"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological dimension. Is there a class of shapes that have a Hausdorff dimension less than their topological dimension, or is this impossible? If there is such a shape, what are common examples of them? If this is impossible, why?
2026-03-25 10:57:13.1774436233
Is it possible to have a Hausdorff dimension less than the topological dimension?
683 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in DIMENSION-THEORY-ANALYSIS
- Codimension of intersection of zero sets of polynomials.
- How many points define a sphere of unknown radius?
- Some problems related to unirational varieties
- Generate uniformly distributed points in n-dimensional sphere
- Dimension of solutions of EDP
- Does the boundary of the Mandelbrot set $M$ have empty interior?
- A one-dimensional Peano continuum that is not embeddable into $\mathbb{R}^3$
- Embedding preference orders in 2D Euclidean space
- Can a variety "of dimension $\geqslant 1$" be finite?
- Splitting $\mathbb{R}^n$ into two subspaces
Related Questions in HAUSDORFF-MEASURE
- example; $H^t(K)=\infty$, and $H^s(K)=0$ for all $s > t$
- Computing Lebesgue and Hausdorff Integrals
- Integral and measures on manifolds
- Equality of Hausdorff dimension
- Hausdorff Dimension of Julia set of $z^2+2$?
- Why do we need to calculate dimensions?
- Is Hausdorff outer measure sigma finite when restrict on a set with same dimension
- Upper Bounding Hausdorff Measure
- Why is Hausdorff measure Borel regular?
- Relationship between the induced measure on an orbit and Hausdorff measure on the orbit
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
The shapes you are asking about do not exist. The reason is:
Theorem. (Sznirelman) For every metric space $X$, the Hausdorff dimension of $X$ is $\ge$ the covering dimension of $X$.
See for instance section VII.2 of
W.Hurewicz, H.Wallman, Dimension Theory, Princeton University Press.