Tiling Theory

Dr. Casey Mann, Faculty Mentor

Informal STEM Shots

Tiling theory is another excellent topic for undergraduate research; it is accessible and abounds with solvable open problems. Dr. Mann is an expert in the area, having authored or co-authored several articles on tilings. Dr. Mann has over 12 years of experience in working with undergraduate students on topics in tiling theory and knot theory. Many of these collaborations have resulted in articles that have been published or are submitted. While open and accessible problems are easy to find in tiling theory (it is not uncommon for students to find their own problems), Dr. Mann has several problems in mind. The problems described next are designed to be accessible, relevant, and impactful.  After a week of learning a few key concepts, the students will be able to begin experimenting and conjecturing.

4266375135_54a7122a5c_m  • Convex Pentagons that Admit i-Block Transitive Tilings of the Plane. It is easily verified that all triangles and quadrilaterals (convex or not) tile the plane. Less obviously, it has also been shown that there are exactly 3 classes of convex hexagons that tile the plane. It has also been shown [18, 45] that convex n-gons with n ≥ 7 admit no tilings of the plane. The only gap in our knowledge pertaining which convex polygons admit tilings of the plane is the case of pentagons. There are 14 known distinct classes of convex pentagons that tile the plane, but it is unknown if there exist any other classes .

The PI and Co-PI of this project developed an algorithm that can determine, for each integer k, all convex pentagons that admit k-block transitive tilings. We conjecture that all pentagons that admit tilings of the plane also admit k-block transitive tilings; if this is true, our algorithm determines all pentagons that tile the plane. The cases k = 1, 2
were handled in mostly by hand, with some assistance from a desktop computer algebra software package. The cases when k > 2 are sufficiently complicated that any sort of by hand approach is impossible. In this research project, the students will work to develop an algorithm for the cases when k > 2 and implement the algorithm on the Hyak computing cluster (see Facilities, Equipment, and Other Resources). If successful, this project will either find previously unknown classes of convex pentagons that admit tilings of the plane (which would be a major achievement) or find that no pentagons exist that admit k-block transitive tilings for relatively small k (say, k = 3, 4, and 5); such a negative result would also be meaningful.

 

3854948263_abe66464d8_z  • Classifying Equilateral Isotoxal Tiles. The Co-PI along with an undergraduate student introduced the idea of equilaterally k-isotoxal tiles. k-Isotoxal tilings are those in which every edge of the tiling is in one of k transitivity classes with respect to the symmetry group of the tiling. Equilaterally k-isotoxal tiles are equilateral tiles that admit only kisotoxal tilings. Such tiles seem to be rare, and designing tiles with this property requires some ingenuity. In [8], several equilaterally 2-, 3-, and 5-isotoxal tiles were discovered. The project proposed here is to continue the work of [8] by classifying all equilaterally k-isotoxal tiles. Further, students could extend the idea and consider dihedrally k-isotoxal or trihedrally k-isotoxal tiles. This project gives students an excellent medium in which to learn the process of conjecture, experimentation, and proof.

 

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