Contextualization

Mathematics is a discipline that goes beyond the four basic operations. Its field of study extends to the real world, providing explanations and answers to practical everyday situations. One of the branches of mathematics that helps us understand the three-dimensional space we live in is Spatial Geometry. Within this branch, we have the concepts of Dihedrals and Trihedrals, which help us better understand the structure of the space around us.

Dihedrals and Trihedrals are spatial geometric figures that are highly relevant in various fields of knowledge, such as Engineering, Architecture, Chemistry, among others. Learning about these concepts is essential to develop a deeper understanding of the world around us.

Introduction to Dihedral and Trihedral

A Dihedral is a figure formed by two half-lines with a common origin and two half-planes with a common origin. We can think of the dihedral as an open book, where the pages represent the half-planes, the spine represents the half-lines, and where the spine meets the pages is the common origin.

A Trihedral is an extension of the dihedral concept. It is a figure formed by three half-lines with a common origin and three half-planes with a common origin. We can think of a trihedral as a corner of a room, where the walls represent the half-planes, the lines where the walls meet represent the half-lines, and the corner where they all meet is the common origin.

Importance of Dihedrals and Trihedrals

The application of dihedral and trihedral concepts is vast and impacts various disciplines. In Engineering and Architecture, for example, these concepts are fundamental for the development and construction of complex three-dimensional structures. In Chemistry, these concepts are used to understand molecular structure and how molecules connect and interact. In the world of Arts, these concepts are used to understand perspective and represent the three-dimensional world on a two-dimensional surface.

Practical Activity

Activity Title: Hands-on - Building with Dihedrals and Trihedrals

Project Objective

The objective of this activity is to apply the concepts of Dihedrals and Trihedrals in the construction of a three-dimensional model. Additionally, the project aims to develop competencies such as teamwork, time management, and problem-solving skills.

Project Description

Each group of 3 to 5 students will be responsible for designing and building a three-dimensional model of a structure, which can be a building, a bridge, a chemical molecule, or any other object with a complex three-dimensional structure. The model must incorporate both dihedrals and trihedrals, and students must be able to identify and explain where and how these figures are used in their construction. Additionally, the team must consider the functionality and aesthetics of their model, taking into account design and engineering aspects.

Required Materials

• Cardboard, cardboard or other similar material to create the model
• Ruler, square, compass
• Office supplies (glue, scissors, tape, etc.)
• Pencils and paper for sketches and calculations

Step by Step

1. Research and Planning: Research complex three-dimensional structures and choose one to be used as the basis for the model. Plan the model, draw sketches, and think about how to incorporate dihedrals and trihedrals into the structure. Establish deadlines and divide tasks among group members.

2. Model Construction: With the planning in hand, start building the model. Remember to record all steps and decisions made during the process. This will be important for the report.

3. Review and Refinement: After completing the construction, review the model. Check the presence and application of dihedral and trihedral concepts and try to improve the model if necessary.

4. Report Writing: Prepare a report according to the instructions in the next section. Take into account all stages of the project, from research and planning to construction and model review.

Project Deliverables

At the end of this project, students must deliver the following items:

1. Three-Dimensional Model: The model built by the students, showing the use of dihedral and trihedral concepts.

2. Written Report: Report with a detailed explanation of each stage of the project, from research and planning to construction and model review.

Description of the Written Document

The written document should include the following sections:

1. Introduction: In this section, students should contextualize the project theme, explain the relevance of dihedral and trihedral concepts, and describe the work's objective.

2. Development: Students should describe in detail the theory behind dihedral and trihedral concepts, explain the activities carried out, present the methodology used to build the model, and discuss the results obtained. That is, describe the functioning of the model, where and how the concepts of dihedrals and trihedrals were applied, the difficulties encountered, and how they were overcome.

3. Conclusions: Students should summarize the main points of the work, state the learnings obtained, and the conclusions drawn about the project. They should also talk about the impact of the project on how they see the use of mathematics in the real world.

4. Bibliography: Students should indicate the sources used, such as books, internet sites, videos, among others.

The project must be completed within a four-week period. Considering the various stages of the project, each participating student must dedicate at least twelve hours to its execution.