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(a) (b) How many window panels are fixed by the identity element of D3? How many panels are unchanged after rotating by 120 degrees?

(a) (b) How many window panels are fixed by the identity element of D3? How many panels are unchanged after rotating by 120 degrees?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In this exercise we consider glass window panels as in the picture below. Each of the seven segments can be made out of blue, green, or yellow glass. Our goal is to understand how many different window panels can be made. The picture has a symmetries of an equilateral triangle. That is, the group D3 acts on colourings of window panels. They can be rotated by 0, 120 or 240 degrees (Ro, R120, R240 respectively), or flipped along three lines (F1, F2, F3). (The three circles in the picture are centred at the points of an equilateral triangle, and their radius is the edge length.) (a) (b) (c) (d) How many window panels are fixed by the identity element of D3? How many panels are unchanged after rotating by 120 degrees? Give an example of a window panel that has a stabiliser of two elements. Use the Counting Theorem to determine the number of different window panels that can be created. Two panels count as different if they can not be made to look alike after rotations and flips.
Transcribed Image Text:In this exercise we consider glass window panels as in the picture below. Each of the seven segments can be made out of blue, green, or yellow glass. Our goal is to understand how many different window panels can be made. The picture has a symmetries of an equilateral triangle. That is, the group D3 acts on colourings of window panels. They can be rotated by 0, 120 or 240 degrees (Ro, R120, R240 respectively), or flipped along three lines (F1, F2, F3). (The three circles in the picture are centred at the points of an equilateral triangle, and their radius is the edge length.) (a) (b) (c) (d) How many window panels are fixed by the identity element of D3? How many panels are unchanged after rotating by 120 degrees? Give an example of a window panel that has a stabiliser of two elements. Use the Counting Theorem to determine the number of different window panels that can be created. Two panels count as different if they can not be made to look alike after rotations and flips.
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