Vortex

# Held Space: art and construction

25 September 2026. Original interactive geometric sculptures, informed by the four artists supplied in the brief and Chapters 59–60 of the local image-steering-accreditation library. This is an artistic implementation, not a new topology theorem or a reproduction of a particular artwork.

## What the references changed

**M.C. Escher: continuity must be experienced.** The museum's account of *Möbius Strip II* (February 1963) identifies nine ants on a partially transparent, single-sided strip. The useful design decision is a traceable path. Our travelling point follows a computed boundary, and the rib construction opens the opaque band to inspection. A one-sided midsurface and a one-component boundary are separate properties, both represented here.

Primary reference: [Escher in The Palace, Möbius Strip II](https://escherinhetpaleis.nl/en/about-escher/masterpieces/mobius-strip-ii). Museum text read; this is not a reproduction of the print or its ants.

**Max Bill: a plane can acquire the weight of an object.** The Pompidou records the evolution from paper and plaster to substantial stone, with the granite version IV dated 1960–61. The original concept and later material versions should not all be dated 1935. In this project, narrow waists alternate with broad faces, and rounded cross sections give the planes readable thickness. This is our interpretation, not a claim that Bill used this parameterization.

Primary reference: [Centre Pompidou, Unendliche Schleife, version IV](https://www.centrepompidou.fr/en/ressources/oeuvre/Rc47MKr). Collection record and curatorial analysis read.

**José de Rivera: the silhouette is not fixed.** David Shayt's Smithsonian account describes polished direct-metal construction and the rotation of *Infinity*. It informed the moving viewpoint and contrasting cool and warm reflections here. The counterposed bands make foreground and enclosure change as one orbits.

Primary reference: [David Shayt, Measuring Infinity](https://amhistory.si.edu/docs/Shayt_Measuring_Infinity_2008.pdf). Selected text on fabrication, finish, and rotation read; no complete historical survey is claimed.

**Charles O. Perry: treat the void as part of the composition.** The artist's *Continuum* page was visually inspected. The Smithsonian describes its open form and rhythm around a central axis. Our interpretation is to make three different relationships to empty space: a dense ruled chamber, a wall opened by unequal windows, and an asymmetric cantilevered gesture.

Primary references: [Perry's Continuum](https://www.charlesperry.com/sculpture/continuum/), [National Air and Space Museum collection record](https://airandspace.si.edu/collection-objects/continuum/nasm_A19780172000). No reference photographs are included or redistributed.

## Revision from the supplied images

The ten images supplied in this conversation are visual references, not instructions embedded in a document. Their artist identities are not inferred here. The images were examined directly; no image files are redistributed in the application.

The strongest correction to the first edition is the direction of construction. The first edition mostly used longitudinal strands tracing narrow bands. Images 1–2 use transverse members and sweeping rims to create volume and changing overlap. Image 7 makes the openings large enough to govern the whole object. Images 4–5 use asymmetry, a far-reaching tip, and a changing metal section. These three ideas define the revised works. The intersecting beams and figurative spirals in the other references remain inspiration, not implemented features.

## Vortex

A three-half-turn Möbius surface is prescribed by a deformed circular core C(u) and a transverse ruling D(u):

    X(u,v) = C(u) + v D(u),  -1 <= v <= 1
    C(u + 2 pi) = C(u)
    D(u + 2 pi) = -D(u)

The angular rotation of D is 1.5 u plus a periodic modulation and a constant phase. This explicitly reverses the transverse coordinate on closure. Its sole boundary therefore makes two circuits. A thick tube follows that full boundary, while 156 straight cylindrical rods join X(u,-1) to X(u,+1). Each rod is individually oriented and instanced; the rods do not follow the core length. All endpoints are checked against the prescribed surface.

Opening changes the core radius; breadth changes the span of the fans; inflection redistributes their rotation. Solid renders a closed thick band, Ribs renders rim plus cross-members, and Edge renders the single rim. A correct Möbius seam does not certify that every wide parameter choice is embedded.

## Aperture

A smooth cylinder-like surface has an elliptical cross-section, a changing radius, a leaning axis, a height-dependent rotation, and waved end lips. Three unequal ellipses are cut out in its unrolled (angle, height) domain before rendering. Two large windows dominate the body; a smaller low window changes the weight of the base.

The rectangle with holes is triangulated using the Earcut implementation bundled in Three.js, then conformingly subdivided three times. Corresponding periodic seam vertices are identified. The midsurface has five boundary components (three windows and two ends), Euler characteristic -3, and no boundary edges at the angular seam. Two offset layers are joined along every boundary, producing one closed orientable shell with Euler characteristic -6, equivalent to genus four. These are mesh-topology results, not claims of a collision-free embedding.

Analytic parameter derivatives supply smooth normals on both offset faces. This avoids normal distortion from the triangulation's unequal triangle shapes. Rib mode samples meridians and clips their intervals against the actual elliptical window equations. Edge mode exposes the five contours. The travelling point follows the upper lip.

## Cantilever

A closed centripetal Catmull–Rom spline passes through 18 authored points. It has a distant projecting tip, a high turn, and two unequal loops passing at different depths. Extend the reach moves the outlying control points. A transported frame closes with two half-turns, and a periodic width function expands a narrow edge into a substantial blade through the lower body.

This is a spatially authored path, not a rotated circular ring. Solid sweeps a rounded rectangular section; Ribs follows 13 longitudinal members; Edge traces two boundaries. The meaning of Ribs is specific to each sculpture and is stated here.

## Presentation and controls

The three works rest on separately rendered plinths. The lowest sampled non-instanced geometry vertex, after the authored rotation and uniform display normalization, is placed at the support height. This is a visual contact construction, not a static-stability calculation. Camera orbit leaves the support fixed. The main sculpture is normalized to fit the stage; the controls' numbers are dimensionless artistic coordinates.

Selecting a work restores its intended starting material and construction. The source is rendered directly with Three.js r153, included with its MIT license. Lighting uses original virtual studio panels, metallic materials, direct lights, and shadow maps; no photographs or generated bitmap sculptures are used. Moving the light rotates the direct lights; the reflection studio stays fixed. PNG export captures the actual current view.

The old source and its research records are retained under `archive-v1/`. The previous chapter-based geometry remains in `geometry.mjs` as a reusable frame and mesh utility. The external research library is unchanged.

## Verification and limits

`node verify.mjs` checks 27 solid configurations: defaults and the eight corners of the three controls for each work. The defaults use the full display subdivision. It checks closed orientable shell incidence, expected Euler characteristics, one connected component, finite coordinates, nonzero triangle areas, and closed trace paths. Aperture's five actual midsurface boundaries are counted independently. Frame parity is checked for Vortex and Cantilever. All six alternate construction views are generated and checked for finite geometry. The 156 rod transforms are independently checked against their intended endpoints.

These checks cover finite implementations. They do not certify all continuous parameter values, collision avoidance, bending energy, static equilibrium, or fabrication readiness. Cast surfaces and assembled rods are different physical constructions. No claim of novelty, historical reconstruction, or measured artistic preference is made.