Symmetry Group Analyzer for Polygons

Math Geometry • Transformations and Symmetry

Written by STEM Calculators Team Published January 25, 2026 Updated February 24, 2026

Symmetry Group Calculator – Dihedral Groups for Polygons \(D_n\)

Analyze the symmetry group of a regular \(n\)-gon: the dihedral group \(D_n\), containing \(n\) rotations and \(n\) reflections (total \(2n\) symmetries). Build the element list, compute products, and view an interactive polygon overlay and Cayley table.

Inputs accept pi, sqrt(2), 1e-3 and basic functions (if math.js is present). Use * for multiplication.

Polygon

Number of sides \(n\): Cayley table: Vertex labels: Show grid: Show axes:

Convention used in computations: \[ D_n=\langle r,s \mid r^n=e,\ s^2=e,\ srs=r^{-1}\rangle \] Elements are listed as \(e,r,r^2,\dots,r^{n-1},\ s,sr,\dots,sr^{n-1}\).

Element tools

Visualize element: Animation progress:

The plot shows the original regular \(n\)-gon (solid) and the transformed image under the selected element (dashed). Drag to pan • wheel/trackpad to zoom • pinch on touch. Units are square.

Multiply (left) \(a\): Multiply (right) \(b\):

Ready

View (square units • interactive)

Solid = original polygon. Dashed = image under the selected symmetry. Use the slider or Play to animate.

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Theory

What is the symmetry group of a regular polygon?

The set of all distance-preserving symmetries of a regular \(n\)-gon forms a group under composition, called the dihedral group \(D_n\). It contains:

\(n\) rotations (including the identity), and
\(n\) reflections.

\[ |D_n| = 2n. \]

Example: a square has \(D_4\) with \(8\) elements (4 rotations + 4 reflections). An octagon has \(D_8\) with \(16\) elements.

Generators and relations (group presentation)

A standard way to describe \(D_n\) is by two generators:

\(r\): rotation by \(2\pi/n\),
\(s\): reflection across a fixed axis.

\[ D_n=\langle r,s \mid r^n=e,\ s^2=e,\ srs=r^{-1}\rangle. \]

The relation \(srs=r^{-1}\) means “reflect, then rotate, then reflect” is the same as rotating the opposite way. It’s the key identity that tells you how rotations and reflections interact.

All elements of \(D_n\)

Every element of \(D_n\) can be written uniquely in one of these forms:

\[ r^k \quad (k=0,1,\dots,n-1), \qquad sr^k \quad (k=0,1,\dots,n-1). \]

The first set are rotations; the second set are reflections (each reflection is a reflection across some axis through the polygon).

Multiplication rules and quick products

Using exponents mod \(n\), the core products become:

\[ r^a r^b = r^{a+b},\qquad r^a (sr^b)=sr^{b-a},\qquad (sr^a) r^b = sr^{a+b},\qquad (sr^a)(sr^b)=r^{b-a}. \]

These follow directly from \(srs=r^{-1}\) (equivalently \(sr^m=r^{-m}s\)).

Orders of elements

The order of an element is the smallest positive integer \(m\) such that \(g^m=e\).

For rotations: \(\operatorname{ord}(r^k)=\dfrac{n}{\gcd(n,k)}\).
Every reflection has order \(2\): \(\operatorname{ord}(sr^k)=2\).

This is why reflections “flip back” after doing them twice, while rotations may need several repeats to return to the identity.

Why the Cayley table matters (and a crystal symmetry tease)

A Cayley table lists all products \(a\cdot b\) in a group. For \(D_n\), it shows a clear block structure: rotations multiply like a cyclic group, while reflections interact by turning products into rotations and vice versa.

Crystal symmetry tease: many symmetry classifications in physics and chemistry use combinations of rotations and reflections. Dihedral groups appear in molecular symmetry (e.g., certain ring-like molecules) and as building blocks of 2D symmetry patterns.

Frequently Asked Questions

What is the symmetry group of a regular polygon?

The symmetries of a regular n-gon form the dihedral group D_n under composition. It contains n rotations and n reflections, so the total number of symmetries is 2n.

What do r and s mean in the dihedral group D_n?

r represents rotation by 360/n degrees, and s represents a reflection across a fixed axis. Every element can be written as r^k or s r^k for k = 0, 1, ..., n-1.

How do you multiply elements in D_n?

Products are simplified using r^n = e, s^2 = e, and s r s = r^(-1), along with reducing exponents modulo n. For example, (s r^a)(s r^b) simplifies to r^(b-a) in D_n.

Why does D_n have 2n elements?

There are n distinct rotations r^k and n distinct reflections s r^k. Together these give exactly 2n different symmetries for a regular n-gon.

What is a Cayley table and why is it useful here?

A Cayley table lists every product a·b in the group for all element pairs. For D_n it makes the rotation/reflection structure clear and helps verify identities quickly.

Symmetry Group Calculator – Dihedral Groups for Polygons \(D_n\)

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