Discrete Deformations of Shifting Piecewise Sets

1. Shifting Piecewise Sets

Shifting Piecewise Sets introduce a captivating paradigm where sets, in Euclidean $n$ - space, undergo a dynamic transformation through a sequence of rigid motions(shifts). Imagine a curve embedded in space, systematically skipping and shifting regions along its domain. This process, executed $k$ times on diminishing sized intervals, gives rise to a fascinating array of “distorted” sets. The following visual attempts to illustrate this concept:

This post aims to present the mathematics of how these shifts work, and investigate some of their general implications. I intend to keep things rather informal, exploring what I have uncovered so far. I hope to add on to this post, as I continue to work on this project. Feedback on my writing, mathematics, and other helpful insights would be greatly appreciated.

1.1. Definitions

Definition 1.1.1.

Define the Extent Encapsulation Vector to be $\vec\ell(k):\mathbb{N}^{n}\to\mathbb{R}^n,$ and Extent Parameter at $i$ to be $\ell_{i}(k):\mathbb{N}\to\mathbb{R}$ such that $$ \vec{\ell}(k)= \langle \ell_{1}(k), \ell_{2}(k),\dots, \ell_{n}(k) \rangle. $$ $\ell_{i}(k)$ denotes a function related to ‘length,’ dependent on $k.$

Definition 1.1.2. Let $I_{i}\subseteq \mathbb{R}$ be defined by $$I_{i}=\bigcup_{z \in \mathbb{Z}}[2z\cdot \ell_{i}(k),(2z+1)\cdot \ell_{i}(k)].$$

The Transfer Confinement Space, $\mathcal{S}^n_{k}\subseteq \mathbb{R}^n,$ at $k$ is denoted, $$ \mathcal{S}^{n}_{k} = \bigtimes_{i=1}^{n} I_{i}. $$ $\mathcal{S}_{k}^{n}$ is effectively the Cartesian product of all intervals, $I_{i},$ specified by their respective Extent Parameter, $\ell_{i}(k).$

Definition 1.1.3. Define a function, $\psi_{k}:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1},$ such that $$\psi_{k}(x_{1},\dots,x_{n},x_{n+1})=\begin{cases} (x_{1}-2\cdot \ell_{1}(k),\dots,x_{n}- 2 \cdot \ell_{n} (k) , x_{n+1} ) \\ \text{ if } (x_{1},\dots,x_{n}) \in \mathcal{S}^n_{k}. \\ (x_{1},\dots,x_{n},x_{n+1}) \\ \text{ if } (x_{1},\dots,x_{n}) \not \in \mathcal{S}^n_{k}. \end{cases} $$ We call $\psi_{k}$ the Shift Modifier at $k.$ If an element, $X \in \mathbb{R}^{n+1}$ lies in the Transfer Confinement Space, $\mathcal{S}_{k}^{n}$ that point gets shifted, as specified by $\psi_{k}.$

Definition 1.14. We define a function $\Psi_{k}:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1},$ such that for a given $X \in \mathbb{R}^{n+1},$ $$\Psi_{k}(X)=\psi_{k} \circ \psi_{k-1} \circ \dots \circ \psi_{2} \circ \psi_{1}(X).$$ We call $\Psi_{k}$ the Spacial Modification Function for $k$ shifts.

2. First Approach with Curve Shifts

The previous definitions will be applied to the seed $X^{2}\subseteq \mathbb{R}^{2},$ a smooth, connected injection with no holes.

A function, $\ell:\mathbb{N} \to \mathbb{R},$ could be defined such that $\ell(k)$ is strictly decreasing at a rate faster than $\frac{1}{k},$ and $\lim_{ n \to \infty }\ell(k)=0.$

Given $\ell,$ construct the Transfer Confinement Space, at $k,$ $\mathcal{S}_{k}$ such that,

$$ \mathcal{S}_{k}=\{x \in \mathbb{R}:(\forall{z}\in\mathbb{Z})(2z\cdot \ell(k)\leq x\leq(2z+1)\cdot \ell(k)) \}. $$

Each variation of $\mathcal{S}_{k}$ contains a set of smaller, closely condensed intervals, as $k$ increases.

The Shift Modifier, at $k,$ $\psi_{k}:\mathbb{R}^{2} \to \mathbb{R}^{2},$ can now be defined.

$$\psi_{k}(x,y) = \begin{cases} (x- 2 \cdot\ell(k),y) \text{ if } x \in \mathcal{S}_{k} \\ (x,y) \text{ if } x \not \in \mathcal{S}_{k}. \end{cases} $$

$\psi_{k}$ relates an element $(x,y)\in \mathbb{R}^{2}$ to $\mathcal{S}_{k},$ such that if the $x$ component lies in the contained intervals, specified by $\mathcal{S}_{k},$ $\psi_{k}$ shifts the entire coordinate by two lengths of $\ell(k)$ to the left. Otherwise, $(x,y)$ is unchanged.

The Spacial Modification Function, $\Psi_{k}: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is constructed from elements in $\{ \psi_{1}, \psi_{2},\dots,\psi_{k} \},$ such that

$$\Psi_{k}(x,y) = \psi_{k} \circ \psi_{k-1} \circ \dots \circ \psi_{2} \circ \psi_{1}(x,y)$$

$\Psi_{k}$ maps a point, $(x,y)\in\mathbb{R}^{2}$ to a new position, by changing the value of $x,$ for $k$ shifts. Considering $X^{2},$ we can define a different mapping of $\Psi_{k},$ for each $k,$ by $X^{2}_{k}\subseteq \mathbb{R}^{2},$ such that

\begin{align*} X_{1}^{2} &= \{ \Psi_{1}(x,y):(x,y)\in X^{2} \} \\ X^{2}_{2} &= \{ \Psi_{2}(x,y):(x,y)\in X^{2} \} \\ X_{3}^{2} &= \{ \Psi_{3}(x,y):(x,y) \in X^{2} \}. \end{align*}

3. Transformations on the Parabola

Now that a consensus has been established on what this shifting algorithm does, some examples are in order. What follows are some observations, when modifying the set containing the parabola.

3.1

Define the seed to be $X^{2}=\{ (x,x^{2}):x \in \mathbb{R} \},$ and Extent Parameter, on $x$ to be $\ell:\mathbb{N}\to \mathbb{R},$ such that $\ell(k)=2^{-(k-1)}.$ The following are visual representations of $X^2, X^{2}_{1},\dots,X^{2}_{6},$ respectively.

alt alt alt alt alt alt

After reorienting this image, and zooming out we observe the following object:

There would appear to be an approaching symmetry about the point $x=-2.$

Furthermore, one can see that the resulting set is a bijection from $\mathbb{R}\to\mathbb{R}^{2}.$

3.2

Define the same seed, but change the Extent Parameter, on $x$ to be $\ell(k)=2^{-k}\cos(\pi k).$ The following are visual representations of $X^{2},X^{2}_{1},\dots,X^{2}_{6},$ respectively.

alt alt alt alt alt alt

Zooming out reveals the following:

The final image maintains some of the structural symmetry observed in the seed.

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