Instructions for joining the curlygons

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  • Place the arms of the two curlygons parallel to each other. Important: the curlygons must have the same orientation (arms bend in the same direction, either clockwise or counter-clockwise).
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  • Hook the notch on the first arm to the middle of the second arm
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  • Wrap the two arms around each other
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  • Hook the notch on the second arm on the first arm
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  • Slightly pull the two arms apart to tighten the knot
Model gallery

Model Instructions

It's easier to add one piece at a time to a growing model, rather that trying to build it out of separate sections.
When following a diagram, it's easier to start at the center and add more curlygons, in a circular manner, towards the periphery.
When describing a model we'll refer to a loop of 3 connected curlygons as a "triangular face" or "triangle". Similarly, we'll call a loop of 4 curlygons a "square", a loop of 5 curlygons a "pentagon", etc.
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A 6 piece model, 8 triangles.
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A 12 piece model, 6 squares and 8 triangles.
Squares are surrounded by triangles and triangles are surrounded by squares.
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ITSPHUN
A 30 piece model, 12 pentagons and 20 triangles.
Pentagons are surrounded by triangles and triangles are surrounded by pentagons..

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Interweaved models

"Double" the curlygons to create superimposed, interweaved models!

  1. Build some model (6 red curlygons in the picture at right)
  2. Insert a new (blue) curlygon on top of each (red) curlygon of the built model:
    - the orientation of the new curlygons is the opposite of the orientation of the old ones
    - the arms of the new curlygons will cross under the arms of the old ones 
  3. Connect the new curlygons; the new connections will always be over the old ones.

What else can we build?

WARNING! The following contains mathematical language which may not be suitable for those with math phobia.
Use your discretion and proceed at your own risk.

While you can connect any two arms of the pieces, including two arms of the same piece, we will talk here only about what can be built when you connect two pieces at most once (i.e., two pieces that are already connected cannot be connected again using two different arms).
We will also assume that, in any completed model, all arms are connected (there are no "free" arms left).

With these restrictions, and assuming a piece represents a graph node, any construct is a model of a graph where every node has degree 4.
When this graph is some connectivity graph associated with a polyhedron, we can view the construct as a model of the polyhedron that captures some connectivity property (and ignores the spatial relations such as lengths and angles).

Viewing each curlygon as a face we can thus model all polyhedra with quadrilateral faces; viewing each curlygon as a vertex we can model all polyhedra where each vertex has degree 4. In the "face" view, the 6 piece model represents a cube and the 12 piece model represents a rhombic dodecahedron; in the "vertex" view these two models represent an octahedron and a cuboctahedron, respectively. In general, a model will represent two dual polyhedra in these two views.

From our point of view, the most interesting graph associated with a polyhedron is the "edge" graph whose nodes are the polyhedra edges; two edges are connected if they share both a face and a vertex. In any polyhedron, every edge is connected to exactly 4 other edges, so the edge graph of any polyhedron can be modeled with the curlygons. In other words, we can build a (connectivity) model of any polyhedron if we take the curlygons to represent the edges of the polyhedron.

In this view, the 6 piece object is a model of the tetrahedron, the 12 piece object is a model of the cube and octahedron  and the 30 piece object is a model of the icosahedron and dodecahedron. In general, duals generate the same model since their edge graphs are isomorphic; every construct can be seen as a model of both some polyhedron and its dual. As illustrated by the 6 piece model, when the modeled polyhedron (here tetrahedron) is self-dual, the model can have a higher symmetry than the polyhedron.

As an interesting observation, there are models with 6,8,9,10,.... pieces (there is no model with 7 pieces).
Indeed there are models with 6 and 8 pieces (a square pyramid in the "edge" view) and we can add a new piece to any model with n pieces, >= 8 as follows:
find a "face" of the model which is not a triangle (such a face always exists) and untie two connections that are on a diagonal (not next to each other). This will free 4 arms of 4 different pieces that can be now connected to the new piece.

Note that there are constructs that are not "edge" models of any polyhedron, the simplest one (10 pieces) is shown below.

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