Calculator Input
Example Data Table
| n | Element A | Element B | A × B | Order(A) | Center |
|---|---|---|---|---|---|
| 5 | r | s | sr^4 | 5 | e |
| 6 | r^2 | sr | sr^5 | 3 | e, r^3 |
| 8 | sr^3 | r^2 | sr^5 | 2 | e, r^4 |
| 7 | r^3 | r^2 | r^5 | 7 | e |
Formula Used
The dihedral group is written as D_n = <r, s | r^n = e, s^2 = e, srs = r^-1>. It contains all rotations and reflections of a regular n-gon.
Group order: |D_n| = 2n
Element forms: every element is either r^k or sr^k, where 0 ≤ k < n.
Multiplication rules:
r^a r^b = r^(a+b mod n)r^a sr^b = sr^(b-a mod n)sr^a r^b = sr^(a+b mod n)sr^a sr^b = r^(b-a mod n)
Element orders:
ord(r^k) = n / gcd(n, k)ord(sr^k) = 2
Center: if n is odd, the center is {e}. If n is even, the center is {e, r^(n/2)}.
How to Use This Calculator
- Enter the number of sides n for the regular polygon.
- Select the type and exponent for element A.
- Select the type and exponent for element B.
- Enter the exponent m to evaluate A^m.
- Press Calculate Dihedral Data to display results above the form.
- Use the CSV button for spreadsheet export and the PDF button for printable output.
FAQs
1. What does a dihedral group represent?
It represents all symmetry operations of a regular polygon. Those operations include rotations around the center and reflections across symmetry axes.
2. Why does Dn have 2n elements?
A regular n-gon has n distinct rotations and n distinct reflections. Adding both sets gives exactly 2n symmetry operations.
3. Why are reflections order 2?
Applying the same reflection twice returns every point to its original position. That means the second application produces the identity element.
4. How is the order of a rotation found?
For a rotation r^k, the order equals n divided by gcd(n, k). This tells you how many repeated applications return the polygon to its start.
5. When is a dihedral group abelian?
For the standard polygon cases with n at least 3, dihedral groups are nonabelian. Rotations and reflections usually do not commute.
6. What does the conjugacy class show?
It groups elements that are equivalent under internal symmetry. In dihedral groups, conjugacy often pairs r^k with r^-k and groups reflections by parity when n is even.
7. Why is the center different for odd and even n?
When n is even, the half-turn r^(n/2) commutes with every element. For odd n, only the identity commutes with everything.
8. What does the Plotly graph visualize?
It plots the original regular polygon and the polygon after applying the selected element A. This helps connect abstract group operations to visible geometric transformations.