Homework 5 Honors 213 Due Friday, February 28 Spring 2020 Symmetry Types of Frieze Patterns Symmetry Reflection across Reflection across 180 degree Indecomposable Type horizontal? vertical? rotation? glide reflection? p111 no no no no p1a1 no no no yes p112 no no yes (no) pm11 no yes no (no) pma2 no yes yes (yes) p1m1 yes no (no) (no) pmm2 yes yes (yes) (no) 1. For n ≥ 3, one possible set of generators for the dihedral group D n is {R, F} (where R gener- ally represents a rotation by 360/n degrees and F represents a reflection across some line). Find a set of relations that these generators satisfy, and use these to write a presentation of D n .(Hint: as usual, do this by trying and comparing examples until you find a pattern.) 2. Identify a set of fundamental symmetries, draw a fundamental region in the figure, and use the table above to classify the symmetry type of the following figure. Assume that the pattern continues infinitely in both horizontal directions, and ignore any minor imperfections.
Transcript
Homework 5 Honors 213Due Friday, February 28 Spring 2020
Symmetry Types of Frieze Patterns
Symmetry Reflection across Reflection across 180 degree IndecomposableType horizontal? vertical? rotation? glide reflection?p111 no no no nop1a1 no no no yesp112 no no yes (no)pm11 no yes no (no)pma2 no yes yes (yes)p1m1 yes no (no) (no)pmm2 yes yes (yes) (no)
1. For n ≥ 3, one possible set of generators for the dihedral group Dn is {R,F} (where R gener-ally represents a rotation by 360/n degrees and F represents a reflection across some line). Finda set of relations that these generators satisfy, and use these to write a presentation of Dn. (Hint:as usual, do this by trying and comparing examples until you find a pattern.)
2. Identify a set of fundamental symmetries, draw a fundamental region in the figure, and usethe table above to classify the symmetry type of the following figure. Assume that the patterncontinues infinitely in both horizontal directions, and ignore any minor imperfections.
3. Identify a generating set of fundamental symmetries, draw a fundamental region in the fig-ure, and use the table above to classify the symmetry type of the following figure. Assume thatthe pattern continues infinitely in both horizontal directions, and ignore any minor imperfec-tions.
4. Identify a generating set of fundamental symmetries, draw a fundamental region in the fig-ure, and use the table above to classify the symmetry type of the following figure. Assume thatthe pattern continues infinitely in both horizontal directions. (Ignore minor imperfections atthe connections of tiles, etc.)
5. Identify a generating set of fundamental symmetries, draw a fundamental region in the fig-ure, and use the table above to classify the symmetry type of the following figure. Assume thatthe pattern continues infinitely in both horizontal directions. (Ignore minor imperfections atthe connections of tiles, etc.)
6. For any symmetry types that you have not used, draw a frieze pattern with that symmetrytype. For each figure that you draw, identify a generating set of fundamental symmetries, drawa fundamental region in the figure, and use the table above to classify the symmetry type of thefollowing figure.
