Math 445 Questions for the Test on Polyhedra and 3D geometry

The questions are more general in some cases than what would be asked on the test, which may cite a specific example with numbers. But some general, conceptual questions will show up either on the written or the oral part.

A. Space Geometry

• Define parallel lines, parallel planes, lines perpendicular to planes, planes perpendicular to planes. Recognize these in examples in polyhedra.
• How is the dihedral angle between two planes defined and how can it be computed in examples, such as in pyramids and the simpler regular polyhedra.

B. Polyhedra

• What is a polyhedron? What are the names of the pieces (vertices, edges, faces)?
• Define "regular polyhedron." Know well the names of the regular polyhedra (Platonic solids) and also know examples of non-regular polyhedra, including examples with regular polygons as faces.
• Be able to explain why there are only 5 regular polyhedra.
• Know or be able to produce quickly the numbers of vertices, faces, edges of any of the Platonic solids.
• Define the concept of the dual of a polyhedron. Know the duals of the Platonic solids and how this is visible in the numbers of vertices, faces and edges.
• Be able to state and justify informally the Euler number theorem for convex polyhedra.

C. Volume and dissections of the cube.

• Be able to explain how to dissect the cube into 3 congruent pyramids, including the lengths of the edges of the pyramids, angles, etc.
• How can this dissection be used to justify the general volume formula for cones and pyramids.
• In what sense is a pyramid a kind of cone?

D. Symmetries of the cube

• Describe the nature and number of the planes of symmetry of the cube. How do these planes cut the face of the cube into 48 right triangles? What are the lengths of the sides of these triangles?
• Explain how to count the number of rotations that are symmetries of the cube. How many of each kind are there?
• Explain how to count all the symmetries of the cube. How many are not rotations? How can these be described (in detail).
• What are the symmetries of the cube the same as the symmetries of the regular octahedron? How do the planes of symmetry cut the faces of the octahedron?

E. Dodecahedron and Icosahedron

• Define a golden rectangle and explain how 3 of them can be used to build an icosahedron. Also tell how this can be used to find the coordinates of the vertices of the icosahedron and to find the edge lengths.
• Tell how there are 5 cubes in the dodecahedron. What are the edges of the cubes and how are the lengths related to the edge length of the dodecahedron.

F. Law of cosines, dot product and equation of a plane in coordinate 3-space

• How can you tell whether line AB is parallel to line CD using coordinates?
• Find the equation of a plane through 3 points.
• Given the equation of a plane, find the point on the plane closest to the origin.
• How can you tell two planes are parallel or perpendicular from their equations? How can you tell whether a line AB is parallel to the plane?
• Explain how the law of cosines for a triangle ABC can be derived from the Pythagorean Theorem applied to an altitude of triangle ABC.
• Explain how the algebraic rules of the dot product give a vector version of the law of cosines that proves a relationship between dot product and the cosine of an angle. (You did both parts in the course.)
• In examples from polyhedra, use the dot product and the equations of plane to find dihedral angles – examples include regular tetrahedra, square based pyramids with triangular sides, octahedral, or a general case when equations are known.
• In space or in the plane, how can you find the center of mass of 2 points, of 3 points? Of 4 points?

G: Cubes and matrices

• If a cube has faces parallel to the coordinate planes, as in recent examples, be able to find the coordinates for the other special points of the cube, including the vertices of the dual octagon and the tetrahedron inside the cube. Be able to find the equations of the planes of the faces of each of these.
• Be able to find the matrix representation and the formula in (x,y,z) for any symmetry of the cube.
• Be able to investigate the composition of two symmetries of the cube either using a model or using matrices. When rotations are composed, do the angles of rotations add as they do in the plane?

H: Grammar and spelling

• Know and use and spell correctly the singular and plural forms of geometry words: Polyhedron and polyhedra, vertex and vertices matrix and matrices.
• Know and spell correctly the names of the polyhedra we have studied.
• Do not use non-existent words (e.g., vertice) or misspell geometry words (e.g., verticies).