Math 445 Take home Problem

The setup

• The sphere S in (x, y, z) space has center O = (0,0,0) and radius 1
• The six points where the x-axis, the y-axis, and the z-axis intersect the sphere S are the vertices of an inscribed octahedron.
• The vertices of the 8 triangular faces of the octahedron form the vertices of 8 congruent spherical triangles that tessellate the sphere S.  We will call these the octant triangles.
• For example, if I = (1, 0, 0), J = (0, 1,0) and K = (0, 0, 1), then IJK are the vertices of an equilateral spherical triangle with (spherical) vertex angles of 90 degrees.
• Let X, Y, Z be the spherical midpoints of the sides of this triangle, namely the great circle arcs JK, KI, IJ. The spherical triangle XYZ is the midpoint triangle of spherical triangle IJK.
• We will call spherical triangle XYZ the midpoint triangle.  We will call the 3 spherical triangles IZY, JXZ, KYZ the corner triangles.

The Problem

Find lots of information about one shape, considered as a spherical figure and also as a polyhedral figure. Specifically, there are 4 sets of questions to answer.

• Show your work.
• Take the time to organize your answers, write them clearly and use enough white space so that they are readable.
• This problem is quite concrete, actually, so you should seriously consider making a model on a sphere or otherwise. Pie plates will be furnished Wednesday to get you started.
• You should not consult other people about this problem. Also, you should not look for answers in reference books, the internet, etc. But you can use your notes and work in the course.

Part 1. Midpoint and Corner Triangles of an Octant Triangle

• Which of these 4 spherical triangles, the midpoint triangle and the 3 corner triangles, are congruent?  Which are equilateral?  Which are isosceles?

• What are the lengths of each of the sides (measured in degrees or radians) of each of these 4 spherical triangles?

• What are the measures of each of the vertex angles (measured in degrees or radians) of each of these 4 spherical triangles?

• What is the area of each of these triangles?  Check that the areas add up to the area of IJK.

• What are the equations of the great circles that form the sides of each of these triangles?

• The spherical segments IX, JY, KZ are concurrent at the center of symmetry C of triangle IJK. What is the spherical distance from C to I? What is the spherical distance from C to I?

Part 2.  Inscribed polyhedron using all the triangles

Consider the midpoint triangles and corner triangles of each of the 8 octant triangles congruent to IJK.  These triangles together tessellate the sphere. You can make an inscribed polyhedron by connecting the vertices of each triangle with rods.  To construct such a structure, you need to compute the chord (3d Euclidean) distance between each of the vertices of the midpoint triangles and each of the vertices of the corner triangles.

[Background: The geodesic domes and hemispheres of Buckminster Fuller are constructed in this way. They are polyhedra that can be inscribed in a sphere. The polyhedron is made of triangles (not all the same, but following a system as in this example.)

• What are these chord lengths?  How many different lengths are there?  How many struts of each length do you need?
• For the polyhedron made of the struts, how many vertices, edges and faces does it have?  What is this number: (number of vertices) – (number of edges) + (number of faces)?

Part 3.  Inscribed polyhedron from the midpoint triangles

Consider the midpoint triangles of each of the 8 octant triangles.  Make a second polyhedron just from the midpoint triangle vertices (i.e. the vertices of the midpoint triangles but not including the vertices of the octahedron). 

• Explain why the faces of this polyhedron are squares and equilateral triangles.
• What is the dihedral angle between one of the squares and one of the triangles? (Hint: This is related to the spherical distance between C and I computed earlier.)
• Again, how many vertices, edges and faces does it have?  What is this number: (number of vertices) – (number of edges) + (number of faces)?
• On the sphere, just using the midpoint vertices as vertices of a tessellation means that you are gluing together the corner triangles in groups of 4 to form equilateral quadrilaterals centered around each octahedron vertex (such as I). What are the angles of this quadrilateral?
• How many great circles overall are needed to form the sides of these midpoint triangles?  (Be careful, sometimes two or more sides of the different triangles will lie on the same great circle.)  How many midpoint vertices lie on each of these great circles?
• What are the equations of each of these great circles? (You did more than this above.)
• What are the poles of the great circles you found in the previous question?  What kind of polyhedron can be formed using these poles as vertices?

Part 4.  Stereographic views and 3d models

Draw a circle on a piece of graph paper.  Draw the x and the y axes.  The interior of the circle is the stereographic image of the "southern" hemisphere of S (where the z coordinate is negative).  The projections of points I and J are on the circle.

• Construct the image by stereographic projection of the 12 corner triangles and the 4 midpoint triangles in the southern hemisphere.
• Bring on Monday some kind of sphere with the midpoint triangles indicated by drawing, rubber bands, tape, etc.
• Extra credit: Make a model of the polyhedron in part 3. Also, extra credit for especially nice models or additional models (meaning either accurate or showing fine craftsmanship).