Flat-Folder: A Crease Pattern Solver

Flat-Folder is software written by Jason S. Ku to compute and analyze valid flat-foldable states of flat-foldable crease patterns, both assigned and unassigned.

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How to use

1. Go to Flat-Folder.

   • Tested to run in Chrome, Firefox, and Safari.
   • Chrome usually runs slightly faster than Firefox and much faster than Safari.
   • You should see the following interface:

   

2. Press "Choose File" to upload a crease pattern in FOLD, SVG, OPX, or CP file formats.

   • Various example files can be found in the ./examples/ folder.
     • You can download them by clicking here.
     • If you have examples files that you'd like me to add (and you have the proper authorization of the creator), please email them to me!
   • The software will probably have trouble if points in the input file are not accurate to single-precision. When importing any file that does not already contain face information, Flat-Folder will look at all the lines imported and compute the length $L$ of the shortest one. Then:
     • if any vertex of the input is closer than $\varepsilon = L/300$ to any other vertex, it will assume they are the same vertex; and
     • if any vertex of the input is closer than $\varepsilon$ to any line, it will assume the vertex is on the line.
   • For SVG format:
     • the import assumes each imported line is an unassigned fold (assignment "U"), unless its "style" attribute contains a "stroke" whose value is one of ["red", "blue", "gray"] corresponding to ["M", "V", "F"] assignments respectively;
     • will also accept values ["#FF0000", "#0000FF", "#808080"].
   • For FOLD format:
     • Import requires two properties:
       • vertices_coords
       • edges_vertices
     • Import can also import two optional properties:
       • edges_assignment (if missing, will assume all edges are unassigned "U")
       • faces_vertices (if missing, will construct its own set of faces from the provided edges)
         • on import, will reorder faces to be increasing by area

3. Once uploaded, Flat-Folder will draw:

   • the crease pattern on left of the display,
   • an x-ray view of the folded crease pattern in the middle of the display, and
   • a red circle behind any vertex of the imported crease pattern that violates either Maekawa or Kawasaki's theorems.
     • For Kawasaki, it checks whether the ((sum of even angles) $- \pi$) is greater than 0.00001.

   

   • Selecting the "Text" option will draw index labels for all the vertices, edges, and faces in the crease pattern. Currently, there is no way in the interface to adjust the font size, so this is only useful for debugging small inputs or by manipulating the text later in an output SVG.

   

4. Press "Fold" to find flat-foldable states of the crease pattern.

   • Flat-Folder will break up the faceOrder variables into disconnected components of variables whose set of solutions are independently assignable from each other.
   • You can limit the number of solutions to find per component by setting the "Limit" option:
     • all is defaut. This will attempt to compute all possible folded states.
     • Alternatively, you can select a number from [1, 10, 100, 1000], which will correspond to the maximum number of solutions to find per component.

5. After computing the overlap graph:

   • Flat-Folder will replace the x-ray view with the overlap graph.

   

   • Selecting the "Text" option will now also draw index labels for all the cells, segments, and points in the overlap graph.

   

   • Clicking on a cell in the overlap graph will highlight:
     • the faces of the crease pattern that overlap the cell (yellow), and
     • the edges of the crease pattern that overlap the segments bounding the cell.

   

   • Clicking on a face of the crease pattern will highlight:

     • the cells of the overlap graph that overlap the face (yellow),
     • the segments of the overlap graph that overlap the edges bounding the face, and
     • the other faces of the crease pattern that overlap the selected face in the folding (blue). Each blue face corresponds to a faceOrder variable (the yellow and blue faces overlap, so much be assigned an order).

     

     • Clicking on one of the blue faces will highlight the features of the corresponding faceOrder variable:
       • its two faces (yellow),
       • its taco-taco constraints (green),
       • its taco-tortilla constraints (red),
       • its tortilla-tortilla constraints (orange), and
       • its transitivity constraints (blue).

     

6. After computing solutions for all components:

   • Flat-Folder will display how many valid flat-folded states were found.
   • If any states were found, Flat-Folder will draw a rendering of the first one on the right of the display.

   

   • Selecting the "Flip" option will redraw the folded state as seen from the other side.

   

   • A "Component" dropdown menu is added to aid in selecting other states.

     • The "none" option hides all display of component information.
     • The "all" option draws every component found on the overlap graph in a randomly assigned color.

     

     • There is one numeric option (zero-indexed) for each component found. Selecting a component will:
       • draw that component on the overlap graph,
       • display the number of states found for that component, and
       • add a numeric input to enter which state to select for that component.

     

     • Changing this number will redraw the flat-folded state based on the change.

     

   • You can change states of each component until you reach a desired state.

   

7. Press "Export" to generate export links to various outputs.

   • Clicking "cp" downloads the crease pattern in FOLD format.
   • Clicking "state" downloads the current folded state in FOLD format.
   • Clicking "img" downloads a snapshot of the current display in SVG format.
   • Clicking "log" downloads a text file of all console output since the most recent file was imported.

   

Algorithm

I gave a talk on the Flat-Folder algorithm at Meiji University in December 2022.

• A one-page abstract can be found on page 5 of this PDF.
• A 25-min recording of the talk is available here.

Existing software like ORIPA and Orihime/Oriedita find flat-foldable states by:

1. constructing an overlap graph of cells, where each cell is a maximal regions of points in the folded image that overlap the same set of crease pattern faces, and
2. finding an ordering of faces in each cell that avoids self-intersection of the paper.

Flat-Folder takes a different approach for step (2) that makes it easier to compress the set of output states. In the following, we take $n$ to be the number of faces in the input crease pattern.

1. First, it finds each pair of faces [f, g] that overlap in the folding and identifies the pair as a Boolean variable that can have one of two possible assignments: either f is over g or g is over f. There are at most $O(n^2)$ variables, which are computed by Flat-Folder in $O(n^4)$ time.

2. Next, it computes all constraints on those variables that must be satisfied to avoid self-intersection. There are four types of constraints:

   • Taco-Taco: A taco-taco constraint occurs when two folded edges [e1, e2] of the crease pattern properly overlap along a segment s of the overlap graph, and of the two faces [f1, g1] adjacent to edge e1 and the two faces [f2, g2] adjacent to edge e2, all of them lie on the same side of segment s in the folding. There are six variables associated with such a constraint:

     • [f1, g1], [f1, f2], [f1, g2], [f2, g1], [f2, g2], [g1, g2]
     • Out of the $2^6 = 64$ possible assignments of these variables, only 16 of them are valid (avoid intersection).
     • There are at most $O(n^2)$ taco-taco constraints.

   • Taco-Tortilla: A taco-tortilla constraint occurs when a folded edge e of the crease pattern adjacent to faces [f1, f2] properly intersects the interior of a third face f3 along some segment s in the overlap graph. Alternatively, e properly overlaps a crease edge e2 that is not folded in the folding (has assignment F), and we let f3 be the face adjacent to e2 that lies on the same side of s in the folding as f1 and f2. There are three variables associated with such a constraint:

     • [f1, f2], [f1, f3], [f2, f3]
     • Out of the $2^3 = 8$ possible assignments of these variables, only 4 of them are valid (avoid intersection).
     • There are at most $O(n^2)$ taco-tortilla constraints.

   • Tortilla-Tortilla: A tortilla-tortilla constraint occurs when two crease edges [e1, e2] (has assignment F) of the crease pattern properly overlap along a segment s of the overlap graph, and of the two faces [f1, g1] adjacent to edge e1 and the two faces [f2, g2] adjacent to edge e2, f1 and f2 lie on one side of s, and g1 and g2 lie on the other side of s. There are two variables associated with such a constraint:

     • [f1, f2], [g1, g2]
     • Out of the $2^2 = 4$ possible assignments of these variables, only 2 of them are valid (avoid intersection).
     • There are at most $O(n^2)$ tortilla-tortilla constraints.

   • Transitivity: A transitivity constraint occurs when three faces [f1, f2, f3] all mutually overlap the same cell of in the overlap graph. There are three variables associated with such a constraint:

     • [f1, f2], [f1, f3], [f2, f3]
     • Out of the $2^3 = 8$ possible assignments of these variables, only 6 of them are valid (avoid intersection).
     • There are at most $O(n^3)$ transitivity constraints.

   The variables and constraints form a bipartite constraint graph with one vertex for each variable and constraint, with an edge between a variable and a constraint if the constraint is associated with the variable. This graph has size $O(n^3)$ and is computed by Flat-Folder in $O(n^5)$ time.

3. If the crease pattern is mountain/valley assigned, each edge assignments forces the assignment of one Boolean variable. Flat-Folder makes these assignments, and assigns any variables that can be infered from those assignments according to the constraints. This step takes $O(n^3)$ time.

4. After removing any the variables that may have been assigned in the last step from the constraint graph, the graph may be disconnected into multiple unconnected components. If it is, variables in one component cannot have any effect on variables in another component, so their assignments are independent. This step takes $O(n^3)$ time.

5. Lastly, Flat-Folder finds valid solutions for each connected component of variables via a brute-force search. This step can take exponental time $2^{O(n^2)}$ but if each component has only a polynomial number of solutions, solving each connected component independently can implicitly represent an exponential number of folded states in polynomial space.