Two Constructions, Two Separation Behaviours
The Sierpinski gasket and the four-corner Cantor dust are generated by repeatedly retaining smaller copies of an initial shape. Their dimensions can be calculated from the number of copies and their contraction ratios, provided the construction satisfies an appropriate separation condition. The examples also illustrate an important distinction: the gasket’s first-level pieces touch, while the dust’s first-level pieces are a positive distance apart. Both satisfy the open set condition, which is enough for the Moran formula.
Work in the Euclidean plane \(\mathbb{R}^2\). For \(s\geq 0\), let \(\mathcal{H}^s\) denote \(s\)-dimensional Hausdorff measure, and use the Euclidean metric to define diameters. Hausdorff dimension is the critical exponent at which Hausdorff measure changes from infinite to zero: for a set \(K\), \(\dim_H K=\inf\{s\geq0:\mathcal{H}^s(K)=0\}\). Each \(\mathcal{H}^s\), restricted to Borel sets, is a measure on \((\mathbb{R}^2,\mathcal{B}_{\mathbb{R}^2})\); it is not a finite measure on the whole plane. The sets considered below are compact.
The Dimension Equation
The earlier material on similarity dimension and the Moran formula establishes the following result: for a finite system of similarities on \(\mathbb{R}^2\) satisfying the open set condition, the Hausdorff dimension of its attractor is the unique \(s\geq0\) satisfying \(\sum_{i=1}^N r_i^s=1\). This is the version used here; no claim is made that this equation determines dimension when the separation hypothesis fails.
In the equal-ratio case, where every ratio is \(r\), the equation is \(Nr^s=1\). The next observation justifies that this equation has exactly one nonnegative solution whenever \(N\geq2\) and \(0<r<1\).
Proof. Since \(0<r<1\), \(\ln r<0\). Taking logarithms in \(Nr^s=1\) gives \(\ln N+s\ln r=0\), so any solution must be \(s=-\ln N/\ln r=\ln N/\ln(1/r)\), which is positive because both \(\ln N\) and \(\ln(1/r)\) are positive. Substitution gives \(r^s=\exp(s\ln r)=\exp(-\ln N)=1/N\), hence \(Nr^s=1\). The logarithmic equation forces every solution to equal this value, proving uniqueness. \(\square\)
Worked Example: The Sierpinski Gasket
Let \(v_1,v_2,v_3\) be the vertices of an equilateral triangle of side length \(1\). Define \(S_i(x)=(x+v_i)/2\) for \(i=1,2,3\). Each map is a similarity with ratio \(1/2\). The attractor is the Sierpinski gasket. First verify the open set condition, then calculate its Hausdorff dimension.
Take \(U\) to be the interior of the original triangle. Every point \(x\in U\) has barycentric coordinates \((\lambda_1,\lambda_2,\lambda_3)\), where all \(\lambda_j>0\) and \(\lambda_1+\lambda_2+\lambda_3=1\). Under \(S_i\), the new \(i\)-th barycentric coordinate is \((1+\lambda_i)/2\), and the other two coordinates are \(\lambda_j/2\). All three coordinates remain positive, so \(S_i(U)\subset U\). Also, every point of \(S_i(U)\) has \(i\)-th coordinate greater than \(1/2\). If a point belonged to both \(S_i(U)\) and \(S_j(U)\) for distinct \(i,j\), its \(i\)-th and \(j\)-th coordinates would both be greater than \(1/2\), contradicting that the three coordinates sum to \(1\). Thus the three open images are pairwise disjoint.
The Moran equation is \(3(1/2)^s=1\). Solving gives
For the decimal, \(\ln3\approx1.098612\) and \(\ln2\approx0.693147\), so their quotient is approximately \(1.584962\). Check: \(2^s=3\) by the exact logarithmic formula, and therefore \(3(1/2)^s=3/3=1\), as required. The first-level compact gasket pieces meet at the midpoints of the original triangle’s sides, so they are not strongly separated. Their touching does not obstruct the open set condition, which concerns the disjoint open images \(S_i(U)\), not disjointness of the compact pieces.
Worked Example: The Four-Corner Cantor Dust
Start with the square \([0,1]^2\) and use four similarities of ratio \(1/3\), one mapping the square into each corner subsquare. Explicitly, take \(S_{a,b}(x,y)=((x+2a)/3,(y+2b)/3)\), where \(a,b\in\{0,1\}\). The attractor is the four-corner Cantor dust. Verify separation and find its dimension.
Use \(U=(0,1)^2\). Each image \(S_{a,b}(U)\) is one of the four open squares with coordinate intervals either \((0,1/3)\) or \((2/3,1)\) in each coordinate. Each image lies in \(U\). For two distinct choices of \((a,b)\), at least one coordinate interval differs; the two intervals in that coordinate are disjoint. Hence the open images are pairwise disjoint, and the open set condition holds. More strongly, the compact first-level pieces lie in the corresponding closed corner squares. If two such squares differ in the first coordinate, their first-coordinate intervals have gap \(2/3-1/3=1/3\); if they differ in the second coordinate, the second-coordinate intervals have the same gap. Thus the compact pieces are pairwise separated by positive distance.
There are \(4\) copies, each with ratio \(1/3\), so the dimension equation is \(4(1/3)^s=1\). Therefore
Using \(\ln4\approx1.386294\) and \(\ln3\approx1.098612\), the quotient is approximately \(1.261859\). Check: the exact formula gives \(3^s=4\), so \(4(1/3)^s=4/4=1\). This satisfies the Moran equation, and the verified open set condition licenses its use for the attractor’s Hausdorff dimension.
Checking the Critical Exponents by Scales
The formula can be checked against the natural covers at each construction stage. A stage-\(n\) piece is the image obtained by composing \(n\) of the defining similarities. The pieces cover the attractor, and their diameters shrink geometrically. The calculation below checks the total \(s\)-power of the diameters at the proposed critical exponent. It is a consistency check on the formula, not a replacement for the lower-dimension conclusion supplied by the Moran formula.
Worked Example: Level Covers at the Critical Dimensions
For the gasket, stage \(n\) has \(3^n\) equilateral triangles of side length \(2^{-n}\). An equilateral triangle’s diameter is its side length, so the sum of the \(s\)-powers of these diameters is \(3^n(2^{-n})^s\). At its critical exponent \(s=\ln3/\ln2\), the identity \(2^s=3\) gives
For the dust, stage \(n\) has \(4^n\) squares of side length \(3^{-n}\), hence diameter \(\sqrt{2}\,3^{-n}\). At \(t=\ln4/\ln3\), the identity \(3^t=4\) gives
In each expression the dependence on \(n\) cancels at the proposed critical exponent. In particular, for the gasket the stage-\(n\) sum is exactly \(1\); for the dust it is exactly \((\sqrt{2})^t\). The latter constant is the diameter factor from using square diameters rather than side lengths and does not change the exponent. These identities confirm the scale balance in both dimension calculations.
What Separation Does—and a Common Error
For both constructions, the copy count and ratio determine the candidate exponent. The separation check determines whether the Moran formula applies. For the gasket, the open images inside the triangle’s interior are disjoint, even though the compact pieces touch. For the dust, even the compact pieces are positively separated. It would therefore be incorrect to demand strong separation in order to use the open set condition.
A related computational mistake is to use the number of surviving copies as the contraction ratio. For the gasket, \(3\) is the number of copies and \(1/2\) is the ratio; the equation is \(3(1/2)^s=1\), not \(3(3/2)^s=1\). The latter would have no nonnegative solution, since \(3(3/2)^s\geq3\) for all \(s\geq0\). For the dust, the corresponding data are \(4\) and \(1/3\), giving \(4(1/3)^s=1\). Keeping the count and the scale in their separate roles is essential.
The two dimensions are different because the constructions retain different proportions of their initial shapes at every stage: the gasket has three half-size copies, while the dust has four one-third-size copies. The separation condition is not a cosmetic detail. Without it, the equation still produces a similarity-dimension candidate, but the dimension conclusion need not follow; overlaps are treated separately in the later material.
Check Your Understanding
For each question, distinguish the numerical dimension calculation from the separation hypothesis that permits it.
- For the three-map gasket construction, verify the open set condition using the barycentric-coordinate description of the images of the triangle interior.
- Why does the fact that two compact gasket pieces meet at a vertex not contradict the open set condition?
- Write the Moran equation for the four-corner dust and solve it for its Hausdorff dimension.
- At the gasket’s critical exponent \(s\), simplify the stage-\(n\) sum \(3^n(2^{-n})^s\).
- State which of the two constructions has pairwise positively separated compact first-level pieces, and justify the separation using its coordinate intervals.