# 1. Shifting Piecewise Sets

*Shifting Piecewise Sets* introduce a captivating paradigm
where sets, in Euclidean
- 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
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 Vectorto be andExtent Parameterat to be such that denotes a function related to ‘length,’ dependent on

Definition 1.1.2.Let be defined byThe

Transfer Confinement Space,at is denoted, is effectively the Cartesian product ofallintervals, specified by their respectiveExtent Parameter,

Definition 1.1.3.Define a function, such that We call theShift Modifierat If an element, lies in theTransfer Confinement Space,that point gets shifted, as specified by

Definition 1.14.We define a function such that for a given We call theSpacial Modification Functionfor shifts.

# 2. First Approach with Curve Shifts

The previous definitions will be applied to the *seed*
a smooth, connected injection with no holes.

A function, could be defined such that is strictly decreasing at a rate faster than and

Given
construct the *Transfer Confinement Space,* at
such that,

Each variation of contains a set of smaller, closely condensed intervals, as increases.

The *Shift Modifier,* at
can now be defined.

relates an element to such that if the component lies in the contained intervals, specified by shifts the entire coordinate by two lengths of to the left. Otherwise, is unchanged.

The *Spacial Modification Function,*
is constructed from elements in
such that

maps a point, to a new position, by changing the value of for shifts. Considering we can define a different mapping of for each by such that

# 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
and *Extent Parameter,* on
to be
such that
The following are visual representations of
respectively.

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

There would appear to be an approaching symmetry about the point

Furthermore, one can see that the resulting set is a bijection from

## 3.2

Define the same *seed,* but change the *Extent
Parameter,* on
to be
The following are visual representations of
respectively.

Zooming out reveals the following:

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