# Simplifying f=uRuR in D3: A Beginner's Guide

• morrowcosom
In summary, the problem is asking to simplify the expression "u R u R" in the dihedral group D3. The relevant equations for D3 state that u^2 = 1 and Rn = 1. Additionally, the standard form for elements in D3 is Rk for rotations and u Rk for reflections, where k is between 1 and n-1. Using these equations and the associativity property, the solution is found to be R (the identity element).
morrowcosom

## Homework Statement

In D3, simplify f = u R u R

## The Attempt at a Solution

u(Ru)R=(uR)uR
u(uR^2)R=(uR)u2
uR=R

I am doing this for independent study, and the website I am using says the solution is 1=R

I just got started on dihedral groups and have no idea what in the heck to do. How do you tell which is a rotation or reflection before the solution arises and/or arrive with the above answer of R=1

Thanks

I have no idea what the notation "u R u R" means. Are u and R members of the group or something else?

I have no idea what the notation "u R u R" means. Are u and R members of the group or something else?

That is the way the problem is stated. Here are some relevant equations and an example problem.

Relevant equations:
To describe the group operation of Dn, we need only note the following rules (called "relations"):
u^2 = 1 (in fact, uk^2 = 1 for every reflection uk)
Rn = 1 (so R-1 = R^n-1)
R u = u R-1 = u R^n-1

To make calculations unambiguous, we agree that every element of Dn will have a standard form:

Rotations: Rk (or 1 for the identity = R0)
Reflections: u Rk (or just u itself if k = 0)
Since Rn = 1, we can take k between 1 and n-1

Example problem:
Example 1: In D3, the symmetry group of the equilateral triangle, calculate a = u R2 u.

Here is a sequence of steps using the relations in D3 along with associativity:
a = u R2 u = (u R2) u = u (R2 u)
= u (R R u) = u R (u R2)
(note we are using relation 3 above, for n = 3)
= u (R u) R2 = u u R4
(now use relations 1 and 2)
= u 2 R4 = R
So we conclude: a = u R2 u = R in D3

## 1. What is the purpose of simplifying f=uRuR in D3?

The purpose of simplifying f=uRuR in D3 is to make the code more concise and readable, while still achieving the same functionality. This can help improve efficiency and organization in your D3 projects.

## 2. What does f=uRuR stand for in D3?

The notation f=uRuR in D3 stands for "function equals update, enter, remove, update, remove". This is a shorthand way of representing the different stages of data manipulation and visualization in D3.

## 3. How can simplifying f=uRuR make my D3 code more efficient?

By simplifying f=uRuR, you can reduce the number of lines of code needed to achieve the same result. This can make your code more streamlined and easier to maintain. It can also help identify any potential errors or redundancies in your code.

## 4. Are there any downsides to simplifying f=uRuR in D3?

While simplifying f=uRuR can improve code efficiency, it may also make your code less customizable. By using shorthand notation, it may be more difficult to make specific changes to your code or incorporate more complex functionality.

## 5. Can beginners in D3 benefit from simplifying f=uRuR?

Yes, beginners in D3 can definitely benefit from simplifying f=uRuR. By learning and implementing this shorthand notation early on, it can help in understanding and writing more complex D3 code in the future. It can also make it easier to follow tutorials and examples that use this notation.

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