As mathematics demonstrates and as the popular YouTube channel Veritasium illustrates, an endless pattern that never repeats itself is feasible. According to the video's description, simple laws of geometry showed that 5-fold symmetry, as well as crystals without a periodic structure, were impossible. However, they could be used to create an infinite pattern that never repeats itself.
The crystal display panel used by Apple in some of its products such as the iPhone and iPad displays uses a grid of columns and rows of pixels to display images. The first model of iPhone had a 320 by 480 pixel display while the latest model has a 468 by 714 pixel screen. The number of columns in the grid is equal to the number of rows, which means there are 72 columns in the screen. A single pixel is made up of three sub-pixels, each one representing one of the primary colors - red, green, or blue. Pixels are placed next to each other with no space in between, which means that the image cannot have any gaps or holes. There can be no overlap between pixels, otherwise a white spot would appear where they did not belong.
Since every part of the image is needed to create a complete picture, computer programs use algorithms to determine what portion of the image should go into each cell of the grid. The iPhone's camera uses a 5-tap algorithm for caching.
This indicates that the patterns exhibit translational symmetry when they are repeated endlessly. The entire pattern may be picked up and changed in numerous ways without causing any alteration. It can be translated in any direction without changing its appearance.
Non-repeating patterns include checks, flags, and sequences such as stripes or dots. Repeating patterns include cycles and rows. A cycle is a pattern that appears again after one repetition, while a row is a pattern that appears only once before moving on to the next line or page. Patterns that do not repeat indefinitely are called finite.
Non-repeating patterns were very popular in Victorian-era needlework. They're also useful for graphic design because they allow you to create unique elements such as logos or product packaging without using duplicate pieces of material.
In mathematics, a non-repeating sequence is a sequence of numbers or symbols for which no pair of consecutive terms are the same. Such sequences were first studied by Leonhard Euler who used them to prove some of his mathematical results. Non-repeating sequences have applications in both pure and applied mathematics. In physics, they are important in dynamical systems theory.
In computer science, a non-repeating sequence is an infinite string of characters where no two consecutive copies of the same character occur.
A periodic tiling is one that has a recurring pattern. Regular tilings with regular polygonal tiles all of the same form and semiregular tilings with regular tiles of more than one shape with every corner similarly placed are two examples. Non-periodic tiling is one that lacks a recurring pattern. A checkered flag is used in some sports to indicate a non-periodic finish to a race.
All regular tilings are also semiregular, but not all semiregular tilings are regular. The Penrose tiling is an example of a semiregular tiling that is not regular. It contains copies of itself along with other shapes. There are also semi-regular tilings that do not have this property, for example the square-and-diamond tiling which consists only of squares and diamonds.
Non-periodic tilings may or may not be repetitive, depending on how you define repetition. If you require that each tile appear the same number of times as any other tile, then no finite non-periodic tiling can be repeated exactly twice. However, if finitely many instances of a tile are forbidden, then it is possible to make periodic versions of the tiling by adding more tiles that contain the original tile along with some additional points. For example, a version of the Penrose tiling can be made by allowing pairs of opposite corners to overlap by 1 unit, forming a "double corner".