Metametric
Continuum

01 · SEPARATION

A seam opens.
A surface unfolds.

The film begins with a catenoid: the hourglass-shaped minimal surface familiar from a soap film stretched between two circular wire rings. The displayed portion goes all the way around its waist. Its upper and lower circular rims remain open.

“Joined around the waist” does not mean a closed surface without boundary. This catenoid has two boundary circles and the topology of an annulus, just like a cylinder’s side or a flat washer. A loop around its waist cannot shrink to a point while staying on the surface.

A seam then separates from one rim to the other. As the slit widens and the surface begins to unfold, that complete circuit is broken. The cut-open sheet has the topology of a disk: it has one continuous boundary and no non-shrinkable loop. A disk need not be flat or circular; here the word describes how the surface is connected.

The widening is an artistic reveal: the film reduces angular coverage from a full turn to the original patch width while the surface begins its catenoid–helicoid transformation. These motions overlap, so there is no stop between separation and unfolding. Every intermediate piece is still a minimal surface, but the displayed area decreases. It is not a claim that an unbroken sheet can open this way while preserving all its measurements.

A cut can change the space of possible paths before the surface changes its underlying curvature.

02 · GEOMETRY

A different shape.
The same inner geometry.

The unfolding begins as the seam opens and continues after its width settles. This movement passes between a patch of a catenoid, an hourglass-like surface, and a patch of a helicoid, a surface that spirals like a sweeping ramp. These belong to a classical associate family of minimal surfaces.

Here C and S describe the catenoid and helicoid patches; the angle θ carries one into the other. Once the angular coverage has settled, the surprising fact is that the surface changes its arrangement in space while preserving its intrinsic metric—the rule for measuring distances along it.

During the fixed-patch portion, the parameters u and v label the same points throughout the movement. Corresponding curves keep their lengths, corresponding regions keep their areas, and angles remain unchanged. A tiny surveyor living on the surface could measure paths and triangles without detecting the change. An observer outside sees a very different form.

Appearance in the surrounding world can change while the geometry experienced on the surface remains the same.

These surfaces also have zero mean curvature. Their two principal curvatures—the strongest and weakest normal bending at a point—balance one another:

“Minimal” means that area is stationary under small variations fixing the boundary. It does not mean every such surface is the global smallest-area solution. Gaussian curvature, which multiplies the two principal curvatures, is preserved point by point here too:

The metric-preserving portion of this movement uses the fixed, cut-open patch reached after separation. It does not claim that the complete catenoid and the complete helicoid are globally the same surface.

03 · IDENTIFICATION

A small seam.
A fundamental change.

The spiral begins to bend around, opening into an almost complete ring. This begins a further artistic extension: the associate-family restrictions on lengths, areas and curvature are released.

While a slit remains, the sheet is topologically a disk. That word describes its connections, not its outline: a rectangle, a curved patch and this slit ring can all have disk topology. None yet allows an uninterrupted circuit around the central opening.

When the matching ends join, the surface becomes an annulus—a flat ring, like a washer. A path can now go all the way around its opening, and that loop cannot shrink to a point while remaining on the surface.

Mathematicians call the joining an identification, or a gluing: points that were distinct are now treated as the same points. It is a precise construction of a new space. It is not a topology-preserving deformation of the original sheet.

The positions can move continuously even though the topology changes at the instant of contact. At that moment, the map from the original patch ceases to be one-to-one. Two separate edges arrive at the same place.

04 · CURVATURE

The ring gathers form.

The annulus now rounds into a deepening channel. This is a change of geometry that can preserve topology. So long as its two circular rims remain separate, the flat ring and the curved channel are both annuli.

Something that looks like volume begins to appear. But the surface itself remains two-dimensional. It curves through three-dimensional space; until the final opening is sealed, the surface alone does not bound a closed interior.

This rounding cannot preserve every internal measurement. A flat annulus has zero Gaussian curvature; a round torus has positive curvature on its outer side and negative curvature on its inner side. Pure bending without stretching cannot create that change in Gaussian curvature. The film allows the necessary stretching.

The distinction is useful: two objects can have the same topology while differing in their lengths, areas and curvature. A dramatic visual transformation need not create a new kind of topological surface.

05 · CLOSURE

A second joining.
A torus emerges.

The channel’s remaining boundary circles meet. Identifying them closes the surface into a torus. This is another topological transition: after the initial cut and the first joining, a second joining removes the remaining boundary.

The torus has two independent kinds of non-shrinkable loop: one travels around the main ring; the other travels around the tube’s cross section. The annulus had only the first. It also had two boundary circles. The completed torus has none.

Here S¹ denotes a circle, and × means the Cartesian product. The two factors are separate circles: one angle locates a point around the main ring, and the other locates it around the tube. Each circle is one-dimensional; their product is a two-dimensional surface. The closed surface bounds a three-dimensional region, the solid torus:

D² denotes a filled two-dimensional disk, including its circular edge: all points (x, y) with x² + y² ≤ 1. The superscript counts dimensions. Taking its product with a circle describes the filled tube carried around the main ring. The surface and the region it bounds are different mathematical objects. The central opening through the doughnut remains connected to the outside; it is not the enclosed interior of the tube.

06 · WHAT SURVIVES

Continuity has more than one meaning.

An invariant is a property unchanged under a specified kind of transformation. There is no single promise that every mathematical quantity must survive every change.

*The catenoid starts with two circular rims. After the seam opens, the cut patch has one continuous boundary loop, including its corners. “Independent loops” here means the first Betti number, β₁: independent one-dimensional homology cycles, represented in these examples by non-shrinkable loops.

The Euler characteristic, χ, can be computed from vertices minus edges plus faces in a suitable finite decomposition. For a connected, compact, orientable surface with g handles and b boundary components:

The annulus and torus both have χ = 0, yet they are not topologically equivalent: their boundaries and independent loop counts differ. One invariant rarely tells the entire story.

Across the intended surfaces, connectedness, orientability and dimension remain shared properties. The surface remains one connected, two-sided, two-dimensional object. Those are genuine similarities, but they do not make all stages homeomorphic—equivalent by a continuous, reversible correspondence of points.

Constant total area could be imposed as an additional constraint by scaling each stage appropriately. This film does not impose it during separation or after the associate-family movement. Positive enclosed volume cannot be conserved from the opening catenoid, whose two open rims prevent it from enclosing an interior of its own.

There is a stronger limit on preserving minimality: no smooth compact surface without boundary can have zero mean curvature everywhere in ordinary Euclidean three-space. On a minimal surface, its coordinate functions are harmonic; on a compact connected surface without boundary, those functions would have to be constant. The object would collapse to a point. The catenoid and associate family’s minimal-surface condition must therefore be relinquished before reaching the closed torus.

Geometry and topology meet again in the Gauss–Bonnet theorem. For a smooth closed surface, total Gaussian curvature is 2πχ. The completed torus has χ = 0, so its positive and negative Gaussian curvature balance in the integral. For surfaces with boundary, boundary turning contributes too; the closed-surface formula cannot simply be applied to the earlier patch.

07 · RETURN

The same journey, read backwards.

After a brief pause, the torus opens into a channel, relaxes into an annulus, separates at the ring seam and folds back into the cut catenoid patch. As the unfolding reverses, the initial slit narrows in the same flowing movement until its edges meet again, restoring the circumferentially complete catenoid. The return gathers pace, then slows as the seam closes. The joined form rests across the loop’s boundary before the next separation begins.

In the mathematics of genuinely glued material, reversing a join requires cutting it apart. The film retraces a geometric map, allowing touching edges to separate again. It is an illustration of constructions and relationships, rather than a physical simulation of an unbroken sheet.

Some changes preserve measurements. Some preserve connections. At a joining, a new space can begin.

Topology helps us recognize a shared structure beneath different appearances—and recognize when a small-looking event changes that structure. Geometry reveals what can remain measurable and familiar throughout a transformation. Metametric Continuum moves between those viewpoints.