Tessellation examples from squares pokemon tessellation

The Intricate World of Tessellations: Exploring Examples with Squares from Pokémon to Higher Dimensions

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Tessellations are fascinating geometric patterns that have captivated mathematicians, artists, and enthusiasts alike for centuries. At its core, a tessellation is simply the tiling of a plane using one or more shapes without any gaps or overlaps between them.

In this article, we'll delve into some remarkable examples of tessellations, starting with the humble square – an shape that may seem simple at first glance but can give rise to breathtakingly complex and intricate designs. We'll also explore how these patterns have been used in popular culture, specifically through the beloved franchise Pokémon.

Regular Tessellations: The Building Blocks

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One type of tessellation is called regular tessillation, where all tiles are identical shapes. This means that every single tile in a regular tessilation is exactly the same shape and size as its neighbors surrounding it around their common edge or corner point never varying even though they may appear differently arranged depending to viewer's perspective due mostly how many sides each individual component has been divided into during construction process itself.

In our case, let's consider an example using squares. When we arrange multiple identical square tiles in a repeating pattern without any gaps between them (meaning no spaces where one tile doesn't touch another), we create what is known as a 'square tessellation.' This particular arrangement falls under the broader category of regular tessellations due to its symmetrical properties and use of only one type of shape.

Semi-Reglar Tessallation: Adding Variety with More Shapes

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Now let's mix things up slightly by introducing more than just squares into our tessilation game. We can achieve this through a process called semi-regular tessellation, where we still maintain the same basic principles as before (no gaps or overlaps between tiles), but now each tile has its own unique shape.

To illustrate this concept further, let's imagine that instead of using only square tiles for our tiling pattern, we also introduce equilateral triangles and rhombuses into the mix. By carefully arranging these different shapes in a repeating manner without any spaces between them (i.e., ensuring that every vertex or corner point where two adjacent polygons meet up perfectly), we can create stunning semi-regular tessellations.

Aperiodic Tessallation: When Patterns Go Haywire

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Finally, let's venture into the realm of truly unpredictable and fascinating patterns – aperiodic tilings. Unlike their regular and semi-regular counterparts, which always repeat themselves in some form or another when viewed over large enough distances (i.e., they exhibit translational symmetry), these tessellations refuse to follow any discernible rules.

In other words, if we were to magnify an area of a periodic tiling ad infinitum, no matter how many times we zoomed in using our hypothetical super-powerful microscopes or telescopes, we would always see the same repeating pattern staring back at us – tiles arranged exactly as they were before. However, with aperiodic tilings things take an entirely different turn...

For example, imagine that instead of squares and triangles forming your tessilation pattern you had hexagons (six-sided polygons) in various sizes and shapes all jumbled together randomly without any apparent order or structure to their arrangement at first glance.

However, upon closer inspection – say if we were to zoom into just one small section using our magic magnifying glass – things might start looking slightly more familiar. Perhaps within this seemingly chaotic sea of hexagons there existed tiny repeating patterns themselves? Maybe even smaller ones nested inside those larger shapes ad infinitum!

This is essentially what happens with aperiodic tilings: no matter how many times you zoom in or out using your most powerful magnifying glasses, the pattern never repeats itself exactly again – it's like staring into an ever-changing kaleidoscope of geometric wonder.

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And so concludes our journey through some amazing examples of tessellations, from simple square patterns to mind-bogglingly complex and unpredictable aperiodic tilings. Whether you're a seasoned mathematician or just someone who enjoys playing with shapes on paper (or computer screen), there's no denying the sheer beauty and fascination that these geometric marvels have to offer.

So next time you come across an intricate design in nature, architecture, art – or even in your favorite video game like Pokémon! – take a closer look. You might just spot some hidden tessellations waiting to be discovered...

Tessellation examples from squares pokemon tessellation