Sierpinski Triangle

About Sierpinski Triangle

In 1915, Polish mathematician Waclaw Sierpinski published a paper describing a curve that is simultaneously a line, a surface, and a point set with properties that defied the geometric intuitions of his era. The construction he outlined was deceptively simple: begin with an equilateral triangle, connect the midpoints of each side to form four smaller equilateral triangles, remove the central inverted triangle, then repeat the process on each remaining triangle, infinitely. The resulting object -- now called the Sierpinski triangle, Sierpinski gasket, or Sierpinski sieve -- became a foundational example in fractal geometry decades before Benoit Mandelbrot coined the term "fractal" in 1975.

The Sierpinski triangle's Hausdorff dimension is log(3)/log(2), approximately 1.585 -- a value that places it between a one-dimensional line and a two-dimensional plane. At each iteration, three copies of the figure at half scale compose the whole, giving a self-similarity ratio of 1/2 with a replication factor of 3. After infinite iterations, the total area of the remaining triangles converges to zero, while the total perimeter of all boundaries diverges to infinity. This paradox -- a geometric object that is visible, structured, and infinitely detailed yet has no measurable area -- exemplifies why nineteenth-century mathematicians called such constructions "monsters" and why twentieth-century mathematicians recognized them as doorways into a richer understanding of dimension and measure.

Sierpinski himself was working in the tradition of Georg Cantor and Giuseppe Peano, mathematicians who had already demonstrated that naive assumptions about continuity, dimension, and cardinality could be spectacularly wrong. Cantor's middle-third set (1883), which removes the central third of a line segment recursively, is the one-dimensional analogue of the Sierpinski triangle and shares its property of being an uncountable set with zero Lebesgue measure. Sierpinski's contribution extended these ideas into two dimensions and showed that pathological sets were not isolated curiosities but members of a vast family of self-similar structures obeying precise mathematical laws.

What makes the Sierpinski triangle especially significant in the history of mathematics is the sheer number of independent routes that converge on it. The same figure emerges from Pascal's triangle when entries are colored by parity (odd vs. even), from the solution graph of the Tower of Hanoi puzzle, from Stephen Wolfram's elementary cellular automaton Rule 90, from the chaos game algorithm described by Michael Barnsley in 1988, and from the addresses of positions reachable in certain combinatorial games. No other fractal arises from so many unrelated starting points, suggesting that the Sierpinski triangle is not merely an invention of Sierpinski but a deep structural attractor embedded in the fabric of discrete mathematics itself.

The triangle's three-dimensional generalization, the Sierpinski tetrahedron (or tetrix), applies the same recursive removal to a regular tetrahedron, producing an object whose Hausdorff dimension is exactly 2 -- the dimension of a flat surface, despite the object being embedded in three-dimensional space. Karl Menger's sponge (1926) extends the principle to cubes, producing a three-dimensional fractal of dimension log(20)/log(3), approximately 2.727. These generalizations demonstrate that the Sierpinski construction is not bound to any particular dimension but is a universal recursive operation applicable to any regular polytope. The Sierpinski triangle is the simplest member of this family, the two-dimensional seed from which an entire hierarchy of fractal objects grows.

Sierpinski's original 1915 paper, titled "Sur une courbe dont tout point est un point de ramification" (On a curve every point of which is a ramification point), was published in the Comptes Rendus de l'Academie des Sciences in Paris. The title itself reveals what interested Sierpinski most: the topological property of universal ramification, meaning that every point of the set is a branching point from which the curve splits into multiple paths. This property distinguishes the Sierpinski triangle from smoother fractals like the Koch snowflake and places it in a class of maximally branched topological spaces that later became important in the study of continua and dendrites.

Mathematical Properties

The Sierpinski triangle is defined by the iterated function system (IFS) consisting of three affine transformations, each contracting toward one vertex of an equilateral triangle by a factor of 1/2. Formally, if the vertices of the initial triangle are v1, v2, v3, the three maps are f_i(x) = (x + v_i)/2. The unique compact set S satisfying S = f_1(S) union f_2(S) union f_3(S) is the Sierpinski triangle, by the Hutchinson-Barnsley theorem (1981).

The Hausdorff dimension is log(3)/log(2) = 1.58496..., derived from the self-similarity equation: three copies at scale factor 1/2. This places the figure strictly between dimension 1 (a curve) and dimension 2 (a filled region). The topological dimension remains 1 -- the Sierpinski triangle is a topological curve in the sense that removing any single point disconnects it into at most three components. The gap between topological dimension 1 and Hausdorff dimension 1.585 is characteristic of fractal sets.

At iteration n, the number of remaining filled triangles is 3^n and each has side length (1/2)^n times the original. The total remaining area is (3/4)^n times the original area, which converges to 0 as n approaches infinity. The total perimeter at iteration n grows as 3 * (3/2)^n * (original side / 2^n), simplified to a quantity proportional to (3/2)^n, which diverges. The Sierpinski triangle thus has zero Lebesgue measure (zero area) but infinite one-dimensional Hausdorff measure (infinite total boundary length).

The connection to Pascal's triangle modulo 2 is precise: color entry C(n,k) black if it is odd, white if even. The resulting pattern of black cells, when viewed as a bitmap, converges to the Sierpinski triangle as the number of rows increases. This follows from Lucas' theorem (1878), which states that C(n,k) mod p = product of C(n_i, k_i) mod p, where n_i and k_i are the digits of n and k in base p. For p=2, C(n,k) is odd if and only if every binary digit of k is less than or equal to the corresponding digit of n -- the condition that defines membership in the Sierpinski triangle when n and k are interpreted as coordinates.

The chaos game convergence theorem guarantees that a random walk choosing uniformly among the three vertices, advancing halfway at each step, visits every neighborhood of the Sierpinski triangle with probability 1, regardless of starting point. The complement of the Sierpinski triangle (the removed regions) forms an open set of full Lebesgue measure, yet the random walk never enters it after the initial transient. Barnsley proved this by showing the IFS is contractive and the attractor is the unique fixed point of the Hutchinson operator acting on the space of compact subsets.

Occurrences in Nature

The Sierpinski triangle and its close approximations appear in biological, geological, and physical systems where growth or force distribution follows branching rules with consistent scaling ratios.

Bone microstructure, particularly in trabecular (spongy) bone, exhibits fractal branching patterns that approximate the Sierpinski triangle at two to four levels of recursion. Trabecular struts subdivide into smaller struts at roughly half scale, distributing mechanical load across a hierarchical network. Studies using micro-CT imaging have measured fractal dimensions of 1.4-1.7 in healthy trabecular bone, a range that brackets the Sierpinski triangle's theoretical 1.585. Osteoporosis reduces this fractal dimension by removing finer-scale struts, effectively interrupting the recursive subdivision.

The respiratory system of mammals provides another biological approximation. The bronchial tree branches into progressively smaller airways, with each generation roughly half the diameter of the previous one and three primary branches at each major division point in many species. Ewald Weibel's morphometric studies of the human lung (1963) measured 23 generations of branching, and the two-dimensional projection of this branching tree onto a sagittal plane closely resembles a Sierpinski triangle. The fractal dimension of the bronchial tree's cross-section has been measured at approximately 1.57, within 1% of the Sierpinski value.

In physics, the Sierpinski gasket serves as a model substrate for studying anomalous diffusion, vibration modes, and electrical conduction on fractal lattices. Rammal and Toulouse (1983) analyzed random walks on the Sierpinski gasket and showed that the diffusion exponent differs from the Euclidean case, leading to sub-diffusive behavior where the mean-square displacement grows as t^(2/d_w) with the walk dimension d_w = log(5)/log(2) approximately 2.322. This means particles diffusing on a Sierpinski-like substrate spread more slowly than on a flat plane, a prediction that has been experimentally verified in porous media and percolation clusters.

Molecular chemistry has produced literal Sierpinski triangles at the nanometer scale. In 2015, a team led by Michael Newkome at the University of Akron assembled Sierpinski triangles from bis-terpyridine building blocks coordinated with ruthenium and iron ions, creating fractal structures with three levels of self-similarity visible in scanning tunneling microscope images. Each level doubled the edge length while tripling the number of triangular subunits, matching the Sierpinski construction rule exactly. These molecular fractals demonstrated that the Sierpinski geometry can emerge from chemical self-assembly when bonding angles and lengths satisfy the correct ratios.

Seashell pigmentation patterns in certain species of cone snails (genus Conus) display triangular cellular automaton patterns that approximate the Sierpinski triangle. Hans Meinhardt's reaction-diffusion models (1995) showed that these patterns arise when pigment-secreting cells at the shell's growing edge follow activation-inhibition rules equivalent to Rule 90 or similar elementary automata, producing nested triangular motifs across the shell surface.

Electrical discharge patterns in dielectric materials, known as Lichtenberg figures, can approximate the Sierpinski triangle under specific conditions. When a high-voltage pulse is applied to a thin insulating sheet between grounded electrodes arranged in a triangular configuration, the branching discharge channels propagate inward from three directions and subdivide at consistent angles, producing a triangular fractal pattern. The fractal dimension of Lichtenberg figures in acrylic blocks has been measured at 1.5-1.7 depending on voltage and material thickness (Niemeyer, Pietronero, and Wiesmann, 1984). While not all Lichtenberg figures are Sierpinski triangles, the triangular boundary conditions can steer the discharge into Sierpinski-like geometries, demonstrating that the pattern can emerge from electrostatic energy minimization as readily as from biological growth rules or mathematical iteration.

River delta systems viewed from satellite imagery occasionally display triangular branching patterns with self-similar subdivision. The Selenga River delta entering Lake Baikal in Siberia, for example, branches into three primary channels that each subdivide further into secondary and tertiary channels, creating a fan-shaped fractal with measured dimension near 1.5. While river deltas are shaped by sediment transport rather than geometric construction rules, the physical constraints of flow splitting in a confined triangular basin produce convergent geometry with the Sierpinski construction.

Architectural Use

The earliest known architectural use of Sierpinski-like triangular subdivision appears in Cosmati pavements, the ornamental marble floors created by Roman marble workers (marmorarii) between roughly 1100 and 1300 CE. Cosmati masters -- families including the Cosmati, Vassalletto, and Laurentius -- inlaid geometric patterns using cut pieces of porphyry, serpentine, and gilt glass into white marble floors. Several surviving pavements in Roman basilicas, including Santa Maria in Cosmedin and the Cathedral of Anagni, contain triangular motifs subdivided to three or four levels of recursion, producing patterns that closely approximate the Sierpinski triangle. The technique, called opus sectile, required precise cutting at each scale, and the recursive subdivisions served both decorative and symbolic purposes, representing the divine order perceived in geometric progression.

Ethiopian Orthodox Christian processional crosses, carved from single pieces of brass or iron, frequently incorporate triangular lattice patterns that mirror the Sierpinski triangle's self-similar structure. Crosses from the Lalibela region (12th-13th century) show triangular cutouts nested within larger triangular frameworks, sometimes to three levels of recursion. These are not abstract decorations but liturgical objects carried during religious processions, where the fractal openwork both reduced weight and created visually striking patterns of light and shadow.

In South Indian temple architecture, the gopuram (entrance tower) of Dravidian temples sometimes employs recursive triangular forms in the arrangement of sculptural niches. The Brihadeeswara Temple in Thanjavur (completed 1010 CE) and the Meenakshi Temple in Madurai (rebuilt 16th-17th century) both contain stone lattice screens (jali work) with triangular subdivisions reaching two to three levels of recursion. The function is both structural (distributing load across the lattice) and aesthetic (creating dappled light in the interior). Tamil kolam floor drawings extend this tradition into ephemeral domestic art, where women draw recursive triangular patterns in rice powder at household thresholds each morning.

Modern architecture has embraced the Sierpinski triangle explicitly. The Menger sponge sculpture at MIT (1995) and the Sierpinski-inspired facade of the Serpentine Pavilion by Bjarke Ingels Group (2016) use fractal subdivision to create structures that are lightweight, visually complex, and structurally distributed. In structural engineering, Sierpinski truss designs have been studied for their favorable strength-to-weight ratio: the recursive removal of material follows the same paths where stress is lowest, producing frames that are both lighter and more efficient than solid triangular trusses of the same external dimensions.

Antenna design has adopted the Sierpinski triangle for practical electromagnetic reasons. In 1998, Carles Puente and colleagues at the Polytechnic University of Catalonia demonstrated that a Sierpinski gasket antenna exhibits multiband behavior, resonating at frequencies whose ratios correspond to the fractal's scaling factor. A standard monopole antenna resonates at a single frequency determined by its physical length, but the Sierpinski antenna's self-similar geometry creates resonance at multiple harmonically related bands without requiring separate elements for each band. This property has been exploited in mobile phone antenna design, where a single Sierpinski element can cover GSM, WiFi, and Bluetooth frequency bands simultaneously, reducing the number of discrete antennas needed inside the device housing.

Construction Method

The classical construction begins with an equilateral triangle of side length s. Label the vertices A, B, and C. At iteration 0, the triangle is solid, with area (sqrt(3)/4) * s^2.

Iteration 1: Find the midpoints of each side -- M_AB (midpoint of AB), M_BC (midpoint of BC), and M_AC (midpoint of AC). Connect these midpoints to form an inner triangle M_AB-M_BC-M_AC. This inner triangle is equilateral with side length s/2 and is inverted relative to the original. Remove this central triangle, leaving three smaller equilateral triangles, each with side length s/2, positioned at the three corners of the original.

Iteration 2: Apply the same midpoint-subdivision-and-removal process to each of the three remaining triangles. Each one yields three sub-triangles of side length s/4, for a total of 9 filled triangles. The area is now (3/4)^2 times the original.

Iteration n: The figure contains 3^n filled triangles, each with side length s/2^n. Total remaining area = (3/4)^n * original area. As n approaches infinity, the area converges to zero and the boundary length diverges.

The chaos game algorithm (Barnsley, 1988) offers an alternative probabilistic construction. Place three vertices of an equilateral triangle at fixed positions -- for example, (0,0), (1,0), and (0.5, sqrt(3)/2). Choose any starting point p_0 in the plane. At each step, randomly select one of the three vertices (with equal probability 1/3), and set p_(n+1) = midpoint of p_n and the chosen vertex. Plot each point after a burn-in period of approximately 10-20 steps to eliminate transient dependence on the starting position. After several thousand iterations, the plotted points fill in the Sierpinski triangle with arbitrary precision.

Pascal's triangle modulo 2 provides a purely arithmetic construction. Write out Pascal's triangle row by row, computing each entry as the sum of the two entries above it, then reduce every entry modulo 2 (replacing even numbers with 0 and odd numbers with 1). Coloring the 1s black and 0s white produces a bitmap that converges to the Sierpinski triangle as the number of rows increases. After 2^k rows, the pattern matches the Sierpinski triangle at k levels of recursion exactly. This works because C(n,k) mod 2 = 1 if and only if every bit of k in binary is less than or equal to the corresponding bit of n -- the bitwise AND condition that defines the Sierpinski set.

Cellular automaton Rule 90 generates the pattern dynamically. Start with a single black cell on an infinite white row. At each time step, a cell becomes black if exactly one of its two neighbors was black in the previous step (XOR rule). The spacetime diagram -- plotting successive rows downward -- produces the Sierpinski triangle. This was documented by Stephen Wolfram in 1983 and connects the Sierpinski triangle to the broader study of computation, emergence, and the boundary between order and chaos in discrete dynamical systems.

The Tower of Hanoi state graph also encodes the Sierpinski triangle. The puzzle with n disks has 2^n - 1 legal states and 3^n - 3 transitions. The graph of states reachable from the initial configuration, drawn with each group of moves for the smallest disk forming the edges of a triangle, is isomorphic to the Sierpinski triangle at n levels of recursion. This connection was first noted by Ian Stewart and provides a bridge between combinatorial game theory and fractal geometry.

Spiritual Meaning

In Hindu philosophy, the recursive triangular form resonates with the concept of Brahman manifesting through layers of maya (illusion) into increasingly differentiated forms. The downward-pointing triangle (trikona) in Shakta tantra represents Shakti, the creative feminine principle, while the upward-pointing triangle represents Shiva, pure consciousness. The Sierpinski triangle's nesting of triangles within triangles mirrors the tantric teaching that every manifestation contains within it the seed of the whole -- that the macrocosm is encoded in the microcosm. Tamil kolam practitioners describe their recursive triangular patterns as representations of this same principle: each morning's drawing at the threshold recreates the act of cosmic manifestation in miniature.

In the Ethiopian Orthodox tradition, the Sierpinski-like openwork of processional crosses carries Trinitarian symbolism. The three-fold recursion -- three triangles emerging from one, then three from each of those -- mirrors the theological formula of the Trinity: three persons in one God, each fully containing the divine nature. The fractal cross thus becomes a visual meditation on the mystery of divine unity-in-multiplicity. The light passing through the lattice during processions was interpreted as divine light (birhan) filtering through the ordered structure of creation.

Western esoteric traditions, particularly Hermeticism and Freemasonry, have long assigned spiritual meaning to the equilateral triangle as a symbol of harmony, completion, and the triune nature of reality (body-mind-spirit, salt-mercury-sulfur, Father-Son-Holy Spirit). The Sierpinski triangle extends this symbolism into infinite depth. Each removal of the central triangle creates a void surrounded by three smaller wholes, a geometric parable about how emptiness and form depend on each other -- a teaching that parallels the Buddhist concept of sunyata (emptiness) as the ground from which all phenomena arise.

The chaos game's spiritual resonance deserves separate attention. The fact that a completely random process, given nothing more than three reference points and a halving rule, inevitably produces the Sierpinski triangle has struck contemplatives across traditions as a mathematical demonstration of the Hermetic maxim "as above, so below." The structure is not imposed from outside but emerges from within the process itself, much as dharma in the Hindu and Buddhist sense is not an external law but the inherent ordering principle of reality. The randomness of each individual step and the determinism of the emergent pattern together model the interplay between free will and cosmic order that preoccupies philosophical traditions from Stoicism to Vedanta.

From the perspective of the Satyori framework, the Sierpinski triangle illustrates the principle that transformation operates through self-similar processes at every scale of existence. The work of clearing samskaras (deep impressions) follows a recursive pattern: resolving one layer reveals three more beneath it, each at finer resolution, until the practitioner reaches a level of clarity where the remaining material has measure zero -- present as a pattern but no longer obstructing awareness. The triangle's infinite boundary and zero area serve as a precise metaphor for this process: the detail is endless, but the substance that once filled the space has been released.

Significance

Sierpinski's 1915 paper arrived at a moment when mathematics was undergoing a foundational crisis. Cantor's set theory, Russell's paradox, and Hilbert's program were forcing mathematicians to reexamine assumptions that had seemed unshakeable for centuries. The Sierpinski triangle contributed to this upheaval by demonstrating that well-defined geometric processes could produce objects with fractional dimension -- a concept that had no place in Euclidean geometry or standard topology. It was one of the first concrete examples of what Felix Hausdorff would later formalize as Hausdorff dimension, a metric that assigns non-integer values to sets whose complexity falls between conventional dimensions.

The triangle's significance deepened in the 1970s and 1980s when Mandelbrot's fractal geometry program showed that self-similar structures were not mathematical pathologies but accurate descriptions of natural phenomena. Mandelbrot cited the Sierpinski triangle as a canonical example of exact self-similarity, contrasting it with the statistical self-similarity found in coastlines, mountains, and clouds. This shift reframed the triangle from a "monster curve" into a teaching tool, a bridge between pure mathematics and physical observation.

In computer science, the Sierpinski triangle's appearance in Rule 90 cellular automata (documented by Wolfram in 1983) demonstrated that extreme visual complexity can emerge from the simplest possible rules -- a single cell with two states following a three-cell neighborhood function. This insight fed directly into Wolfram's later work on computational irreducibility and the idea that simple programs can generate structures of arbitrary complexity, a theme that runs through his 2002 book A New Kind of Science.

The triangle also appears as a fundamental structure in chaos theory through Barnsley's chaos game, published in Fractals Everywhere (1988). The algorithm works as follows: place a point anywhere in the plane, randomly choose one of three vertices of an equilateral triangle, move halfway toward that vertex, plot the new point, and repeat. After discarding the first few transient points, the plotted points converge on the Sierpinski triangle with probability one, regardless of the starting position. This result stunned many mathematicians because it showed that deterministic structure (a precisely defined fractal) could emerge from a purely random process -- a theme that resonates with phenomena in statistical mechanics, where macroscopic order arises from microscopic randomness.

In education, the Sierpinski triangle has become the standard introductory example in courses on fractal geometry, dynamical systems, and mathematical visualization. Its accessibility -- the construction can be drawn by a child, yet its properties challenge professional mathematicians -- makes it a bridge between recreational mathematics and research-level topology. The tower of Hanoi connection, discovered by Ian Stewart, brought the triangle into combinatorial game theory, while its role as a Julia set for specific rational maps connects it to complex dynamics. Few mathematical objects span this many subfields while remaining intuitive enough to sketch on a napkin.

Connections

The Sierpinski triangle connects to the Mandelbrot set through the shared framework of fractal geometry: both are self-similar structures defined by simple recursive rules, though the Mandelbrot set exhibits statistical self-similarity while the Sierpinski triangle displays exact self-similarity at every magnification. Mandelbrot himself used the Sierpinski triangle as a pedagogical stepping stone toward the more complex fractals generated by iteration in the complex plane.

The relationship to Fibonacci numbers emerges through combinatorial channels. When Pascal's triangle is reduced modulo 2, the resulting binary pattern of odd and even entries reproduces the Sierpinski triangle exactly. Pascal's triangle also encodes binomial coefficients, which are intimately connected to Fibonacci numbers through diagonal summation (each Fibonacci number equals the sum of specific diagonal entries). The Sierpinski pattern therefore sits at a junction point between number theory, combinatorics, and fractal geometry.

The Flower of Life and the Sierpinski triangle share the property of emerging from the recursive application of a single geometric operation -- circle packing in one case, triangle subdivision in the other. Both patterns appear in medieval sacred architecture, and both encode information about spatial tessellation and symmetry. The Cosmati pavements of 12th- and 13th-century Roman churches contain both patterns in close proximity, suggesting that medieval craftsmen recognized a kinship between recursive circular and triangular motifs.

The Platonic solids connect through the Sierpinski triangle's three-dimensional extension: the Sierpinski tetrahedron (or tetrix), which applies the same recursive removal process to a regular tetrahedron. The resulting object has a Hausdorff dimension of 2 -- the same as a flat surface -- despite being embedded in three-dimensional space. Each face of the Sierpinski tetrahedron is a Sierpinski triangle, making the tetrahedron (the simplest Platonic solid) the natural host for this fractal in three dimensions.

In Hindu and South Indian traditions, the Sri Yantra and kolam floor patterns share the Sierpinski triangle's recursive nesting logic. Tamil kolam designs, drawn daily at household thresholds, frequently employ self-similar triangular subdivisions as symbols of completeness and infinite renewal. The parallel is structural rather than coincidental: both the mathematical fractal and the ritual pattern encode the idea that complexity and wholeness emerge from the repetition of a simple generative act.

The Celtic knot tradition shares the Sierpinski triangle's emphasis on interconnection without beginning or end. While Celtic knotwork achieves this through interlaced curves, the Sierpinski triangle achieves it through nested self-reference: every part contains the whole, and the whole is nothing but the arrangement of its parts. Both motifs have been used across their respective traditions to symbolize eternity, the interpenetration of scales, and the unity underlying apparent multiplicity.

The Koch snowflake sits at the opposite end of the fractal spectrum from the Sierpinski triangle in topological terms. Both are constructed by recursive operations on a triangle, but the Koch snowflake adds material to its boundary while the Sierpinski triangle removes material from its interior. The result is a striking contrast: the Koch snowflake encloses a finite area with an infinite perimeter, while the Sierpinski triangle has a perimeter that diverges to infinity around an area that converges to zero. Sierpinski's interest in universal ramification points marks his triangle as maximally branched, while Helge von Koch's curve (1904) is everywhere continuous but nowhere differentiable -- two different ways for a deterministic process to produce structure that defies classical calculus.

The Julia set connection runs through complex dynamics. Certain rational maps in the complex plane produce Julia sets whose underlying skeleton is the Sierpinski triangle, the most famous being the family of maps studied by Curt McMullen and others in the 1980s and 1990s. This places the Sierpinski triangle inside the broader landscape of attractors generated by iteration on the Riemann sphere, alongside the Mandelbrot set and the entire zoo of complex fractals. The same recursive logic that builds the Sierpinski triangle by removal in real space generates it as a basin boundary in complex space -- evidence that the figure is a genuine mathematical attractor rather than an artifact of any single construction.

The Sierpinski triangle's three-dimensional cousin, the Sierpinski tetrahedron, makes the tetrahedron the natural seed for fractal extension into solid space. Each face of the Sierpinski tetrahedron is a Sierpinski triangle, so the entire object can be understood as four interlocked copies of the two-dimensional fractal. The Hausdorff dimension of the Sierpinski tetrahedron is exactly 2, meaning that despite occupying three-dimensional space it has the dimensional measure of a flat surface -- one of the cleanest demonstrations that integer dimensions are an approximation rather than a fundamental property of geometric objects.

Further Reading