Two Mathematicians

The work and its context

Two Mathematicians makes a repeated pattern feel simultaneously inevitable and contingent. A grid gives each local event its place, but the line that describes that event may be exact, wavering, densely hatched or almost absent. In the five works examined here, the same field of attention moves between large empty cells, intricately worked junctions and a nearly continuous surface of strokes. The viewer can recognise order before understanding how it has been made. ^{1 2 3 4 5}

The project appeared through Art Blocks Factory in February 2022 under the credit BY MA. Its artist statement identifies Islamic geometric patterns as the subject and contrasts precise construction with the appearance of an informal draft. The current Art Blocks artist profile and the BYMA website identify Armaghan Fatemi, whose artist dossier situates the project within a wider practice. The legacy project credit is retained here rather than silently replaced by the current profile name. ^{6 7}

This is an interactive drawing system rather than an autonomous animation. The finished pattern remains still until the viewer changes the view. Zoom reveals the character of the marks; the “i” key adds construction lines and, for the historical pattern families that carry one, a site label. Those controls turn the picture into something that can be inspected at more than one level. AB5D holds #106, a blue drawing whose dense hatching makes pale paths emerge through the field. ^{8 5}

Conception and development

The project's public repository, linked directly from the BYMA website, explains two approaches. One reconstructs particular existing patterns and moves selected points symmetrically to create related designs. The other constructs a pattern within a polygonal layout by intersecting lines drawn from its sides. The repository names Eric Broug's Islamic Geometric Patterns as a reference for the first approach and Craig Kaplan's research for the second. These acknowledgments establish a documented technical lineage; Broug's book was not independently inspected for this essay. ^{7 9}

Kaplan's Computer Generated Islamic Star Patterns describes a procedure using a tiling to organise repeated motifs, together with software for producing and rendering them. It discusses Hankin's earlier work as one part of that construction tradition. The relevance to Two Mathematicians lies in the separation between the geometry of a pattern and its rendered appearance. A network can be drawn as delicate lines, broad interlacing bands or textured regions without becoming a different underlying construction. ¹⁰

Historical reconstruction should not be mistaken for proof of how every original artisan worked. The project's statement gives a specific early date for Hankin, while the bibliography of the linked Kaplan paper cites a 1925 publication. This research did not establish the original paper intended by the artist's date. The essay therefore identifies the method and its documented route through Kaplan without relying on that unresolved bibliographic detail. ^{6 10}

The artists' stated contrast between exactness and informality is implemented as two rendering archetypes. The public features call them Sober and Tipsy. Those names describe treatments applied within a carefully bounded system. The irregular marks still obey geometry and seeded choices. Their looseness is composed into the drawing procedure, so the work can evoke a draft without becoming an unfinished version of a more correct image. ^{6 8}

How the work was constructed

The complete indexed program contains 37,584 bytes and uses p5.js 1.0.0. It reads the last seven hexadecimal characters of the token hash as a seed. A compact generator advances that seed through integer multiplication and a bit mask, producing values used by the configuration and drawing routines. Weighted choices select a pattern family, rendering treatment, colours, fills and line styles. These weights describe the program's possibilities, not audited frequencies across the 300 minted works. ⁸

Five construction functions carry historical site names: Córdoba, the Cappella Palatina, Kharraqan, Ibn Tulun and the Abd al-Samad complex. They contain coordinate sets, line connections and rules for moving selected points. Reflections and rotations complete related regions around the initial construction. The labels document the program's associations; this inspection does not independently authenticate each historical attribution or the dates embedded in the code. ⁸

The other two families begin with square or hexagonal templates. Lines extend from side midpoints at a chosen angle, and intersections determine vertices in the resulting pattern. Alternative connection modes change which neighbouring sides participate. A small change in angle can therefore alter several related shapes together while preserving the rule that joins one tile to the next. The program supplies ranges for these angle choices, constraining the transformations before rendering begins. ⁸

Repetition is organised separately. A tiling routine determines scale and placement from the viewing dimensions and the desired number of units. Some rows or columns are offset for hexagonal arrangements. Additional fill tables decide which regions receive one treatment and which receive another. The result is more varied than stamping one decorated polygon across the screen: the geometry repeats, but the distribution of filled and unfilled regions can have a larger rhythm. ⁸

Line expansion gives the network width. The script calculates offsets around connected edges and constructs polygons for the resulting bands. Where interlacing is enabled, selected connections are adjusted to produce crossings. Background-coloured passages and differently drawn edges can make a network appear to pass over and under itself. This is a geometric drawing convention, not a simulation of woven material. ⁸

The surface treatments include dots, lines, dashed lines, solid regions and nested outlines. Crisp rendering can also use waves and crosses. In the rough mode, straight segments become paired Bézier strokes with displaced control points and bowed paths. Dots can become small irregular loops. The repository explicitly credits Rough.js as a basis for the hatching machinery. These transformations explain why a precise repeated network can look as though it has been drawn with a hand that never quite repeats the same gesture. ^{8 11}

Colour remains restrained. The program defines a small set of foreground and ground combinations, mostly coloured marks on a light beige field, with several alternatives on a dark ground. A separate overlay colour distinguishes the construction grid. The limits make changes in stroke and fill conspicuous: a blue field of hatching and a black field of nested outlines are not merely alternative palettes for an otherwise identical visual experience. ^{8 2 5}

Rendering occurs in off-screen buffers before the visible view is composed. The default drawing is static. Zoom interpolates between scales over a short interval, using the already generated image rather than choosing another pattern. The “i” control adds the tiling overlay and available label. On #106, both the additional dashed grid and enlargement after repeated up-arrow presses were visually verified. No audio implementation was found. The canvas follows the viewing window's dimensions, so the shape of the viewing area also affects how much of the pattern is shown. ^{8 5}

The mint

The current index records 300 minted works and a maximum of 300. The first token is dated 26 January 2022 at 10:25:29, before the public release listed by Art Blocks on 8 February. Activation is recorded on 3 February. The scheduled start is 17:00:00 on 8 February, and the second mint is recorded twenty-three seconds later. The early token, activation and scheduled release are distinct events. Full timestamps and offsets remain in the source record. ⁶

Completion is recorded on 5 July 2022 at 00:40:04. The present price setting is 0.14 ETH, with a fixed-price sales note. Neither that setting nor the current maximum independently reconstructs the original announcement or subsequent configuration changes. This essay has not separated administrative, artist and public transactions, and does not convert the time between index fields into a verified public sellout duration. ⁶

Copyright and licensing

The current artwork licence label is CC BY-NC-SA 4.0. The project record supplies no separate terms link, so the corresponding Creative Commons deed is linked independently. It permits sharing and adaptation with attribution, a noncommercial restriction and the share-alike condition for adaptations. This is the label observed in the current record; a complete history of earlier artwork terms has not been reconstructed. ^{6 12}

The artist-linked source repository declares ISC in its package metadata. That is meaningful evidence and should not be collapsed into a claim that no software licence exists. However, no separate full licence file appeared in the inspected repository tree, and the repository was not proven to be the exact source revision of the deployed bundle. The metadata declaration is therefore reported with those limits. The original program is analysed here without being republished in the evidence package. ⁹

The dependencies and acknowledged code have their own records. The p5.js 1.0.0 repository includes LGPL 2.1. The current Rough.js repository carries an MIT licence naming Preet Shihn, while the project's hatching file identifies Rough.js as its basis without pinning the upstream version. These observations establish distinct sources of rights information; they do not resolve every historical incorporation detail. AB5D's CC0 dedication concerns the original essay prose. ^{13 11 14}

Close reading of five outputs

The comparison uses editions 0, 1, 10 and 20 as an early-edition group, together with the verified AB5D holding #106. Their full token identifiers are 242000000, 242000001, 242000010, 242000020 and 242000106 on Ethereum contract 0xa7d8d9ef8d8ce8992df33d8b8cf4aebabd5bd270. All five official originals were inspected in a 1280 by 720 viewing area. The sample supports comparison, not a claim to represent the whole edition. ^{1 2 3 4 5}

In #0, pale purple lines describe large, nearly circular polygonal cells on cream. Fine waves run obliquely through their interiors. Small three-pointed gaps remain where the cells meet, making the surrounding space unusually active. The larger forms seem to press together, while the narrow gaps keep them distinct. Because the waves repeat at a much finer scale, the picture offers two rhythms at once: the slow alternation of cells and gaps, and the close vibration of their infill. ¹

The thin rectangular frame gives that repeated field an edge without making the geometry appear to end there. Cropped cells continue into the boundary, suggesting a larger pattern of which this is one view. The precision of the outlines makes the waves read as a chosen texture rather than a failure of the geometry. This is a useful starting point for the series because construction and embellishment remain readily separable to the eye. ¹

In #1, large empty hexagons occupy most of the light field. Their connecting bands are densely worked, with small concentric hexagons at junctions and repeated contours inside the links. The wavering doubled edges make those links feel less mechanical than the regular arrangement initially suggests. Looking across the whole image, the empty cells dominate. Looking closely, attention collects at the nodes and narrow bands. The distribution of labour within the picture is uneven: elaborate marks frame considerable silence. ²

In #10, fine pink lines on charcoal describe angular, stepped forms crossed by diagonals. Small polygonal pockets carry a mesh of crosses, while larger regions remain dark and comparatively open. The colour contrast is low enough that the network does not overwhelm the ground. Its abrupt changes of direction become the main event. Compared with #1, the picture feels less like cells joined by decorated links and more like several routes passing through one another. The indexed features identify the Kharraqan family and interlacing, but the interpretation of those routes comes from the viewed drawing. ³

In #20, black hatching fills cells separated by broad pale passages. Dotted lines run within those passages, giving the otherwise empty bands their own delicate rhythm. The varying angles of the hatching help one region detach from the next, even though the whole surface uses the same restrained pair of colours. The image is particularly effective at reversing figure and ground: the filled cells can appear to be the objects, then the pale crossing paths take over as the continuous structure. ⁴

AB5D's #106 shares the generated hexagonal family with #0 and #20, but its blue hatching produces a different balance. Closely spaced, almost vertical strokes cover much of the cream field. Pale angular paths cut through them, with no strong dark contour separating each region. The slight irregularity of the strokes becomes more evident under enlargement. At the initial scale, their density can suggest a continuous blue fabric; closer inspection restores the individual marks and the gaps between them. ⁵

Pressing “i” on #106 adds faint dashed construction lines across that surface. The overlay does not replace the drawing or settle every visual ambiguity. It offers another network through which to read it. Enlarging the view then changes the relation between the construction and the hatching without generating a new token image. This interaction is modest but consequential: the viewer can move between the pattern as an enveloping field and the pattern as a set of decisions about edges, intervals and marks. ^{8 5}

Significance and limitations

Two Mathematicians treats geometry as a source of both discipline and expressive variation. Its different surfaces remain answerable to a shared construction, yet the character of the line changes what that construction feels like. Empty hexagons, densely hatched cells and dark interlacing routes reward different speeds of looking. The controls extend that comparison into the act of viewing, allowing the supporting grid to become visible without making it the only meaningful account of the work. ^{8 1 2 3 4 5}

Several questions remain open: the exact relationship between the public repository and the deployed bundle, historical mint settings, the unresolved early Hankin citation, and the full provenance of incorporated software. The site's named architectural references would also benefit from comparison with independent historical documentation. Those limits distinguish what has been observed from what is still to be established. The examined works already demonstrate a specific achievement: a repeated geometric system can carry visibly different kinds of attention, from measured outline to the restless accumulation of strokes. ^{6 8 9 10 11}