OrigasmiQue

1. OrigasmiQue Problem setters and judges: Christian Colombo Jean Paul Ebejer Karl Fenech Gordon Pace Chris Porter

2. Origami: The Art of Paper Folding

3. Paper Folding Conundrums • In folding and unfolding a sheet of paper we can mark lines and intersection points. What points can be calculated in this manner? Is it possible to trisect an angle? • Is it possible to fold a piece of paper is such a manner that the perimeter of the resulting shape is bigger than that of the original rectangle?

4. This Year’s Challenge • Given: • A 2x1 sheet of paper; and • A 2-dimensional shape (the target shape).

5. This Year’s Challenge • Given: • A 2x1 sheet of paper; and • A 2-dimensional shape (the target shape). • Expected: • A sequence of instructions (move/translate the paper, rotate it around the origin or fold the paper along the x-axis); • which, if applied to the sheet of paper starting at the origin results in a shape similar to the target one.

6. This Year’s Challenge • Given: • A 2x1 sheet of paper; and • A 2-dimensional shape (the target shape). • Expected: • A sequence of instructions (move/translate the paper, rotate it around the origin or fold the paper along the x-axis); • which, if applied to the sheet of paper starting at the origin results in a shape similar to the target one. • More: • In case of a tie, the solution with the least number of instructions wins. • A secondary prize for the fold resulting in the most original shape will be given.

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38. Comparing Solutions • A grid (per 1/64 units) is overlaid over the target shape and the given solution. • Let: • c be the number of correctly hit points (points both in the solution and the target shape); • w be the number of incorrectly hit points (points in the solution but not in the target shape); • a be the number of points (the area) of the target shape. • The score of such a solution is:

39. Comparing Solutions • A grid (per 1/64 units) is overlaid over the target shape and the given solution. • Let: • c be the number of correctly hit points (points both in the solution and the target shape); • w be the number of incorrectly hit points (points in the solution but not in the target shape); • a be the number of points (the area) of the target shape. • The score of such a solution is: The count of correctly hit points has to be larger than that of incorrectly hit ones

40. Comparing Solutions • A grid (per 1/64 units) is overlaid over the target shape and the given solution. • Let: • c be the number of correctly hit points (points both in the solution and the target shape); • w be the number of incorrectly hit points (points in the solution but not in the target shape); • a be the number of points (the area) of the target shape. • The score of such a solution is: The score is never below zero

41. Comparing Solutions • A grid (per 1/64 units) is overlaid over the target shape and the given solution. • Let: • c be the number of correctly hit points (points both in the solution and the target shape); • w be the number of incorrectly hit points (points in the solution but not in the target shape); • a be the number of points (the area) of the target shape. • The score of such a solution is: The score is normalised according to the size of the target shape

42. Comparing Solutions • A grid (per 1/64 units) is overlaid over the target shape and the given solution. • Let: • c be the number of correctly hit points (points both in the solution and the target shape); • w be the number of incorrectly hit points (points in the solution but not in the target shape); • a be the number of points (the area) of the target shape. • The score of such a solution is: Score may range between 0 and 1

43. Some Observations • Most submissions were implementations of either a bounding box or a convex hull algorithm. There was also a single machine learning approach.

44. Some Observations • Most submissions were implementations of either a bounding box or a convex hull algorithm. There was also a single machine learning approach.

45. Some Observations • Most submissions were implementations of either a bounding box or a convex hull algorithm. There was also a single machine learning approach.

46. Some Observations • Most submissions were implementations of either a bounding box or a convex hull algorithm. There was also a single machine learning approach.

47. Some Observations • Most submissions were implementations of either a bounding box or a convex hull algorithm. There was also a single machine learning approach.

48. Some Observations • Most submissions were implementations of either a bounding box or a convex hull algorithm. There was also a single machine learning approach. • Convex hull is intrinsically better than bounding box, • But still overapproximatesshapes that are concave or have holes. • When the area of these holes exceeds that of the polygons, the convex hull algorithm drops to zero scores.

49. The Solutions (the ones who answered) • 7 Java solutions, 1 C++ solution, 1 C# solution. • Some teams used a bounding box solution: • Team #3(ARIA) • Team #7 (FusRoHuhs) • Team #6(Programoids) – done via a convex hull calculation • Team #1 (Abstergo) used this for shapes with 6 or more sides • A common convex-hull implementation was to apply for a number of times: Take the next edge of the given shape, translate it to the origin, rotate and fold. • Team #2(DCoders) applied this on each edge of the target shape. • Team #1(Abstergo)applied this on the convex hull of the target shape, and performed the step 2*number of sides times. • Team #8(Team *) iterated this step until the areas of the target shape and the generated one matched.

50. Team #15: Ixaris • Two algorithms running on multiple threads were implemented: • Convex hull solution; • A greedy search, adding T;R;F parts to the solution.