Outline of Polyhedra Unit
Regular Tetrahedra
Distances, cross-sections, and planes of symmetry
- If length of side = s, then find slant height (altitude of a face)
- Find shape of cross section by a plane of symmetry (an isosceles triangle)
- Find length of altitude of tetrahedron
- Radius of incircle and circumcircle
- Reminder of barycenter of 4 points and what it says about center of tetrahedron
- Find a plane which cuts a square cross-section
Angles
- What are the dihedral angles between the faces?
Symmetries
- What planes of symmetry and how many?
- What rotations and how many?
- What other symmetries and how many?
Cubes
Distances, cross-sections, and planes of symmetry
- If length of side = s, then find lengths of diagonal of face and diagonal
of cube.
- Find shapes of cross section by a plane of symmetry (more than one shape)
- Radius of incircle and circumcircle
- How to inscribe a regular tetrahedron in a cube
- Find a plane which cuts a hexagonal cross-section
Symmetries
- What planes of symmetry and how many?
- What rotations and how many?
- What other symmetries and how many?
- How do these symmetries relate to symmetries of inscribed tetrahedron?
Other shapes in the cube: What are the numbers, dimensions and angles?
- Regular tetrahedron
- Stella Octangula
- Dissect into 3 congruent square-base pyramids (top vertex of pyramid = vertex
of cube).
- Dissect into 6 congruent square-base pyramids (top vertex of pyramid = center
of cube)
Space-filling polyhedra
- Regular tetrahedra in tetrahedra
- Space-filling shapes from the cube
General tetrahedra
Other Platonic solids: relation with cubes, golden ratio, duality