Assignment 7 (Due Wednesday 11/19)

Practice:  Learn how to read off the barycentric coordinates of a point P given lattice information or ratio information as in the practice problems in Lab 7 and on the web.  There will be some trapezoid practice also.  There will be more information on the web for study suggestions for the quiz next week.

(7.1 – 10 points) Finding barycentric coordinates from cevian ratios

Given a triangle ABC and points A' on BC and B' on CA, let P be the intersection of AA' and BB'.  Suppose the ratio BA'/A'C = 2/3 and ratio CB'/B'A = 1/5.

• What are the barycentric coordinates of P?
• If C' is the intersection of CP with AB, what is the ratio AC'/C'B?
• If Q is the intersection of line A'B' with line AB, what is the ratio AQ/QB?

(7.2 – 15 points) Plotting points from barycentric coordinates

Construct an equilateral triangle ABC and construct the points P, Q, R if

• the barycentric coordinates of P are (1/7, 4/7, 2/7)
• the barycentric coordinates of Q are (2/(3+sqrt (2)), 1/(3+sqrt (2)), sqrt(2)/(3+sqrt (2)))
• the barycentric coordinates of R are (-1/3, 6/3, -2/3)

(7.3 – 10 points) Coordinate and cubes

In (x,y,z) space, the 6 points e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) and –e1, -e2, -e3 form the vertices of a regular octahedron.  The faces are equilateral triangles.  The centers of these triangles are the vertices of a cube.

• What are the coordinates of the vertices of this cube?
• What are the coordinates of the vertices of the tetrahedron formed from 4 vertices of the cube (two choices of tetrahedron)?
• What are the coordinates of the vertices of the octahedron whose vertices are the midpoints of the edges of the tetrahedron?

(7.4 – 20 points) Roof of a dodecahedron

This problem is about the "roof" on a square base that is used to construct a dodecahedron.

Setup: Constructing the roof

• Take two regular pentagons with side s and diagonal length d.  We have shown earlier that d/s = golden ratio.
• Cut each pentagon along one diagonal so that the 2 pieces are an isosceles triangle and an isosceles trapezoid.
• Take a square of side d.  Attach to two opposite sides of the square the trapezoids (along the side of length d).  Attach to the other two opposite sides the triangles.  These attached polygons fit together to form a pentahedron (5-sided polyhedron) with a square base.  It resembles a tent or a roof.  We will call it the roof.

Problem 1.  What is the height of the two vertices of the roof above the square base, i.e., what is the distance of each of these points from the square base?

Problem 2: Find the exact dihedral angle (and a decimal degree approximation) of 2 dihedral angles in the roof. 

• the angle that the triangle makes with the square base and
• the angle that the trapezoid makes with the square base. 

Problem 3:  Use your answer to Problem 2 to show why 6 of these roof pieces can be attached to the faces of a cube to form a regular dodecahedron.

Hints and suggestions:  You can use a coordinate-free approach, but it is also OK to use coordinates, with the base in the (x, y, 0) plane with sides parallel to the axes, so that the heights are z-coordinates.  You also have the distance formula available.

(7.5 – 10 to 20 points) Seeking Archimedean Polyhedra

Use a search engine such as Google or the Math Archives or the Math Forum or other means, and find a site that gives a good explanation, with examples, of what is meant by an Archimedean Polyhedron (some may spell it Archimedian) and with other information about polyhedra.

• Send an email to the Math 444 list with the web address of the page.
• In the email, write a one-paragraph review telling what you found valuable (or not so valuable) in the site.
• Name and describe your favorite Archimedean polyhedron, giving info about faces, edges, vertices so that one could recognize it from your description
• Optional extra credit: make a model of your polyhedron.  You can ask for some cardboard shapes by email.  Be sure to say how many and what kind.

Note:  It may be difficult to find 31 different sites, but more credit will given for originality in finding the site, interesting and insightful reviews, and clear descriptions.  In other words, original comments about the same site are OK, but finding good, different sites is storngly encouraged. All this information will be collected on a web page for the class.