Shapes and Patterns in Everyday Life
Explore the fascinating world of shapes and their patterns.
Claudio Bravo, Auguste Hébert, Diego Izquierdo, Benoit Loisel
― 4 min read
Table of Contents
In our everyday life, we come across various Shapes like squares, circles, and triangles. These shapes can be simple, like a paper cut-out, or complicated, like the beautiful designs we see in nature and art. Imagine if we could arrange these shapes in interesting ways to create new designs. That’s what we will explore here!
Tessellations?
What areTessellations are Patterns made by repeating shapes. Think of a tiled floor: each tile fits perfectly with the others without any gaps or overlaps. The shapes used in tessellations can be regular, like squares or hexagons, or irregular, like free-form shapes that fit together in a unique way. These patterns can be found in art, architecture, and even in nature.
Shapes in a Space
Let’s dive deeper into how these shapes exist in a space. Imagine you have a large piece of paper, and you want to cover it entirely with shapes. Depending on your choice, some shapes will fit nicely together, while others might leave gaps. The way shapes fit together in a space brings us to the idea of "weight" in geometry. It’s like having a bag of different shapes and figuring out how heavy each shape feels when placed together.
The Importance of Shape Relationships
Now, let’s talk about how shapes can relate to one another. If you have a triangle and a square, you can place them next to each other, but they won’t fit perfectly. On the other hand, if you have two squares, they can stack up or align perfectly without any issues. Understanding these relationships helps us play with designs and patterns.
Reflecting Shapes
Another fun aspect of shapes is Reflections. This is like looking in a mirror. If you shine a light on a shape, the reflection gives you a new perspective. Artists often use reflections to create stunning visuals. Imagine how a shiny lake reflects the trees and sky. This principle is similar when working with shapes in patterns.
Expanding Shapes
When we mention expanding shapes, think of blowing up a balloon. The shape grows while keeping its original form. This idea is used in tessellations too. By expanding or shrinking shapes while keeping their proportions, we can fill a space efficiently. When these shapes are stretched out, they can create unique patterns that amaze the eye.
Simple Shapes, Complex Patterns
One of the most exciting things about shapes is using simple shapes to create complex patterns. For example, a simple triangle can be repeated in different colors and orientations to create a mesmerizing design. This is similar to how a small piece of music can be transformed into a symphony.
Symmetry
The Beauty ofSymmetry is another important concept in the world of shapes. When one side of a shape mirrors the other, like in a butterfly, we have symmetry. Many artists and architects take advantage of symmetry to create visually appealing designs. Think of a perfectly balanced building or a beautiful piece of artwork.
Making Your Patterns
So, how can you create your patterns? Begin by picking a shape you like. You can use paper cutouts to experiment. Place the shapes next to each other and see how they fit. Play with colors, sizes, and orientations. Don't be afraid to mix different shapes. The more you play, the more interesting patterns you can create!
Nature’s Tessellations
Have you ever noticed how honeycomb is made up of hexagons? Nature is the ultimate artist when it comes to tessellations. From the scales of a fish to the petals of a flower, shapes and patterns are everywhere! Observing these natural designs can inspire you in your own creations.
Patterns in Art and Architecture
Throughout history, artists and architects have used tessellations in their designs. Think of intricate tile work in mosques or the works of famous artists who played with shapes and patterns. These designs not only beautify spaces but also tell stories and convey cultural significance.
Conclusion: Embrace the Shapes!
As you can see, the world of shapes and patterns is vast and vibrant. Whether you are an artist, a student, or just someone who appreciates beauty, understanding shapes opens the door to endless possibilities. So grab some paper, get creative, and embrace the wonderful world of shapes!
Title: Tessellations of an affine apartment by affine weight polytopes
Abstract: Let $\A$ be a finite dimensional vector space and $\Phi$ be a finite root system in $\A$. To this data is associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study "affine weight polytopes" associated to these data. We prove that these polytopes tesselate $\A$. We also prove a kind of "mixed" tessellation, involving the affine weight polytopes and the poly-simplical structure on $\A$.
Authors: Claudio Bravo, Auguste Hébert, Diego Izquierdo, Benoit Loisel
Last Update: 2024-11-15 00:00:00
Language: English
Source URL: https://arxiv.org/abs/2411.10282
Source PDF: https://arxiv.org/pdf/2411.10282
Licence: https://creativecommons.org/licenses/by/4.0/
Changes: This summary was created with assistance from AI and may have inaccuracies. For accurate information, please refer to the original source documents linked here.
Thank you to arxiv for use of its open access interoperability.