Pentalining: A New Way to Create Modular 5-fold Patterns Lars Eriksson Mariefred, Sweden; lars@rixn.se Abstract The subject is modular 5-fold patterns. Breaking boundaries can sometimes enable discovery of new insights. Here I break the edge rules to be able to create tile motifs that are more common among historical 5-fold patterns. I introduce two approaches to a concept I call Pentalining. This concept came to me from an Islamic geometric pattern in a Moroccan restaurant. It looked very traditional, but its appearance was deceptive. I found several uncommon shapes. To be able to tile it, I had to break the edge rule. Figure 1: A pattern in the restaurant in the Royal Mansour Hotel, in Casablanca, Morocco [13]. Introduction Manmade 5-fold patterns have existed for centuries, especially for Islamic geometric patterns. In Islamic geometric art it was important for the old masters to keep their know-how a secret. Only a few instructional sources, like the Topkapı scrolls [12] have survived into our days. Throughout the years, Westerners have published research about 5-fold patterns, like Kepler [10], Bourgoin [3] about Islamic geometric art, and Hankin [1, 9] who defined the tiling method he called the “polygons in contact.” Later work includes Lu and Steinhart’s paper [11], where they defined a tile set, the Girih tiles (which is renamed in my work as the Core 5 tile set [7]). The key publication in the field of modular patterns using tiling (polygons in contact) is Bonner’s book from 2017 [2]. Another key contributor worth mentioning is Peter R. Cromwell [6]. My published work is an off-shoot from Bonner’s work, as I push the boundaries of this research field to include a redefinition of Hankin’s “polygon in contact” method into what I call the edge rule concept [7] and non-equilateral tile sides [8]. My objective is to continue with the next step on this journey, the non- midpoint crossings of the lines. This paper is a slight sidetrack from this objective. It is about breaking the edge rule of the tiles in a way so that when tiling a pattern, it still passes as a traditional pattern. Edge Rule Tiling in this context is about piecing tiles together so that sides, with the same length, match. The motif of each tile interacts with another by a set of rules for how the lines are to “flow” over to adjacent tiles to ensure that the final pattern become seamless. I call these rules the edge rules. The main components of these edge rules are the number and angle of the crossing lines. This paper builds on this edge rule idea from previous published papers [7, 8]. There, you can familiarize yourself with the tiling concept, edge rule, and the names of the tiles in the tile sets reused here.