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Homotopic cliff shed construction · a visual analogy

Three ways to build
a cliff shed.

Imagine a small climbers’ shelter halfway up a cliff. Airlift it, haul materials by rope, or build in place. The intended result stays the same; the work changes.

Can one way of reaching a goal be continuously changed into another?

Same drawn start. Same drawn finish.Schematic
A path between the ground and a cliff shelter Two dashed reference paths and a solid interpolated path share their first and last points. Numbered dots show four corresponding drawing vertices. This is not a physical route map. IMAGINED CLIFF START SHELTER
Solid: current blend. Dashed: chosen reference drawings.
Dots match by position in the sequence, not by equivalent work. Geometry is illustrative, with no physical scale.

Explore the drawing

Choose two approaches

0%
HelicopterRope & pulley

Drag or use arrow keys. This changes the drawing, not project progress.

Paused at the starting drawing.

A smooth picture is not a feasible plan.

The two endpoints stay fixed by construction. Intermediate drawings do not establish that crews could switch between these methods.

What the four dots stand for

The correspondence is supplied for the drawing.

  1. 1 · Start on the ground
  2. 2 · Establish an approach
  3. 3 · Bring work to the cliff
  4. 4 · Complete the shelter

Three imagined approaches

01 / AIR

Helicopter Assembly

Assemble modules on the ground, wait for a weather window, then airlift and install them.

Questions left open: lift capacity, wind limits, access and installation loads.

02 / ROPE

Rope & Pulley System

Establish a base camp, install a pulley system, haul materials and build on the cliff.

Questions left open: anchor design, hauling loads, crew access and staging.

03 / CLIMB

Climbing Construction

Stage a team, establish a route, build a platform and assemble the shelter in place.

Questions left open: fall protection, rescue, platform stability and exposure.

What survives the change?

A question to investigate,
not a set of green ticks.

The drawing contains no engineering, cost or time model. These original prototype claims remain unmodelled:

Storm protection
Unknown · not modelled
Cliff safety
Unknown · not modelled
Winter deadline
Unknown · not modelled
$50,000 budget
Unknown · not modelled

The homotopy intuition—and its limit

In mathematics, a homotopy is a continuous deformation between maps within a specified space. For paths, one can require the endpoints to remain fixed. Here, corresponding vertices are linearly interpolated: H(s, t) = (1 − t)A(s) + tB(s), where s runs along the equally partitioned segments and t is the blend. This gives a homotopy of these drawn paths in the unrestricted plane, with the same first and last coordinates.

A space of feasible project plans has not been defined. Nor have allowable transitions, obstacles, resources, costs or deadlines. So this is not a proof that the construction approaches are homotopic as feasible plans, or that safety, budget or completion dates are preserved. Matching the second dot in two drawings does not make “Weather Window” and “Pulley Install” interchangeable activities.

The useful project question is: what would have to remain true while a team changed method? A stronger model would need explicit feasible states and constraints, and would test every intermediate state against them.

Explore the companion task-field experimentSixteen tasks, supplied affinities and a reproducible spatial layout

Optional second experiment · heuristic spatial grouping

What happens when related task dots attract?

The earlier prototype also let sixteen construction tasks move under attraction and repulsion. This version keeps their names, types and supplied priorities. It turns those labels into a repeatable drawing rule, with no claim that a construction sequence emerges.

Sixteen numbered construction task dots Task dots move under heuristic layout forces. Select a dot with a click or Enter or use the task selector. Dashed links show nearby positive-affinity pairs, not construction dependencies.
Dashed links = positive affinity and separation below 100 drawing units. Larger dots with a pale ring = supplied “critical” priority. Neither size nor distance represents duration or construction risk.
0.5

0.1–2.0. Reset after changing strength to compare from identical starting positions. Repulsion remains active at every setting.

Paused. Seed 190 fixes the starting positions.

0 / 600 steps

Drill anchor points

Foundation / anchoring · supplied priority: critical.

  • Foundation / anchoring
  • Structural
  • Weatherproofing
  • Safety systems

The supplied drawing rules

Within 150 drawing units, same-type pairs attract with weight 1.2; foundation–structure pairs with 0.8; structure–weatherproofing with 0.6; other safety pairs with 0.4. Remaining pairs mildly repel (−0.3). A pair involving a supplied critical task has 1.5 times that influence. Short-range repulsion limits overlap.

All positions update from the same prior snapshot, in fixed steps with damping, speed caps and a bounded square. Seed 190 reproduces the start. The same strength and number of steps reproduce the layout. The 600-step stop is an illustration limit, not evidence of convergence.

Attraction is not a dependency

A foundation dot drawing a structural dot closer does not enforce “foundation before structure.” There are no directed prerequisites, start or finish times, crew capacities, loads or completion checks. These are proximity links, not a schedule or construction phases.

Spatial grouping does not prove feasible construction, optimal scheduling or real project self-organisation. The original “optimal phases” and “no central scheduler needed” claims are withdrawn. This experiment can prompt questions about relationships that a separate planning model would need to encode.