Finnish Aurora Kaleidoscope

Move your mouse over the black box to see the Northern Lights pattern form. Adjust the number of mirrors (symmetry order) with the slider.

The Math and Physics of the Aurora Kaleidoscope

This visualization beautifully connects **mathematics** and **physics**. The colorful patterns you see are an artistic representation of the **Aurora Borealis**, a physical phenomenon caused by charged particles from the sun interacting with Earth's magnetic field. While we aren't simulating the exact physics, we are using the principles of **symmetry**, a key concept in both fields, to create the visual effect.

The kaleidoscope effect is based on two fundamental geometric transformations: **reflection** and **rotation**. By taking a base pattern (the points you move with your mouse) and repeatedly applying these transformations, we generate a symmetrical design.

Mathematical Transformations

A point $(x, y)$ can be transformed using the following rules:

**Reflection:** The code reflects points across the x-axis, which can be expressed with the transformation matrix: $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$ This gives us the reflected point $(x', y') = (x, -y)$.
**Rotation:** A point is rotated around the origin by an angle $\theta$ (in degrees). The new coordinates $(x', y')$ are calculated using the rotation formulas: $$ x' = x \cos(\theta) - y \sin(\theta) $$ $$ y' = x \sin(\theta) + y \cos(\theta) $$

The visualization works by taking your mouse's position, reflecting it, and then rotating both the original point and its reflection to create all the symmetrical copies. The number of mirrors you set with the slider determines the **rotational symmetry**, or the number of identical "slices" that make up the whole pattern.

Number of Mirrors (Symmetry Order): 6

Quiz: Test Your Knowledge

Using the rotation formulas above, what is the new coordinate of a point located at (2, 3) after a rotation of 90 degrees?

Adjust the slider to a **symmetry order of 3**. What is the angle between the two closest mirror lines, in degrees?

Understanding Symmetry Order and Mirror Angle

It's easy to get confused between the number of mirrors and the angle you see in each repeating segment of the pattern.

The **Symmetry Order ($n$)** is the number of identical "slices" that make up the whole circle. This is determined by the number of mirrors you set with the slider.

The **Angle between two closest mirror lines ($\theta$)** is the total angle of the circle ($360^\circ$) divided by the number of slices. This relationship is always true: $$ \theta = \frac{360^\circ}{n} $$

For example, if you set the **Symmetry Order to 3**, the circle is divided into 3 equal slices. Therefore, the angle between each mirror is $\theta = 360^\circ / 3 = 120^\circ$. If you set the order to 6, the angle is $60^\circ$.

Solution for Quiz Question 1

Let's solve the first quiz question step-by-step using the rotation formulas. The question asks for the new coordinates of a point at $(2, 3)$ after a $90^\circ$ rotation.

The given values are:

Initial point: $(x, y) = (2, 3)$
Rotation angle: $\theta = 90^\circ$

First, recall the values of sine and cosine for $90^\circ$: $$ \sin(90^\circ) = 1 $$ $$ \cos(90^\circ) = 0 $$

Now, let's plug these values into the rotation formulas: $$ x' = x \cos(\theta) - y \sin(\theta) = (2) \cos(90^\circ) - (3) \sin(90^\circ) $$ $$ x' = (2)(0) - (3)(1) = 0 - 3 = -3 $$ $$ y' = x \sin(\theta) + y \cos(\theta) = (2) \sin(90^\circ) + (3) \cos(90^\circ) $$ $$ y' = (2)(1) + (3)(0) = 2 + 0 = 2 $$

The new coordinate of the point is $(-3, 2)$, which corresponds to **Option B**.