Contextualization

Throughout human history, a variety of civilizations around the world have used polygons to create fascinating works of art and elegant architectural structures, as well as to solve practical problems. Think of the pyramids of Ancient Egypt, the beautiful Islamic mosaics, the intricate web of roads that formed the Roman Empire, and so on. The influence of these geometrically rich shapes can still be seen today, in everything from the way we design our buildings to how we play video games.

Polygons are not just two-dimensional figures we see in math books. They are the basis for many things in our world, from architecture to art, through engineering, computer science, and graphic design. Polygons are used to model objects in computer programs, such as games and animation software. In engineering and architecture, polygons help create stable and efficient structures. In art, they form the basis of many forms of visual expression. Each of these uses requires a deep understanding of the angles of polygons, which leads us to our main topic.

Introduction

Polygons are flat figures bounded by line segments, which we call sides. The angle formed between two consecutive sides of a polygon is called an internal angle. And the angle formed between one side of a polygon and the extension of the next side is called an external angle. The study of these angles is fundamental to the structural understanding of polygons.

Regular polygons are those that have all sides and angles equal. And this is where mathematics really shines! There is a simple relationship between the number of sides of a regular polygon, its internal angle, and the external angle. And this relationship is expressed by the following formulas: sum of internal angles = (n-2)180° and each internal angle = (n-2)180°/n, where n represents the number of sides of the polygon.

In this project, you will explore all these concepts in a practical and engaging way. You will investigate how the angles of regular polygons relate to the creation of mosaics and tiling art, the mathematics behind architecture and art, computer modeling, and space exploration. Additionally, you will delve into concepts of geometry, algebra, arts, and sciences. Get ready for a journey of mathematical discoveries!

Practical Activity

Activity Title: Polygons in Art and Real Life: A Practical Investigation

Project Objective:

Investigate the application of polygon angles in the creation of mosaics and tiling, in architecture, art, computer modeling, and space exploration. Delve into the concepts of geometry, algebra, arts, and sciences.

Detailed Project Description:

The class will be divided into groups of 3 to 5 students to carry out the activity. The project should be carried out over 4 weeks, as an estimate of the amount of work needed for each group to develop the project.

Each group will choose one of the suggested themes: Mosaics and Tiling, Architecture, Art, Computer Modeling, or Space Exploration. They will then carry out a series of activities related to the chosen theme, in order to explore the importance of polygon angles in each field.

Required Materials:

• Graph paper
• Ruler
• Compass
• Colored pencils
• Computer with internet access

Detailed Step-by-Step for Activity Execution:

1. Theoretical Research (1 week)

Research and study on the chosen theme and its relationship with polygon angles. Each group must prepare a mini report, describing the theory on the theme, and explaining the relationship between the chosen theme and polygon angles.

2. Practical Activity (2 weeks)

Depending on the chosen theme, the group must develop a practical activity.

• Mosaics and Tiling Theme: creation of a mosaic or tiling using polygons.
• Architecture Theme: development of a simple architectural project where the application of polygon angles is essential.
• Art Theme: creation of an artwork based on polygons.
• Computer Modeling Theme: use of a 3D modeling software to create an object using polygons.
• Space Exploration Theme: creation of a model of a spaceship or space station using polygons.

3. Final Report (1 week)

Preparation of a final report with all the information collected in the theoretical research, description and photos of the practical activity, and a conclusion. The report should be divided into four sections: introduction, development, conclusions, and bibliography used.

Project Delivery:

At the end of the project, the groups must deliver:

• Mini report of the theoretical research.
• Practical activity, which can be a physical object or a digital file, depending on the theme.
• Final report, including the sections of introduction, development, conclusions, and bibliography.

The reports should be written clearly and concisely, focusing on the description of the research and development process of the practical activity, the application of the concepts learned in the activity, the results obtained, the conclusions drawn from these results, and the bibliography used.