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# Only Those Who Attempt the Absurd Will Achieve the Impossible

[GitHub](https://github.com/HectorTablero/QED-Scripts/tree/main/Magazine%204/Only%20those%20who%20attempt%20the%20absurd%20will%20achieve%20the%20impossible) (Open Source, MIT License) | [Live Demo](https://qed.mat.uam.es/revista/articulo/intentar-lo-absurdo)

**Date:** July 2026
**Technologies:** JavaScript, Tailwind CSS

---

## Overview

The tessellation element reuses a repo-wide tiling engine (`Shared/tiling.js`) that models a tile as a polygon with edge pairings — whatever is deformed on one free edge is carried, through the paired isometry, onto its partner — and fills the plane by breadth-first search over the group those isometries generate. Every pairing isometry is a $2\times3$ affine map $g: p \mapsto Mp + t$ with $M$ a rotation or reflection matrix; see [`Portfolio/shared.md`](../shared.md) for the exact matrices used.

## EscherTessellation

### Description

Lets the reader drag control points on a square or hexagonal tile's free edges and watch the paired edges deform in consequence, then tiles the plane with the result, mirroring how Escher converted regular grids into interlocking lizards and birds.

### Implementation

`EscherTessellation` is a thin subclass of the shared `TileEditorBase`, supplying only its own defaults and captions; all of the deformation, pairing, and plane-filling logic lives in the shared `DeformableTile`/`Aff` classes:

```javascript
::: js EscherTessellation
"square", "rotation", 6, 3.4
escher1
autoPlay, allowFullscreen, openControls, h=180
:::
```

Two pairing rules are offered — translation (opposite edges) and rotation (adjacent edges paired about their shared vertex) — both verified to preserve the tile's area exactly under arbitrary deformation, which is what keeps the deformed tile still tiling.

### Technical Considerations

The plane-filling patch is found by BFS over the pairing isometries and their inverses, deduplicated on the transform itself, so it needs no separate knowledge of which wallpaper group a given pairing happens to generate.

## MobiusStrip

### Description

Renders a Möbius strip (or, for comparison, an ordinary twist-free annulus) with a walking ant and a paint-fill demonstration, both making concrete Escher's claim that the strip has only one surface.

### Implementation

The surface is parametrised directly, with $u \in [0, 2\pi)$ running along the strip and $v \in [-1,1]$ across its width. Writing $k = \tfrac12$ for the half-twist rate and $w = v \cdot w_{\text{half}}$ for the physical half-width at parameter $v$, the embedding is

$$\text{point}(u,v) = \big(r\cos u,\ r\sin u,\ w\sin(ku)\big), \qquad r = 1 + w\cos(ku),$$

a circular centerline of radius 1 with a cross-section that both moves radially ($w\cos(ku)$) and lifts out of plane ($w\sin(ku)$) as it twists by half a turn per full lap ($k=\tfrac12 \Rightarrow$ a $\pi$ twist over $u:0\to2\pi$). Setting $k=0$ instead collapses this to a flat, twist-free annulus for comparison. The result is projected with a fixed perspective and drawn as depth-sorted, Lambert-shaded quads — no 3D library involved. The ant's lap counter and the paint fill both key off the surface's true period: because $\cos(ku)$ and $\sin(ku)$ only repeat once $ku$ advances by $2\pi$, i.e. $u$ advances by $4\pi/(2k)=4\pi$ when $k=\tfrac12$, a half-twisted strip only closes up after two circuits ($4\pi$) versus one ($2\pi$) for the plain annulus — the parametrization's own period _is_ the one-sidedness proof, not an illustration of it:

```javascript
::: js MobiusStrip
1, 60, 0.35, 320
mobius1
autoPlay, allowPause, autoPauseOnScroll, allowFullscreen, openControls, h=170
:::
```

Painting only marks the half of the strip the ant is _currently_ standing on, so full coverage is only reached if the ant genuinely completes both laps while the brush is down — the actual demonstration, not a shortcut to it.

### Technical Considerations

The mesh is depth-sorted by the painter's algorithm every frame; the sort itself only needs to reflect the current camera orientation, which the reader controls by dragging.