To address the questions regarding Escherian metamorphosis:

Question 1

The answer is yes. For any two tiles ( A ) and ( B ) in ( \mathbb{R}^2 ) and any ( \epsilon > 0 ), there exists a strong ( \epsilon )-metamorphism. This relies on the path-connectedness of the space of compact connected shapes (with non-empty interior) under the Hausdorff distance. By continuously deforming ...

Short answer

• For the version without the area-scaling in the adjacency condition (your “ε‑metamorphism”), the answer is yes in every dimension n ≥ 2. One can give an explicit construction and a linear-in-1/ε length bound of the form length ≤ C · max{diam(A), diam(B)} · dH(A,B)/ε, with C an absolute constant (and the same proof adapts to ℝn with a dimension-dependent constant).

• For the st...

Short answer to Question 1: “No”.
There are monohedral tiles A and B for which no tiling of the plane can contain both of them, hence no ε-metamorphism – strong or weak – can exist, no matter how large ε is. A concrete pair of such tiles is described below. Once Question 1 is answered in the negative, Question 2 is moot for that pair; for tiles that do admit a metamorphosis the length functio...

Here is a short self-contained answer to both your questions. In fact nothing “deep” is hiding behind the Metamorphosis woodcuts – one can always interpolate two tiles by a finite “strip-subdivision’’ argument – and one shows that the optimal length is indeed of order

[ \ell_{A,B}(\e);\approx;\frac{\max{\diam A,;\diam B},\d_H(A,B)}{\e},. ]

1. **Existence of stron...

This is a fascinating and well-formulated question. The heart of the matter lies in a conflict between the geometric continuity suggested by the Hausdorff metric and the discrete, combinatorial nature of tilings.

Let's break down the questions.

Question 1: Is there always a strong ε-metamorphism?

The answer is almost certainly no.

The fundamental obstacle is the **combinatorial rigidit...

First, recall the definitions:

• A shape is a compact connected set that is the closure of its in...

Throughout we are working in the Euclidean plane.
For a set X write Δ(X)=diam X and a(X)=area X.

1. A very large square is “ε–close’’ to every bounded set

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Lemma 1 For every bounded set E⊂ℝ² and every ε>0 there exists R=R(E,ε)>0 such that the square

    Q(R)= [0,R]×[0,R]

satisfies

    d_H(E , Q(R))   <   ε·a(Q(R))/2. ...

Is "Escherian Metamorphosis" Always Possible?

This question explores a fascinating mathematical formalization of Escher's "Metamorphosis" artwork, where one tiling pattern gradually transforms into another across a plane.

Understanding the Question

The core question asks whether, for any two shapes that can each tile the plane individually (tiles A and B), we can create a gradual transitio...

This is a fascinating question that formalizes Escher's artistic concept of metamorphosis between tessellating patterns. Let me address both questions:

Question 1: Existence of Strong ε-metamorphosis

The answer appears to be yes, though I cannot locate a specific theorem stating this. Here's why this should be true:

Key Observations:

1. Both shapes tile the plane: Since and ...