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Brilliant

3D Geometry

via Brilliant

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Overview

Explore the fundamental concepts of three-dimensional geometry: What strangely-shaped 3D pieces can result from slicing up 3D polyhedra with planes? What flat polygons can fold up into 3D shapes? If you're running around on the surface of a cube-world, what's the shortest path between two opposite corners? (The answer to this last one might surprise you.)

In this course, you'll stretch problem-solving techniques from flat figures into a third-dimension and explore some mathematical ideas and techniques completely unique to 3D geometry. For example, you'll investigate and learn how to apply Euler's facet counting formula, a formula which describes a surprising algebraic relationship that relates the number of corners, edges, and faces that any polyhedron can have.
To succeed at this course, you should already have some familiarity with the basics of 2D geometry. Additionally, some algebra is used in this course, but nothing beyond the level of Algebra I.

Syllabus

  • Introduction: Explore various ways of thinking about shapes in 3D.
    • Cuts Through Shapes: Think about 3D shapes by cutting them into pieces.
    • Surfaces of Shapes: Fold and fold again to transform 2D shapes into 3D.
    • Pieces of 3D: Slice, extrude and transform 3D shapes into different configurations.
  • Nets and Paths: Fold and unfold 3D shapes to see how they fit together.
    • Introduction to Nets: Fold up nets to make 2D shapes into 3D. Unfold them to see how the faces relate.
    • Nets of a Cube: What nets can successfully fold up to make a cube?
    • Exploring Cubes: Explore the faces of a cube and use nets to see how they relate.
    • Platonic Solids: Discover how many of these symmetrical solids can be constructed.
    • Lines Through Cubes: Apply the Pythagorean theorem to 3D distances.
    • 3D Shortest Distance: How can the shortest distance on the surface of a 3D shape be found?
    • Strings and Ants: Puzzle out these 3D distance problems by unfolding the shapes.
  • Cuts and Cross Sections: Slice 3D shapes into pieces and see what happens.
    • Introduction to Cross Sections: Think like an MRI machine as you slice through these shapes.
    • Building Intuition: How do cross sections relate to the shape they come from?
    • Cross Sections of Cubes: Explore the variety of shapes that can be obtained just by slicing up a cube.
    • Predicting Solids: Can the cross sections of a solid reveal its full shape?
    • Halves of Solids: How many ways are there to cut a 3D solid in half?
    • Other Fractions of Cubes: Stretch and test your understanding with these cube fraction puzzles.
  • Facet Counting: Find, understand, and prove Euler's formula about the pieces of polyhedra.
    • Vertices, Edges, and Faces: Is there a pattern here?
    • Uniform Vertex Configurations: Examine polyhedra that have the same polygons in the same order at every vertex.
    • Cutting Solids: Keep cutting solids and see what happens to the shapes as they transform.
    • Euler's Formula: Discover the formula that describes the relationship between faces, edges, and vertices.
    • Proving Euler's Formula: See why Euler's formula must always be true.
    • Duality: Explore the connections between dual polyhedra and the ways they relate.

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