Untangle
Reposition vertices of a scrambled planar graph until no two connecting line segments cross or intersect each other.
HISTORICAL ORIGIN LORE
The Lineage of Untangle
Conceived as Planarity by computer scientist John Tantalo in 2004, inspired by Kuratowski’s theorem and Wagner’s theorem in topological graph theory (which states that a finite graph is planar if and only if it does not contain a subgraph homeomorphic to K5 or K3,3). The puzzle translates graph theory into a soothing visual spatial exercise.
SYSTEM MECHANICS & RULES
Rules & Cognitive Architecture
A set of vertices (nodes) interconnected by straight-line edges (strings).
Initially, edges overlap and cross one another, displayed with highlight indicators.
Drag vertices across the canvas to reposition them.
The puzzle is solved when every single vertex has been positioned such that zero edges intersect.
Trained Mental Models
- ▸Planar graph embedding intuition and topological untangling
- ▸Convex hull boundary sorting and concentric cluster separation
- ▸High-dimensional spatial relaxation through direct manipulation
INCLUDED FREE FOREVER
The Three Base Tiers
Gentle
8–10 vertices arranged with low chromatic degree; simple perimeter dragging resolves crossings.
Standard
14–18 vertices with complex internal cycles requiring layered interior/exterior ring separation.
Deep
22–26 vertices with dense edge matrices requiring systematic convex hull perimeter unfolding.
BEYOND+ MASTER EDITIONS
The Master Variants for Untangle
K14 Complete Web
High-density planar embedding with 14 heavily connected hub vertices
Pushes planar boundary theorems to maximum density, generating intricate webs with 14 vertices and over 36 non-crossing edges that form interlocking geometric polyhedra.
- ·Maximal planar graph triangulator
- ·Real-time edge-intersection segment detection
- ·Boyer-Myrvold linear-time planarity testing
Polyhedral Torus Network
Multi-tiered circular geometries representing 3D polyhedral projections
Draws inspiration from 3D polyhedra (icosahedral and dodecahedral skeletal nets) projected onto a 2D plane, requiring concentric circular layout thinking to achieve zero intersections.
- ·3D-to-2D Schlegel projection nets
- ·Concentric orbit vertex clustering
- ·Force-directed relaxation physics preview
FREQUENTLY ASKED QUESTIONS
Questions on Untangle Architecture
Are all generated Untangle graphs guaranteed to be planar?
Yes. Every generated graph starts from a randomized planar triangulation before vertex coordinates are scrambled.
How are intersections indicated?
Intersecting edges subtly glow with a warm amber hairline, which turns into calm ivory once the crossing is cleared.
Is the touch response responsive on 120Hz displays?
Yes. Touch gesture handling is driven by native Android pointer events with zero input latency.
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Play Untangle on Android
Available directly on Google Play with Gentle, Standard, and Deep tiers included free forever.