Fractal Symmetry Explorer

Math Geometry • Transformations and Symmetry

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

Fractal Symmetry Calculator – Self-Similarity in Mandelbrot/Sierpinski

Explore symmetry and self-similarity in classic fractals. Render the Mandelbrot set (escape-time iterations on the complex plane) or a Sierpiński triangle (iterated subdivision), then zoom/pan and toggle symmetry overlays.

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

Fractal & rendering

Fractal: Iterations / depth: Render quality: Color scheme: Symmetry overlay:

Canvas is interactive: drag to pan • wheel/trackpad to zoom • pinch on touch (if supported). Use “Reset view” to return to the default framing.

View (center & zoom)

Center \(c_x\) (real): Center \(c_y\) (imag): Zoom (higher = closer): Escape radius (bailout):

Animation progress:

Animation shows an interpolation between the default view and your current view. (Useful to “feel” self-similarity when you zoom in/out.)

Ready

Fractal view (zoomable • interactive)

Tip: For Mandelbrot symmetry, try mirroring the center \(c_y\to -c_y\). For Sierpiński, toggle symmetry overlay to see the \(D_3\) axes.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory

Fractals, self-similarity, and symmetry

A fractal is a structure with detail at many scales. The key idea is self-similarity: parts of the object resemble the whole, either exactly (mathematically perfect) or approximately (in many natural/complex systems).

A symmetry is a transformation (like rotation or reflection) that leaves an object unchanged. Fractals can have symmetries too, and those symmetries can persist across zoom levels—hence the intuition of “infinite symmetry.”

Mandelbrot set: escape-time iteration and conjugate symmetry

The Mandelbrot set is defined on the complex plane. For each complex number \(c\), define a sequence:

\[ z_0 = 0,\qquad z_{n+1} = z_n^2 + c. \]

If the sequence remains bounded, then \(c\) is in the Mandelbrot set. In practice, we use an escape test: if \(|z_n| > R\) for some bailout radius \(R\ge 2\), the orbit will diverge and we say it “escapes.”

\[ \text{If }|z_n|>R\ (\text{often }R=2),\ \text{then }c\notin M. \]

Symmetry: The Mandelbrot set is symmetric about the real axis. This comes from complex conjugation: if \(z_{n+1}=z_n^2+c\), then taking conjugates gives \(\overline{z}_{n+1} = \overline{z}_n^{\,2} + \overline{c}\). Therefore, \(c\in M\Rightarrow \overline{c}\in M\), which is reflection across \(\operatorname{Im}(c)=0\).

Zooming often reveals repeated “mini-Mandelbrots” and filament patterns—self-similarity that can be striking even though it’s not perfectly uniform everywhere.

Sierpiński triangle: exact self-similarity and dihedral symmetry \(D_3\)

The Sierpiński triangle begins with an equilateral triangle. Each step subdivides it into 4 congruent triangles and removes the middle one; the process repeats on the remaining 3.

This fractal has exact self-similarity: each corner contains a scaled copy of the entire figure, scaled by a factor \(1/2\). After depth \(d\), the number of smallest triangles is:

\[ N(d)=3^d. \]

Symmetry: Because the construction treats the three corners equally, the Sierpiński triangle has the same symmetries as an equilateral triangle. Its symmetry group is the dihedral group \(D_3\) (order \(6\)): 3 rotations (\(0^\circ,120^\circ,240^\circ\)) and 3 reflections (across the three axes through a vertex and the opposite side midpoint).

Zoom as a mathematical idea

Zooming is a scaling transformation: you rescale the coordinate system to look at a smaller region in more detail. In fractals, rescaling often uncovers structures that resemble the original—this is the geometric heart of self-similarity.

In this explorer, the “Symmetry overlay” draws symmetry axes that should remain meaningful regardless of zoom level.

Frequently Asked Questions

What does the Fractal Symmetry Explorer show for the Mandelbrot set?

It renders the Mandelbrot set by iterating z_{n+1} = z_n^2 + c and coloring points by how quickly the orbit escapes beyond the chosen escape radius. Changing center (cx, cy), zoom, and iterations lets you examine self-similar regions at different scales.

Why is the Mandelbrot set symmetric about the real axis?

The Mandelbrot set is symmetric across Im(c) = 0 because complex conjugation preserves the iteration rule: if c is in the set, then its conjugate c-bar is also in the set. Mirroring cy to -cy produces the reflected view.

What does depth mean for the Sierpinski triangle?

Depth is the number of subdivision steps applied to the starting equilateral triangle. Each step keeps 3 smaller corner triangles, so the number of smallest triangles grows as 3^depth.

What symmetry does the Sierpinski triangle have?

Because the construction treats the three corners equally, it has the same symmetries as an equilateral triangle: 3 rotations (0, 120, 240 degrees) and 3 reflections across the three symmetry axes. The symmetry overlay highlights these axes while you zoom.

How do render quality and iterations affect what I see?

Higher render quality increases pixel detail, and higher iterations improve boundary accuracy for the Mandelbrot set and add finer structure for both fractals. If performance slows down, reduce quality or iterations/depth and render again.

Fractal Symmetry Calculator – Self-Similarity in Mandelbrot/Sierpinski

Rate this calculator

Frequently Asked Questions

Related calculators