js@na, at a square aspect ratio with zero external assets, and it carries the CC BY-NC-SA 4.0 licence, the only share-alike licence in this batch.¹⁶

Conception and development

The conception is a childhood toy carried past its physical limits. Bremer's own account, published on her Visual Cinnamon site on October 20, 2021, three weeks before release, traces the line directly: she loved Spirograph kits as a kid and hated how one slipped pen ruined a drawing; in 2016 she built a web page to draw perfect coded spirographs; in March 2020 she bought an AxiDraw pen plotter and returned to the forms physically; and wanting to understand the symmetry math she "came across the paper by Frank A. Farris about 'Wheels on Wheels on Wheels'" and adjusted its math into her own code.² When she discovered Art Blocks in July 2021 she knew at once what her project would be, and during July and August 2021 she extended the plotter code into Rotae.²

One design decision deserves emphasis: she first tried to mimic the pen-plotter look digitally, "but then made digitally" gave way, on advice, to embracing what a plotter cannot do: gradients, vibrant poppy colors, changing line thickness, different kinds of lines.² The released series shows that pivot everywhere, from translucent rainbow mandalas to flat graphic ribbons.⁷ The description's care constraint, paths that "won't turn into chaos, but an unpredictable, yet fully symmetric path that closes on itself", is the Farris construction working as intended: symmetry is chosen first, and radii and frequencies are then solved backwards so the curve closes quickly instead of orbiting hundreds of times.¹²

Bremer also published the rarest design statistics herself: 142 Rotaes (27%) carry the "sparkle" feature of tiny circles along the line; backgrounds divide into even (59 works, 11%), gradient (279, 53%), dots (160, 30%, which become stars and nebulae in the space palette), and pop (30, 6%, only for the space, sunset and dark rainbow palettes); palettes include space, sunset, fog, taxi cab, hot/cold, antique, doogle, one hue, viridis and 80's pastel.² The registry interview field is empty, but this self-published making-of is richer than most interviews.¹

How the work was constructed

The full artist script was retrieved from the official indexed project record at the Art Blocks mainnet GraphQL endpoint, filtered by the exact slug rotae-by-nadieh-bremer and corroborated against contract 0xa7d8d9ef8d8ce8992df33d8b8cf4aebabd5bd270 and project ID 194. The retrieved script is 11,808 bytes with SHA-256 digest fc6b0ee77c0cc4a49955fb35449288f1f0e5f7002bd0139c4aa7cc6cc6d604a3 (recorded in evidence.json; the script itself is not redistributed). The indexed runtime is js@na: compact minified JavaScript with no library at all, writing straight to a canvas 2D context.¹⁸

The randomness is a hand-written xorshift128 generator seeded with four 32-bit words parsed from hexadecimal characters 2 through 33 of the chain hash.⁸ Parameter generation follows the published recipe exactly. The wheel count rolls an integer from three to eight; the symmetry order rolls two to nine; and each wheel's frequency is assembled from a sign, a random multiple of the symmetry order, and an offset drawn from a table of coprime sets, one row per symmetry, with re-rolls whenever a frequency collides with a previous wheel's.⁸ That coprime table, [[1],[1,2],[1,3],[1,2,3,4],[1,5],[1,2,3,4,5,6],[1,3,5,7],[1,2,4,5,7,8]] indexed by symmetry, is the Farris result in code form: it is what guarantees the closed, quickly-closing, fully symmetric path the description promises.¹⁸² Radii roll between 0.1 and 1.0 and are then normalized against a shared budget so no single wheel dominates.⁸

Color is unusually serious for this batch: the script implements full OKLab and CIELAB conversion matrices in inline constants, interpolating gradients in perceptually uniform space rather than in raw RGB.⁸ Style rolls then choose line treatments, palettes filtered per style family, and the sparkle and background features Bremer documented.⁸² Rendering is two-phase, exactly as described: the whole path is drawn at once, then after a five-second timeout an interval timer redraws the path point by point over about twenty seconds, revealing how the curve is traced.⁸ The live renderer confirmed all of it for token 194000267: the reported hash matched the chain record, the complete shape was already present four seconds after load, the frame differed at ten and twenty-five seconds as the redraw progressed, the animation had settled into byte-identical frames by fifty seconds, and no audio elements exist in the page.⁹ There are no interaction handlers beyond a window-resize listener.⁸

The mint

The dated record, observed in the current Art Blocks index on September 18, 2026, spans about ten weeks.¹¹⁰ Edition 0 (token 194000000) was minted on October 23, 2021 at 05:57:59 UTC, thirteen days before the project was activated on November 5 at 20:59:39 UTC, a pre-activation artist mint; Bremer's own site cites this token's hash, 0xd6eb1c46...c679, as her worked example, and it matches the chain record inspected here.¹¹⁰² The registry's sale-start timestamp reads November 9, 2021 at 17:00:00 UTC, the official collection page carries "Released Nov 9, 2021, 5:00 PM UTC", and the second token followed eight minutes and ten seconds later at 17:08:10 UTC.¹¹¹¹⁰

At 529 editions this is the largest series in the batch, and it sold like a mid-size Factory release of its moment: 363 editions minted in November 2021, 98 in December, and the final 67 in January 2022, closing on January 11, 2022 at 12:19:46 UTC.¹⁰ The minted count equals the maximum of 529, and the project is marked complete, locked and active, with the paused flag false.¹ The currently indexed price is 0.1 ETH per token; no dated 2021 source inspected here states the original mint price, so that figure is reported as the present registry value and the launch price remains unverified.¹ The AB5D collection's edition 267 was minted twelve days into the public sale, on November 21, 2021.¹⁰¹²

Copyright and licensing

The project registry reports the artwork licence label "CC BY-NC-SA 4.0" for Rotae, with no linked terms URL in the record.¹ This essay therefore categorizes the artwork licence as creative-commons, variant CC BY-NC-SA 4.0, the only share-alike licence in this batch.¹ The CC BY-NC-SA 4.0 deed grants the freedoms to Share, copy and redistribute the material in any medium or format, and to Adapt, remix, transform and build upon the material, under three conditions: Attribution, giving appropriate credit with a link to the licence and an indication of changes; NonCommercial, no use for commercial purposes; and ShareAlike, contributions based on the material must be distributed under the same licence as the original. The deed also forbids applying additional legal or technological restrictions.⁶

The ShareAlike condition is the substantive difference from the CC BY-NC 4.0 and CC BY-NC-ND 4.0 licences carried by peers in this batch: a collector or artist may publish remixes of a Rotae, but only by licensing the remix identically, which keeps derivatives inside the same non-commercial commons.¹⁶ That choice sits comfortably with a project whose own making-of page openly teaches its mathematics and even walks readers through extracting a token's hash.² The script carries no licence header or copyright comment of its own, so no code licence is reported for this package; the absence is recorded rather than resolved.⁸ The work declares no third-party runtime dependency: the registry lists zero external assets and the script is library-free JavaScript.¹⁸ Owning a Rotae token conveys no copyright by itself, and no conflicting licence statements were found in the inspected sources. Images in this essay are referenced through canonical Art Blocks token pages and media rather than republished.¹⁰

Close reading of examples

Five outputs were inspected directly through their platform media, and the header was additionally exercised in the live renderer.⁹⁷

Edition 267 (token 194000267, minted November 21, 2021), the AB5D-held header, is the translucent-ribbon family at its most floral: on a pale off-white ground, an eight-petal flower built from overlapping gradient strokes in burnt orange, brick red and teal, the overlaps darkening like layered watercolor where the path crosses itself near the hub. In the live view the complete flower appears at once, then after the promised few seconds the line begins to retrace itself, and the wheel logic becomes visible as the nib sweeps petal after petal; the redraw finished and the frame settled still within the minute.⁹⁷

Edition 0 (token 194000000, the pre-activation artist mint of October 23, 2021) is the collection's graphic opposite and its own citation: Bremer's site quotes this exact hash as her teaching example.¹⁰² It lands the taxi cab palette she documented: a saturated yellow ground with bold flat ribbons in solid black and white, no transparency, no gradient, a poster-like starburst that shows how much range the style rolls carry.²⁷

Edition 1 (token 194000001, the first public mint, eight minutes after the sale opened) shows the minimal line style: a single pale cyan-white ribbon on a bright cyan ground, weaving a four-pointed star with a small woven knot at its center.¹⁰⁷ Edition 100 (token 194000100) is the dark-rainbow mandala family: on deep navy, translucent strokes in orange, pink, teal and violet build an eight-pointed star whose crossings bloom into soft secondary colors.⁷ Edition 300 (token 194000300) shows the wide-ribbon style on a hot pink ground, cream and pink bands looping into a flower, the same wheel grammar at poster scale.⁷

Significance and limitations

Rotae is the batch's most thoroughly documented project: the artist published the mathematics (Farris's wheels-on-wheels symmetry construction), the development history from childhood Spirograph to pen plotter to Art Blocks, the design statistics for every major feature, and even a worked hash example, and every checkable claim in that account verified against the script and the chain record inspected here.⁸¹⁰² It is also a case study in medium translation: a plotter aesthetic deliberately abandoned for digital-native color, and a Fourier-flavored machine presented as a toy, with the artist herself noting that more than two wheels are no longer spirographs and quoting Piter Pasma's joking name for the family, "Polyepihyperderpflardioids."² Within the batch it carries the only share-alike licence, matching a project whose author openly teaches its own method.¹⁶

Unresolved research questions, flagged for review:

• The original mint price at the 2021 sale is not established by any dated 2021 source inspected here; only the current registry value of 0.1 ETH is recorded.¹
• No registry interview exists for this project (artist_interview null); the artist's own project page served as the primary conception source instead.¹²
• The palette percentages Bremer publishes sum across only the palettes she lists; the complete palette table was not independently recomputed from the script for all 529 outputs.²
• The live probe reported a canvas backing store of 300 pixels at read time in the headless environment, smaller than the expected viewport-scaled size; the visual output was unaffected, but the discrepancy is recorded rather than explained.⁹