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# A Mathematical Foundation for Foundation Paper Pieceable Quilts

• Mackenzie Leake
• Gilbert Bernstein
• Abe Davis
• Maneesh Agrawala

✻ Stanford University, ✧ UC Berkeley, ✱ Cornell University

Three foundation paper pieced designs generated in our tool and sewn in colorful fabrics

Abstract: Foundation paper piecing is a popular technique for constructing fabric patchwork quilts using printed paper patterns. But, the construction process imposes constraints on the geometry of the pattern and the order in which the fabric pieces are attached to the quilt. Manually designing foundation paper pieceable patterns that meet all of these constraints is challenging. In this work we mathematically formalize the foundation paper piecing process and use this formalization to develop an algorithm that can automatically check if an input pattern geometry is foundation paper pieceable. Our key insight is that we can represent the geometric pattern design using a certain type of dual hypergraph where nodes represent faces and hyperedges represent seams connecting two or more nodes. We show that determining whether the pattern is paper pieceable is equivalent to checking whether this hypergraph is acyclic, and if it is acyclic, we can apply a leaf-plucking algorithm to the hypergraph to generate viable sewing orders for the pattern geometry. We implement this algorithm in a design tool that allows quilt designers to focus on producing the geometric design of their pattern and lets the tool handle the tedious task of determining whether the pattern is foundation paper pieceable.

Overview:
Fig. 1: Given an input pattern design we encode the geometry as a dual hypergraph, where nodes represent faces and hyperedges represent seams connecting two or more faces. We visualize the hyperedges with colored boundaries (left). In this work we prove that if this hypergraph is acyclic, the pattern design is foundation paper pieceable, and we present a leaf-plucking algorithm that iteratively removes leaf hyperedges, where a node is only contained in that hyperedge, to generate a sewing order for the design, which is the reverse of the order in which we plucked the nodes (center). Our quilt design tool shows the resulting sewing order by numbering the faces (center, Sewing order) and lets users color the faces to visualize the design. Quilters can use foundation paper piecing to sew the quilt by attaching fabric pieces one at a time in the sewing order and precisely construct the quilt top (right).

Paper video: