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 Vector to be and Extent Parameter at to be such that denotes a function related to ‘length,’ dependent on

Definition 1.1.2. Let be defined by

The Transfer Confinement Space, at is denoted, is effectively the Cartesian product of all intervals, specified by their respective Extent Parameter,

Definition 1.1.3. Define a function, such that We call the Shift Modifier at If an element, lies in the Transfer Confinement Space, that point gets shifted, as specified by

Definition 1.14. We define a function such that for a given We call the Spacial Modification Function for 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.