Rotational Symmetry Calculator – Order & Angle for Shapes
A shape has rotational symmetry if rotating it about some point by a non-zero angle
makes it land exactly on itself. The number of times it matches itself during a full \(360^\circ\) turn
is called the order of rotational symmetry.
1) Order \(k\) and smallest rotation angle
If a shape has rotational symmetry of order \(k\ge 2\), then the smallest positive angle that maps the
shape onto itself is:
\[
\theta_{\min}=\frac{360^\circ}{k}.
\]
Examples:
- Square: \(k=4\), \(\theta_{\min}=90^\circ\).
- Equilateral triangle: \(k=3\), \(\theta_{\min}=120^\circ\).
- Rectangle (not a square): \(k=2\), \(\theta_{\min}=180^\circ\).
2) Rotation formula (how points move)
Rotations are done around a center point \(C=(h,k)\). For any point \(P=(x,y)\),
rotate by angle \(\theta\) (in radians) using:
\[
\begin{aligned}
x' &= h + (x-h)\cos\theta - (y-k)\sin\theta,\\
y' &= k + (x-h)\sin\theta + (y-k)\cos\theta.
\end{aligned}
\]
3) Why the rotation center is the centroid
If a set of vertices is unchanged by a non-trivial rotation, then its average point must stay fixed.
For a rotation, the only fixed point is the rotation center itself, so the centroid of the vertices
must lie at the rotation center.
This tool computes the vertex centroid:
\[
\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i,\qquad
\bar{y}=\frac{1}{n}\sum_{i=1}^n y_i,
\]
and uses \(C=(\bar{x},\bar{y})\) as the rotation center.
4) How the calculator detects rotational symmetry
In auto-detect mode, the tool tests candidate orders \(k\) and checks whether rotating all vertices by
\(\theta=360^\circ/k\) produces the same polygon.
-
Polygon mode: the vertex order is respected. After rotation, the vertex list should match
the original list up to a cyclic shift.
-
Point-set mode: the tool ignores the vertex order and compares the rotated vertex set to the original set.
This is useful when you only care about symmetry of the points (not the edges).
5) Test mode (order or angle)
In test mode you can input either:
- an order \(k\), and the tool tests \(\theta=360^\circ/k\), or
- a direct angle \(\theta\) in degrees.
To prevent inconsistent settings, the calculator disables one input when the other is used.
6) Regular polygon check (extra insight)
A regular polygon has all sides equal and all vertices equally spaced around its center.
This tool includes an approximate regularity check:
- Side check: consecutive side lengths are nearly equal.
- Radius check: all vertex distances to the centroid are nearly equal.
- Angular spacing check: vertices are nearly spaced by \(2\pi/n\) around the centroid.
If a polygon is regular with \(n\) vertices, then (in polygon mode) the highest rotational symmetry order is typically
\(k=n\), so \(\theta_{\min}=360^\circ/n\).
7) Rotation wheel (angle selector)
The calculator provides a rotation wheel and slider to choose \(\theta\) visually.
- In auto mode, the wheel snaps to the nearest detected valid rotation.
- In test mode, the wheel sets \(\theta\) (and automatically disables \(k\)).
- If you enter \(k\), the wheel locks to \(\theta=360^\circ/k\) to keep the inputs consistent.
8) Reading the graph
- The original polygon is drawn solid.
- The rotated copy is drawn as a dashed overlay (if enabled).
- Play animates rotation from \(0^\circ\) to the selected angle.
- The centroid \(C\) is marked on the plot and acts as the rotation center.
9) Common mistakes
- Vertices not in order (polygon mode expects a proper vertex order around the shape).
- Using a tolerance that is too strict for decimal-heavy coordinates (try \(10^{-5}\) or \(10^{-4}\)).
- Self-intersecting polygons can behave unexpectedly (point-set mode may be more appropriate).
