Famous Fractals
The Sierpiński Triangle, Explained
An equilateral triangle, a simple midpoint rule repeated forever, and zero area remaining — how one Polish mathematician's 1915 curiosity became one of the most recognizable fractals in all of mathematics.
What Is the Sierpiński Triangle?
The Sierpiński triangle — also called the Sierpiński gasket or Sierpiński sieve — is one of mathematics' most iconic self-similar fractals. Its silhouette is immediately recognizable: an equilateral triangle whose interior is riddled with a constellation of triangular holes at every scale, each hole itself surrounded by smaller holes, all the way down to infinitesimal detail.
What makes it so compelling to mathematicians and newcomers alike is the mismatch between its apparent complexity and the extreme simplicity of the rule that produces it. The rule is: take any filled equilateral triangle, connect the midpoints of its three sides to form a smaller central triangle, and remove that central triangle. Then repeat the same operation on every filled triangle that remains. Iterate this forever, and the Sierpiński triangle appears.
Despite its visual richness, the resulting shape has a topological dimension of 1 and a Hausdorff (fractal) dimension of log(3)/log(2) ≈ 1.585 — more complex than a curve but less space-filling than a plane. Its total area, by any measure, is exactly zero.
For context on why fractional dimensions matter, see our pillar guide What Is a Fractal?, and for a broader survey of self-similar geometric shapes, visit The Koch Snowflake.
How Do You Construct the Sierpiński Triangle?
There are three entirely different routes to the same destination, and each one illuminates a different facet of the object's deep structure.
Method 1: The Classical Midpoint Removal
This is the construction Wacław Sierpiński described in his 1915 paper. Start with a solid equilateral triangle. Find the midpoints of all three sides and connect them, dividing the original into four congruent smaller triangles. Remove the central one. You now have three filled triangles. Apply the same rule to each. After n iterations, you have 3n filled triangles, each with side length (1/2)n times the original. The total filled area after step n is (3/4)n times the original — a number that collapses to zero as n approaches infinity. Infinite iterations leave infinite structure but no area whatsoever.
Method 2: The Chaos Game
The chaos game, formalized by mathematician Michael Barnsley, seems at first like it has nothing to do with the classical construction. The algorithm:
- Mark three vertices of a triangle anywhere in a plane.
- Choose a random starting point inside the triangle.
- Roll a three-sided die to pick one vertex at random.
- Move halfway from the current point to that vertex. Plot the new position.
- Repeat from step 3, indefinitely.
After discarding the first few points, the plotted cloud of tens of thousands of random midpoints converges unmistakably to the Sierpiński triangle. Pure randomness, iterated under a deterministic distance rule, sculpts the same fractal that Sierpiński found by orderly removal. This is the hallmark of an iterated function system (IFS): the attractor is fixed regardless of starting conditions.
Method 3: The Self-Referential Equation
The third route is conceptual. The Sierpiński triangle is the unique compact set S satisfying the self-referential equation: S equals three half-sized copies of itself, each placed at one corner. In other words, the whole equals three half-sized copies of itself, perfectly tiled with no overlap. This is the Hutchinson equation, and it encapsulates why every zoom into any sub-triangle reveals an exact replica of the whole — self-similarity is not a coincidence but the definition of the object.
What Is the Fractal Dimension of the Sierpiński Triangle?
Fractal dimension is the rigorous answer to the question: How much space does this shape fill? Ordinary geometry gives clean integer answers — a line is 1-dimensional, a filled square is 2-dimensional. Fractals occupy the uncanny territory in between.
For the Sierpiński triangle, the calculation is elegant. The shape is self-similar: it consists of three copies of itself, each scaled down by a factor of 1/2. The Hausdorff (fractal) dimension d satisfies the equation 3 = 2d, which gives d = log(3)/log(2) ≈ 1.585.
Intuitively: if you double the linear size of the Sierpiński triangle, you get exactly 3 copies, not 4 (as a solid two-dimensional square would give). That shortfall — 3 instead of 4 — is why the dimension falls at 1.585 rather than 2. The fractal is too irregular to fill a plane, but it is also far more intricate than any smooth curve. See the comparison table below for how this places it among classical fractals.
| Fractal | Scaling factor | Self-similar copies | Hausdorff dimension |
|---|---|---|---|
| Cantor set | 1/3 | 2 | log(2)/log(3) ≈ 0.631 |
| Sierpiński triangle | 1/2 | 3 | log(3)/log(2) ≈ 1.585 |
| Koch snowflake boundary | 1/3 | 4 | log(4)/log(3) ≈ 1.262 |
| Menger sponge | 1/3 | 20 | log(20)/log(3) ≈ 2.727 |
Who Was Wacław Sierpiński, and Why Did He Study This Shape?
Wacław Sierpiński (1882–1969) was one of the most prolific Polish mathematicians of the twentieth century, contributing foundational results to set theory, topology, and number theory. He published over 700 papers and 50 books across a career that survived two world wars and the devastation of Warsaw.
Sierpiński introduced the construction in a 1915 paper, and his original goal was not to create a fractal in the modern sense. He was exploring a philosophical puzzle in point-set topology: could one construct a curve such that every single point on it was a branch point — a junction where the curve split in multiple directions? The triangular gasket answered that question in the affirmative.
The pattern itself, however, predates Sierpiński by centuries. The same triangular motif appears in 13th-century Italian mosaic floors in medieval Rome — long before anyone had the vocabulary of set theory to describe what they had carved in marble. Sierpiński gave the pattern its mathematical identity; the craftsmen of Rome simply found it beautiful.
What Is the Hidden Connection Between Pascal's Triangle and the Sierpiński Triangle?
One of the most surprising results in elementary mathematics is that two objects from completely different domains — a number pyramid from combinatorics and a geometric fractal from topology — are secretly the same thing.
The connection works as follows. Take Pascal's triangle — the famous array where each entry is the sum of the two entries above it. Now color each entry based on whether it is odd (dark) or even (light). The visual pattern that emerges, as the triangle grows row by row, converges unmistakably to the Sierpiński triangle.
The mathematical explanation rests on Kummer's theorem, which describes when a binomial coefficient is divisible by a prime p. For p = 2, the odd entries of Pascal's triangle correspond exactly to positions where there are no carrying operations when adding row and column indices in binary. The fractal structure of those positions reproduces the Sierpiński gasket precisely. More concretely, the proportion of odd entries in the first 2n rows grows as 3n/4n, directly encoding the dimension ratio log(3)/log(2).
The same parity rule governs cellular automaton Rule 90: starting from a single active cell and applying the XOR rule (each cell becomes the exclusive-or of its two neighbors) at each time step generates a space-time diagram that is a perfect discrete approximation of the Sierpiński triangle. Pascal mod 2, Rule 90, and the Sierpiński gasket are all the same mathematical object wearing different clothes. The fractal is not a shape so much as a computational principle that recurs wherever self-referential binary logic lives.
Where Does the Sierpiński Triangle Appear Outside the Textbook?
Unlike some mathematical abstractions that remain comfortably theoretical, the Sierpiński triangle keeps materializing in unexpected places across science and engineering.
DNA Nanotechnology
In 2004, researchers published a landmark study demonstrating that DNA tiles can self-assemble into Sierpiński triangles at the nanoscale. Engineered DNA tiles encode an XOR rule in their sticky ends; as tiles bind to neighbors in solution, they compute the rule and grow into a fractal pattern visible under atomic force microscopy. The experiment proved that molecular self-assembly can implement arbitrary computation — with the Sierpiński triangle as the demonstration calculation, and a result that linked fractal mathematics directly to the emerging field of DNA computing.
Fractal Antennas
Sierpiński-geometry antennas exploit the triangle's self-similar edge structure to achieve multi-band resonance: the same physical antenna responds efficiently to multiple frequencies because its fractal boundary repeats the same geometric motif at multiple scales. This design principle has been deployed in compact antennas used in mobile devices and broadband wireless infrastructure, where packing multi-band performance into a small form factor is a core engineering constraint.
Cellular Automata and Computation Theory
Rule 90, mentioned above, is more than a curiosity. It is one of the simplest one-dimensional cellular automata that exhibit complex recursive behavior, and variants of it appear in the analysis of computational universality. Conway's Game of Life replicator patterns also produce Sierpiński-triangle space-time diagrams, linking the fractal to fundamental questions about emergence in simple rule systems. You can explore the computational dimension further in our article on Chaos Theory and Fractals.
Art, Architecture, and Decoration
Long before Sierpiński's paper, decorative artists produced the pattern in Cosmati mosaic floors of medieval Roman churches, and echoes of it appear in various lattice and textile traditions across cultures. The shape's intrinsic visual balance — each level of detail mirroring the whole — makes it satisfying across cultural contexts, consistent with what physicist Richard Taylor calls “fractal fluency”: the measurable physiological comfort humans derive from viewing fractal patterns at mid-complexity (dimensions roughly 1.3–1.8). The Sierpiński triangle, at dimension 1.585, sits comfortably within this calming range.
How Do You Draw the Sierpiński Triangle by Hand?
Five hand-drawn iterations on graph paper are enough to make the structure unmistakably clear. Here is the procedure:
- Draw a large equilateral triangle. On graph paper, a triangle with a base of 16 grid squares provides five comfortable iterations.
- Find the three midpoints of the sides and connect them. You now have four congruent smaller triangles inside the original.
- Leave the central triangle empty — shade it lightly or leave it blank. The three corner triangles remain filled.
- Repeat for each filled triangle. Apply the same midpoint rule to each of the three corner triangles to get nine filled triangles.
- Continue for three to five total iterations. By iteration five, you have 35 = 243 tiny triangles, and the Sierpiński pattern is clearly visible.
Alternatively, try the chaos game version: mark three vertex points, pick any starting dot, and repeatedly plot the midpoint between the current dot and a randomly chosen vertex. After several thousand points the fractal emerges from pure chance — a meditative exercise in emergence that requires nothing more than a pencil and patience, and a direct demonstration that order can arise from randomness.
Frequently asked
What is the Sierpiński triangle?
The Sierpiński triangle — also called the Sierpiński gasket or Sierpiński sieve — is a self-similar fractal built by starting with a solid equilateral triangle and repeatedly removing the smaller central triangle formed by connecting the midpoints of each side. After infinite iterations, the resulting shape has infinite structural detail but zero total area. It was formally described by Polish mathematician Wacław Sierpiński in 1915, though decorative versions appeared in 13th-century Italian Cosmati mosaics centuries earlier. Its Hausdorff (fractal) dimension is approximately 1.585, placing it mathematically between a one-dimensional curve and a two-dimensional plane.
What is the fractal dimension of the Sierpiński triangle, and what does it mean?
The Sierpiński triangle has a Hausdorff dimension of log(3)/log(2) ≈ 1.585. This non-integer value arises because the shape consists of exactly three copies of itself, each scaled by a factor of 1/2 — so doubling the triangle's size gives you 3 copies, not the 4 you would get from a solid two-dimensional square. The dimension of 1.585 measures how completely the fractal fills space: more complex than any smooth curve (dimension 1) but less space-filling than a solid region (dimension 2). As iteration depth increases toward infinity, the total area converges to zero while the boundary complexity grows without bound — a striking example of infinite detail coexisting with zero measure.
How is the Sierpiński triangle connected to Pascal's triangle?
If you take Pascal's triangle and color entries odd versus even, the resulting pattern is a discrete approximation of the Sierpiński triangle that converges to the exact fractal as the number of rows grows toward infinity. The mathematical reason is Kummer's theorem: a binomial coefficient C(m+n, m) is odd if and only if there are no carrying operations when adding m and n in binary. The same parity structure governs cellular automaton Rule 90 — the XOR of neighbors rule — which also generates the Sierpiński triangle, connecting combinatorics, geometry, and computation theory through a single underlying principle of binary self-reference.
What is the chaos game, and why does it produce the Sierpiński triangle?
The chaos game constructs the Sierpiński triangle through pure randomness. Place three vertices of a triangle in the plane and pick a random starting point. Then: (1) randomly choose one of the three vertices; (2) move to the midpoint between the current point and that vertex; (3) plot the new point; and (4) repeat. After discarding the first few points and plotting tens of thousands of iterations, the random scatter converges unmistakably to the Sierpiński triangle. This works because the three midpoint transformations form an iterated function system (IFS) whose unique mathematical attractor is the Sierpiński gasket, regardless of starting position or the random order of steps — a profound demonstration that deterministic structure can emerge from randomness.
Does the Sierpiński triangle appear in real-world science and engineering?
Yes, in several remarkable domains. In DNA nanotechnology, a 2004 study (published in PLOS Biology) demonstrated that engineered DNA tiles encoding an XOR rule self-assemble into Sierpiński triangles visible under atomic force microscopy — the first molecular-scale fractal computation. In antenna engineering, Sierpiński-geometry antennas resonate efficiently at multiple frequencies because of their self-similar edge structure, enabling compact multi-band antenna designs. In cellular automata, Rule 90 and certain Conway's Game of Life configurations produce Sierpiński triangle space-time diagrams. And decoratively, the triangular gasket pattern appears in 13th-century Roman Cosmati floor mosaics, centuries before any mathematical formalization.
How can I draw the Sierpiński triangle by hand?
Five hand-drawn iterations on graph paper are enough to make the structure clearly visible. Start with a large equilateral triangle (a 16-unit base works well). Connect the midpoints of the three sides, dividing the original into four congruent sub-triangles, then shade or remove the central one. Repeat this operation for each of the three remaining filled triangles, producing nine filled triangles. Continue for three to five rounds total — by iteration five you have 35 = 243 tiny triangles arranged in the unmistakable Sierpiński pattern. You can also try the chaos game: label three vertex points and repeatedly mark the midpoint between your current dot and a randomly chosen vertex. After several thousand points the Sierpiński structure emerges entirely from chance.